Chapter 10 Calculus of exponentials, logarithms, and complex fields

A logarithmic potential curve U=ln(y)=log_ey was given by (6.11).  Our first example is the flip or inverse exponential curve y=e^U since that function is so important for making the complex phasor e^-(iù+Ã)t.

Also, the population growth function y=e^t=exp(t) is one of the most used if not the most useful of transcendental functions. Roughly, transcendental means not expressed by finite algebra or constructed by Euclid’s strict rules. (However, like transcendental spirituality, it is easily approximated!) Later in this section we will prove that the exponential is the only function that is equal to its slope or derivative.

             (10.1)

In other words, if e^x is a force or potential curve then F(x) and U(x) are similar, in fact, identical.

            F^math(x) = U(x). if and only if: U(x)=e^x                        (10.2a)

For physicist’s definition (6.9) of force, e^-x is the one for which potential and force are identical.

 F^phys(x)  = U(x). if and only if: U(x)= e^-x                  (10.2b)

For now we use these slope-function relations to construct the exponential curve approximately. Starting from origin (x=0) we use the fact that any positive number to zero power is 1. (e⁰=1) From that point we draw a right triangle made of a unit altitude, a unit base, and a hypotenuse line of slope-1 as indicated in Step-0 of Fig. 9.12. The hypotenuse line gives approximately the points just above and just below x=0.  Then subsequent steps move the right triangle Äx to a point on the previously constructed line to make the next line. Since the slope is equal to the new function value, the base stays fixed at 1, but the altitude grows with the function value and makes the new line and a new point up the e^x-curve.

This approximation is a rough one. It underestimates a concave curve and overestimates convex ones because it puts the next point x+Äx on a tangent from the previous point x. That’s OK only if the curve is pretty straight and tangent slope is about the same at x+Äx. A better approximation uses the tangent halfway between neighboring tangents and extends that new slope to x+Äx to find the next point.

Now if you rotate your y= e^x-graph by 90° you get a logarithm U(y)=-ln(y) graph as shown in Fig. 10.1 (lower right). Each U(y)-curve-normal defines a unit-altitude triangle whose base is the force F(y)=1/y.

The story of e : A tale of great interest

Long ago banks would pay simple interest at some rate r such as r=0.03 (3%) based on a 1 year period. You gave a principal p(0) to the bank and some time t later they would pay you p(t)=(1+r·t)p(0). If you put in $1.00 at rate r=1 (like Israel and Brazil that once had 100% intrest.) you got $2.00 at t=1year.

           

Fig. 10.1 Rough constructions (a) exponential curve y=e^x=exp(x). (b) Log potential. (c) 1/y-Force.

 

Later on fancy banks would pay semester compounded interest  at the half-period and then use  during the last half to figure final payment. Now $1.00 at rate r=1 earns $2.25.

 

Fancier banks would pay trimester compounded interest  at the 1/3^rd-period  or 1^st trimester and then use that to figure the 2^nd trimester and so on. Now $1.00 at rate r=1 earns $2.37.

Still fancier banks would pay quarterly, monthly, weekly, daily, and so on. The race was on to give better earnings at a given interest rate r. Let’s compare some different earnings on $1.00 at rate r=1. At first it looks like you gain a lot by compounding more often. Then earnings slow to a halt just shy of $2.72.

        

That halting point is Euler’s growth constant e=2.718281828459… that we’re after. Let's try huge numbers (m) of multiplications in . (Get out a calculator. Rule & compass is useless now!)

                        p^1/m(1) = 2.7169239322                      for m = 1,000

                        p^1/m(1) = 2.7181459268                      for m = 10,000

                        p^1/m(1) = 2.7182682372                      for m = 100,000

                        p^1/m(1) = 2.7182804693                      for m = 1,000,000                                           (10.3)

                        p^1/m(1) = 2.7182816925                      for m = 10,000,000

                        p^1/m(1) = 2.7182818149                      for m = 100,000,000

                        p^1/m(1) = 2.7182818271                      for m = 1,000,000,000

The solid figures represent numbers that stay the same as we raise m. It’s still a torturous way to find e. We do a Billion (That’s “B” as in “Boy!”) multiplications (m=10⁹) just to get 6 solid figures beyond 2.71.

            A better way expands binomial or its power  for all rates r and times t. We let mr·t=n and m =n/r·t to simplify it for huge multiplication numbers m or n.

                                                                                             (10.4)

A binomial expansion (See page 119) turns exponential function e^r·t into a power series in  with x=1.

We actually save work as multiplication number n gets huge! (“Huge” means “as close to ∞ as you like.”)

              

Huge n makes n(n-1) cancel n² , and n(n-1)(n-2) cancel n³ , and so on. The exponential e^r·t series is born.

   (10.5a)                 (10.5b)

Let’s try it out for r·t=1 to evaluate e to order-o. (The precision order o is the power of highest term used.)

                                    Precision order: (o=1)-e-series = 2.00000 =1+1

                                                            (o=2)-e-series = 2.50000  =1+1+1/2

                                                            (o=3)-e-series = 2.66667  =1+1+1/2+1/6

                                                            (o=4)-e-series = 2.70833  =1+1+1/2+1/6+1/24

                                                            (o=5)-e-series = 2.71667  =1+1+1/2+1/6+1/24+1/120             (10.6)

                                                            (o=6)-e-series = 2.71805  =1+1+1/2+1/6+1/24+1/120+1/720

                                                            (o=7)-e-series = 2.71825

                                                            (o=8)-e-series = 2.71828

Nine terms in series (10.5) give 5-figure accuracy (10.6) and do the work of a million products in (10.3). That’s a million reduced to 8 sums and half-dozen or so divisions. It’s a big savings of arithmetic labor!

Derivatives, rates, and rate equations

Binomial expansions provide ways to find calculus formulas for slope or velocity introduced geometrically in Ch. 1. Soon we will do the same for curvature or acceleration and other higher order calculus concepts.

Suppose someone gives you a plot of formula like x(t)=t² or x(t)=sin4t or an exponential plot of x(t)=e^t that we just did in Fig. 10.1. You should be able to estimate its slope at any point from its x versus t graph. However, a binomial expansion may let you find an exact formula for its slope.

Consider a parabola x(t)=t² for example. Let’s find the slope of a line that goes through point x(t) and a point x(t+Ät) =(t+Ät)² that is a tiny time interval Ät later. Binomial expansion gives Äx=x(t+Ät)-x(t).

                         Äx=x(t+Ät)-x(t)=(t+Ät)²-t²=t²+2t·Ät+(Ät)²-t²=2t·Ät+(Ät)² 

 Slope ratiofollows. If  Ät is tiny we ignore it. Then tangent slope  is the 1^st derivative of x(t)=t².

                     (10.7a)                                (10.7b)

This checks the geometry of parabola 2ëy=x² in Fig. 9.4. Slope is , twice the x-value in units of 2ë. Consider an n-power curve x(t)=Atⁿ. Binomial expansion of Äx=x(t+Ät)-x(t) has n terms, most in +…+.

                         Äx=x(t+Ät)-x(t)=A(t+Ät)ⁿ-Atⁿ=Atⁿ+Ant^n-1·Ät+…+A(Ät)ⁿ-Atⁿ=Ant^n-1·Ät+…+A(Ät)ⁿ

If  Ät is tiny, only 1^st term Ant^n-1 in slope ratio is not tiny-tiny. That 1^st term is 1^st derivative of x(t)=Atⁿ.

   (10.8a)                         (10.8b)

Series for x(t)=Ae^t is unchanged (for r=1) by . It does kill term number-∞, but  is tiny-tiny-tiny anyway.

                        (10.9)

For 100% intrest (r=1), growth rate-of-Ae^t equals Ae^t. Otherwise, growth rate of Ae^rt is proportional to Ae^rt. To state that the growth rate of a function x(t) equals a constant “intrest rate” r times current value of x(t) is to write a differential rate equation whose “solution” is x(t)=Ae^rt. (The constant A is “initial capital” A=x(0).)

                                                                    (10.10)

It is Malthus’s population explosion equation for positive rate r>0! It is radioactive decay equation for r<0.

 

The binomial expansion

High school algebra courses generally contain a treatment of the binomial theorem that is used for our e^r·t expansion after equation (10.4). In case your course missed that (or you weren’t paying attention!) we’ll take a close look at this remarkable formula. The binomial algebra and related Pascal triangle geometry is the basis of so much mathematics and physics that it deserves a book chapter of its own.

            First it helps to work out the first few binomial series (x+y)⁰, (x+y)¹, (x+y)², (x+y)³,… by simply multiplying them together as we did for the e^r·t series that started this discussion. The first examples (x+y)⁰=1 and (x+y)¹=x+y are easy since the 0^th and 1^st powers of a number n are defined to be 1 and n, respectively. The square of a binomial is simple enough, too.

                                                (x+y)²=(x+y)·(x+y)=x²+xy+yx+y²= x²+2xy +y²                                                (1)

You might find it helps to make a table of product terms to do algebraic multiplication of this sort. Just make a box and write one factor ((x+y) in this case) on top and the other ((x+y) again) along the left.

            (2)

The just multiply each thing on top by each thing on the left and add them up to get (1). Try it with (x+y)³.

                                                      (3)

We can continue this process to get (x+y)⁴, (x+y)⁵,…and so forth.

                             (4)

  (5)

After awhile, you might notice a pattern in the numbers or coefficients B_pq of the various power terms x^py^q where the powers p and q must add up to the power n=p+q of (x+y)ⁿ being calculated. These B_pq are called the binomial coefficients of x^py^q and a triangular array pattern in Fig. 1 is called Pascal’s triangle.

            This pattern is like a Ponzi scheme since every number in it except the pinnacle B₀₀=1 is the sum of one or two numbers that lie above it and to either side. (This sum is going on in (2) thru (5) above.) So the pinnacle position q-p=0 on the central vertical triangle axis ends up with the biggest number B_pq for each power-row n=p+q. At n=p+q =10^th row, pinnacle B_5,5 accumulates 252 from 11 spots –5<q-p<+5.

 

Table 1. Binomial combinatorial coefficients up to power n=10

            Gamblers may recognize B₅₅=252 as the number of ways you can get exactly 5 x-cards and 5 y-cards from an n=20 card deck of 10 x-cards and 10 y-cards. More simply, B₅₅=252 is the number of ways to get exactly 5 heads and 5 tails from an n=10 coin tosses, or x⁵y⁵ from an (n=10)-power binomial.

            (x+y)(x+y)(x+y)(x+y)(x+y)(x+y)(x+y)(x+y)(x+y)(x+y)=(x+y)¹⁰=x¹⁰+…252x⁵y⁵+…y¹⁰             (7)

As you go down the line of 10 factors (x+y) you must pick x or y from each factor (x+y) to make just one (n=10)-power term x^py^q with n=p+q. There are 2¹⁰ =1024 such terms. (Just add up the 10^th row of Table 1.)

            (1+1)¹⁰=2¹⁰=1¹⁰+…252·1⁵1⁵+…=1+10+45+120+210+252+210+120+45+10+1=1024                   (8)

Check the other rows, too. (It’s a good to know powers-of-2 in a binary age!)

                        2²=4, 2³=8, 2⁴=16, 2⁵=32, 2⁶=64, 2⁷=128, 2⁸=256, 2⁹=512, 2¹⁰=1024,…                 (9)

            Now suppose, instead of just two things x or y, you could choose n different things {a,b,c,…,x,y,z,..} from each of the n factors in (7). Then the number of ways you may get a given term a·b·c·…·x·y·z·.. having all n different things is the number n!=n·(n-1)·(n-2)·…·2·1 of permutations of n things. Each permutational reordering gives another equal term (a·b=b·a).

So, n! is the “n-nomial coefficient” for a term with n-different factors. However, if we are counting terms x^py^q like a binomial series has with only two different things, the p! permutations of the x things and  the q! permutations of the y things do not count as new terms. Then n! divided by p! and q! gives B_pq.

       examples:

This gives binomial series that follows (10.4) and the Gauss-binomial distribution plotted below.

General power series approximations

Are power series like (10.5) useful for functions other than exponentials? Well, Mr. Maclaurin and Mr. Taylor thought so. Series that bear their names are de rigeur in good math books. (And, in this one, too!)

            Let’s start with a general power series like (10.5) but with arbitrary constant coefficients c₀, c₁, etc.

                                                                                (10.11a)

We derive c₀ by setting time t to an initial time t=0. (Like C-programmers, we count “uh-zero, uh-one, uh-two,..”)

                                                            c₀ = x(0)                                                                                  (10.11b)

So the 0^th coefficient c₀ is initial position x(0). Now we use (10.8b) to find a derivative of each term.

                                                           (10.11c)

Rate of change of position x(t) is velocity v(t). Setting t=0 derives c₁.

                                                            c₁ = v(0)                                                                                  (10.11d)

So the 1^st coefficient c₁ is initial velocity v(0). Now find a 2^nd derivative using (10.8b).

                                                  (10.11c)

Change of velocity v(t) is acceleration a(t). Set t=0 to get c₂.

                                                            c₂ = a(0)                                                                               (10.11d)

So the 2^nd coefficient c₂ is half the initial acceleration a(0). Now a 3^rd derivative:

                                               (10.11e)

Change of acceleration a(t) is jerk j(t). (Jerk is a NASA sanctioned term!) Set t=0 to get c₃.

                                                            c₃ = j(0)                                                                                (10.11f)

So the 3^rd coefficient c₃ is initial jerk j(0) over 3! Now a 4^th derivative:

                                                  (10.11g)

Change of jerk j(t) is inauguration i(t). (If NASA can be silly, so can we!) Set t=0 to get c₄.

                                                            c₄ = i(0)                                                                               (10.11h)

So the 4^th coefficient c₄ is initial inauguration i(0) over 4!. Now a 5^th derivative.

                                                       (10.11i)

Change of inauguration i(t) is revolution r(t). (Ooops! Politically incorrect!) Quick set t=0 to get c₅.

                                                            c₅ = r(0)                                                                               (10.11j)

That’s enough iterations to show the Maclaurin series of any function x(t) that has decent derivatives.

                        …                 (10.12a)

By “decent” we mean the non-exploding types that we can deal with. The following is a list that shows some of the notations used for the higher order derivatives discussed so far.

                                                      (10.12b)

The “dot” notation writes n-derivatives of x(t) by puttting n-dots over x. This may help prevent writer’s cramp. But, j-dot looks, well, kind of jerky. It’s common to use primes () for x-derivatives.

            How good is a power series (10.5) at faking x=e^t beyond t=1listed in (10.6)? We plot various orders of approximation in Fig. 10.2. The 1^st order (2-terms of (10.5a)) is just a straight line of slope 1. A 2^nd order (3-term) parabola, 3^rd order cubic, 4^th order quartic, etc. each peel off x=e^t in sucession. All meet at (t=0,x=1).

 

Fig. 10.2 Comparing x=e^t with its n^th-order approximate power series.

Sine-wave power series

A severe test of power series is their ability to fake sine waves. The derivative and rate equation for the sine function x(t)=sinùt uses expansion x(t+Ät)=sinù(t+Ät). To expand sin(a+b) or cos(a+b) we use Fig. 10.3.

            sin(a+b)= cosa sinb + sina cosb          (10.13a)           cos(a+b)= cosa cosb - sina sinb           (10.13b)

                                    Fig. 10.3 Geometry of sine and cosine expansion identities.

 

Expansion of Äx=x(t+Ät)-x(t) for sine or cosine is easy since sinù·Ät=ù·Ät and cosù·Ät=1 for tiny Ät.

                    

                      (10.14a)                                          (10.14b)

We will need the sine and cosine slope (derivative) formulas that follow from this.

                                      

                (10.15a)                                     (10.15b)

A list of series coefficients in (10.12) for sine x=sin ùt and cosine x=cos ùt is worked out below.

                                  

A sine derivative repeats after four orders: …sin t, cos t, -sin t, -cos t, (again) sin t, cos t, -sin t, -cos t, (etc.) .

The resulting sine and cosine series show this repeat-after-4-pattern of factors 0,1,0,-1 of terms.

                           

                                                            (10.16a)                                                                       (10.16b)

The sine is an odd function to time reversal (sin(-t) =-sin(t)), but cosine is even (cos(-t) =+cos(t)). Thus sine has only odd powers p=1,3,5,… of time and cosine has only even powers p=0,2,4,…. Series plots (10.16) in Fig. 10.4 have highest power or order o=1^st,2^nd,3^rd,4^th,etc. Number n of terms is for sine and  for cosine.

Fig. 10.4 Comparing (a) x=sin t and (b) x=cos t with their n^th-order approximate power series.

 

It takes a 9^th (for sin t) or 10^th (for cos t) order series of 5 terms to get one full oscillation with 5% or better precision. Then 10 terms gives two oscillations, and so on. Fig. 10.4 shows that precision breaks down quite explosively. Polynomials are exponentially degrading approximations of wave motion.

Euler’s theorem and relations

Sine, cosine, and e^rt power series (10.16) and (10.9) lead to an 18^th Century crown jewel of mathematics. It is due to a close relation of these series and the functions they represent. It is hard to imagine, but exponential intrest rate growth and simple harmonic oscillation are related. As it turns out, the relation is quite imaginary!

            Suppose the fancy bankers really went bonkers and made intrest rate r an imaginary number r=iè. Imaginary number  has powers with a repeat-after-4-pattern: i⁰=1, i¹=i, i²=-1, i³=-i, i⁴=1,etc... It fits the pattern leading to cosè and sinè series (10.16). Series (10.9) with imaginary rt=iè joins the (10.16) series.

           

                                 (10.17)

The resulting Euler-DeMoivre Theorem is a beautiful identity and a very powerful tool as we shall see. First and foremost it is a complex wave phasor function that we will use in Unit 4. (Note: è =-ù·t.)

                                                                  (10.18)

Fig. 10.5a plots  in the complex plane, a real-vs-imaginary graph. Fig. 10.5b shows as a complex phasor clock. Real part Reø =x(t) is position. Imaginary part is ù-scaled velocity Imø =v(t)/ù. Conversion of polar-to-Cartesian (10.19a) and vice-versa (10.19b) is on scientific calculators. (Recall cautions at end of Ch. 1.)

   (10.19a)                       (10.19b)

Real part Reø is the “is” (that Clinton sought in 1997) and Imø is what Reø is “gonna-be” in-cycle (as in “gonna be in trouble!” A mantra,“Imagination precedes reality by one quarter” works here as in US corporate world.) Euler expo-sinusoidal identities relate cosè, sinè, and e^±iè. A conjugate ø* reflects i with –i.

                (10.20a)                   (10.20b)

A special case is e^-iπ=-1. (We’ll also use a real π-exponential: e^-π=0.04321.) Other special cases are noted.

,           ,           .                       (10.21)

 

Fig. 10.5 (a) Complex plane. (b) Phasor clock. Cartesian form uses (Reø, Im ø). Polar form uses (|ø|,è).

 

Wages of imaginary intrest: Phasor oscillation dynamics

By now bankers should know what happens when you use imaginary intrest. The accounts oscillate up and down and the imagineering bankers oscillate in and out of the slammer. (At least that was the way until 2001 when the Bush administration passed the No Banker Left on His Behind Act that also outlawed reality.)

            Consider exponential rate equation (10.15) with negative imaginary rate r=-iù.

                                               (10.22a)

It becomes a real 2^nd order equation if we apply the derivative operation to both sides.

                                                                     (10.22b)

It is the Newton-Hooke simple harmonic oscillator equation, but it has the same solution as (10.19) above.

                                       (10.23a)

It combines Newton’s force law F=m·a=m and Hooke’s force law F=-k·x. The ù value repeats (9.9b).

                                                                                (10.23b)

What Good Are Complex Exponentials?

Complex Exponentials are used to describe oscillation, resonance, waves and fields. We don't use them just to be cute! Let’s look at some compelling reasons for using imaginary or complex arithmetic.

Complex numbers provide "automatic  trigonometry"

If you have trouble remembering trigonometric identities then this is a good reason all by itself to use complex numbers. For example, if you're taking a test and you can't remember what is cos(a+b), then just factor ei(a+b) = eiaeib, expand exponentials into eia = cos a + i sin a and multiply them out.

                                    ei(a+b) = eiaeib

            cos(a+b) + i sin(a+b) = (cos a + i sin a) (cos b + i sin b)

            cos(a+b) + i sin(a+b) = [cos a cos b - sin a sin b]+i[sin a cos b + cos a sin b]            (10.24a)

That’s two trig identities for the price of one! The real part gives the cosine relation (10.13b).

                        cos(a+b) = [cos a cos b - sin a sin b]                                                               (10.24b)

The imaginary part gives the sine relation (10.13a).

                        sin(a+b) = [sin a cos b + cos a sin b].                                                              (10.24c)

Complex exponentials Ae-iùt tracks position and velocity using Phasor Clock.

Recall discussion of phasor diagram in Fig. 10.5b. Real and imaginary give phase: position and velocity.

Complex numbers add like vectors.

Physics of wave interference involves the addition or subtraction of oscillating signals. If the signals are represented by complex numbers then you simply add (or subtract) their Cartesian components.

                        zsum = z + z' = (x + iy) + (x' + iy') = (x + x') + i(y + y')

                        zdiff  = z − z' = (x + iy) − (x' + iy') = (x − x') + i(y − y')

Before adding, convert z and z' to Cartesian (x,y) form if given in polar form z=reiö and z'=r'eiö'. Radius r of a vector z is its magnitude or complex absolute value |z|. Square |z|² is proportional to energy or intensity.

                        |z| = r = √(x2 + y2) = √([x - iy][x + iy]) = √(z*z)

We write |z|² as product of z and its complex conjugate z* = x - iy =re-iö to derive radius |zsum| of a vector sum zsum or radius |zdiff| of a difference zdiff. It is an easy way to get the well-known cosine laws.

  (10.25a)

                                                       (10.25b)

Vector diagrams of sum, difference, and product of complex z and z′ are shown in Fig. 10.6.

Fig. 10.6 Parallelogram diagonals are sum zsum=z+z' and difference zdiff=z-z' vectors.

Complex products provide 2D rotation operations.

A product zz' of two complex numbers expressed in Cartesian form as z = x + iy and z'= x'+ iy' is

                                    z z' = (x + iy) (x' + iy') = [xx' - yy'] + i[xy' + yx'].

It is simpler if the numbers are expressed in polar form as z = r eiö and z' = r' eiö'.

                                    z z' = ( reiö )( r'eiö' ) = r r' ei(ö+ö').                                         (10.26)

Note that multiplication results in addition of exponents and a sum of polar angles. Radii multiply to give a product rr' but angles add to give a sum (ö + ö'). You might imagine z rotating vector z' by ö radians or that z' rotates z by ö' radians. Consider in detail a rotational operator eiö on a vector z =(x + iy).

                        eiö·z = (cosö + i sinö)·(x + iy)= x cosö − y sinö + i(x sinö + y cosö )                  (10.27a)

Ch. 5 2-by-2 rotation matrix R_ö (Fig. 5.3d) acts on a 2D vector r to give results precisely similar to eiö·z.

                                                                      (10.27b)

                                              (10.27c)

Complex products set initial values

Phase angle -ùt of phasor e-iùt rotates clockwise with time. Multiplying e-iùt by a complex amplitude A =|A|eiñ sets its phase back by angle ñ and its radius to |A|. Amplitude A is the initial value x(0)=|A|eiñ.

x(t)=Ae-iùt = x(0)e-iùt = |A|eiñe-iùt = |A|e-i(ùt-ñ)                         (10.28)           

Such products set initial values of oscillator clocks. A positive angle ñ is a phase lag since it moves the phasor counter-clockwise and sets its clock back. A negative angle ñ=−|ñ| gives a phase lead.

Complex products provide 2D “dot”(•) and “cross”(x) products.

            Consider any two vectors A=A_x+iA_y and B=B_x+iB_y and their “star” (*)-product A*B.

                                                         (10.29)

Real part is scalar or “dot”(•) product A•B. Imaginary part is vector or “cross”(×) product, but just the Z-component normal to xy-plane. To better understand this math trickery, we rewrite A*B in polar form.

                                              (10.30a)

This matches standard 3D definitions of dot(•) and cross(×) products in Appendix 1.A of this Unit.

                                                                                                  (10.30b)

Expansion (10.24) of Ä-angle  relates reiè forms (10.30) to xy-forms in (10.29).

                                  

           (10.30c)                        (10.30d)

Complex deriviative contains “divergence”(∇•F) and “curl”( ∇xF) of 2D vector field

            By relating (z,z*) to (x=Rez,y=Imz) we may define a z-derivativeand “star” z*-derivative.

                                                                             (10.31)

Derivative chain-rule shows real part of has 2D divergence ∇•F and imaginary part has curl ∇×F.

                                          (10.32)

Now we can invent source-free 2D vector fields that are both zero-divergence and zero-curl by taking any function f(z) and conjugating it (change all i’s to –i) to give f^*(z*) for which . For example, if f(z)=a·z then f^*(z*)=a·z*=a(x-iy) is not a function of z so it has zero z-derivative, hence zero ∇•F and zero |∇×F|.

            F=(F_x,F_y)=(f^*_x,f^*_y)=(a·x,-a·y) has zero divergence:  ∇•F=0 and has zero curl: |∇×F|=0.             (10.32)

A plot of vector field F=(f^*_x,f^*_y) =(a·x,-a·y) in Fig. 10.7 shows a divergence-free laminar (DFL) flow field.

Complex potential ö contains “scalar”( F= ∇Ö) and “vector”( F=∇xA) potentials

            Any DFL flow field F is a gradient of a scalar potential field  Ö  or a curl of a vector potential field A.

                                    F= ∇Ö                                                                                                                  F= ∇×A                                                           

There is a complex potential ö(z)=Ö(x,y)+iA(x,y) whose z-derivative is f(z) and it comes with its complex  conjugate  ö^*(z*)=Ö(x,y)-iA(x,y) whose z*-derivative is the f^*(z*) that we use to plot DFL flow fields F.

                                               (10.33a)                                                      (10.33b)

Derivative  by (10.31) has 2D gradient of scalar Ö and curl of vector A.

                                       (10.34)

Some more math trickery has “vector-A” be just a “Z-component” A=A_ze_z normal to the complex (x,y)-plane. So A(x,y)=A_z(x,y) is treated as a single function of (x,y) like scalar Ö(x,y). Also, a mathematician definition for force field F=+∇Ö replaces our usual physicist’s definition F=-∇Ö of (6.9). (No annoying (-)-sign now!)

            To find ö=Ö+iA we integrate f(z)=a·z to get ö and isolate real (Reö=Ö) and imaginary (Imö=A) parts.

                                                               (10.35a)

Note: either part gives the whole F field. Factors in (10.34) could be oror any (f,j) with f+j=1.

              (10.35b)      (10.35c)

Scalar static potential lines Ö=const. and vector flux potential lines A=const. define a field-net in Fig.10.7.

            Fig.10.7 Complex field f(z)=z of F=(x,-y) vectors on potentials of static Ö=(x²-y²)/2 and flux A=xy.

Complex integrals ∫  f(z)dz  count “flux”( ∫Fxdr) and “vorticity”(  ∫F•dr)

            Integral f(z) (10.35a) between point z₁ and point z₂ in Fig. 10.8 is potential difference Äö=ö(z₂)- ö(z₁) between the end-points. In DFL fields, Äö is independent of the integration path z(t) connecting z₁ and z₂.

                                                (10.36)

The real part ÄÖ of Äö is work done pushing r up a hill in Fig. 10.8. (Now force F= ∇Ö  points up-slope.) Since F=(f^*_x, f^*_y) is plotted using f^*(z*), we set f(z)=(f^*(z*))^* to get real and imaginary parts of f(z)dz.

                    (10.37)

Fig. 10.8 Stereo-3D view of Fig. 10.7(ö(z)=z²/2) plots static potential Ö normal to xy-axes.

 

Real part  sums F projections along path vectors dr to get ÄÖ in (10.36). Imaginary part  sums F projection across dr that is, it sums flux thru surface elements dS=dr×_ normal to dr to get ÄA.

            One power-law field f(z)=azⁿ lacks a power-law potential. It is . Its integral is a logarithmic potential ö(z)=a·ln(z)=a·ln(x+iy). (Recall (6.11).) Use ln(a·b)=ln(a)+ln(b), ln(e^iè)=iè, and z=re^iè.

                                                         (10.38)

Potential a·ln(z) is the field of a line of charge q if a=q is real and a line of current J if a=iJ is imaginary. Fig. 10.9a is a diverging F-field of unit charge (q=1) and Fig. 10.9b is a curling F-field of unit current (J=1). Line charge F-field is like an electric E-field. Line current F-field is like a magnetic B-field of a wire, a vortex.

            F-field and radial streamlines (A=è =const.) diverge normal to equal-Ö circles (Ö=r =const.) in Fig. a. F-field and circular streamlines (A=r =const.) curl clockwise normal to radial equal-Ö lines (Ö=è =const.) in Fig. b. (The clockwise (-i)-sense of rotation results from plotting f^*(z*)=-i/z* as our (*)-convention requires.)

            Stereo-3D potential plots of real-line-source field shown in Fig. 10.10a show mathematical structure of its Ö and A potentials that lets us compare them to imaginary-line-source potentials in Fig. 10.10b. Real part Ö=ln(r) of (10.38) for real (a=1)-source in Fig10.10a is a surface like a morning-glory. Blue-(A=è=const.) -streamlines stream down its throat normal to (Ö=r =const.) level circles.

 

Fig. 10.9 Fields due to a unit Z-line-source normal to center. (a) Real source a=q=1. (b) Imaginary a=iJ=i.

 

            Below that Ö-vs-(x,y)-plot is a 3D A-vs-(x,y)-plot for the same real source in Fig. 10.10a. Imaginary part A=è of (10.38) gives radial steps that are level lines of a single helix or helicoid. Red-(Ö=r =const.)-lines stream up its spiral staircase normal to (A=è=const.) steps. At the top step A=è=π , above the –X-axis, is a “waterfall” of red lines falling by ÄA=2π straight to bottom helical step A=è=-π. This 2πi-fall of complex potential ö(z) by Äö=iÄA=2πi at è=±π equals the loop integral of f(z) from è=-π to è=+π.

                                                                                               (10.39)

Imaginary part ÄA of a loop integral counts real source (“flux”) since loop flux is Im in (10.37). Real part ÄÖ= Re counts imaginary source (“vorticity”) since only that makes work around a loop, that is, perpetual motion! In Fig. 10.10b, Ö and A switch roles to make imaginary-line-source-potentials.

Fig. 10.10(a) Real unit line-source (a=1) with diverging F-field resembling E-field of electric line-charge.

Fig. 10.10(b) Imaginary line-source (a=i) with curling F-field resembling B-field of electric line-current.

Complex derivatives give 2D multipole fields

            Of all integer-power-law field functions f(z)=zⁿ of z, only a/z =az^-1 has a non-power-law multi-valued integral and potential (10.38) and non-zero flux-work-loop integral (10.39). This f(z)=az^-1 is a 2D line monopole field and is its monopole potential of source strength a.

                       (10.40a)                                            (10.40b)

            Now let these two line-sources of equal but opposite source constants +a and –a be located at z=±Ä/2 thus separated by a small interval Ä. This sum (actually difference) of f^1-pole-fields is called a dipole field.

                                             

If interval Ä is tiny and is divided out we get a point-dipole field f^2-pole that is the z-derivative of f^1-pole.

(10. 41a)                                 (10. 41b)

A point-dipole potential ö^2-pole (whose z-derivative is f^2-pole) is a z-derivative of ö^1-pole. Pair (10. 41) looks like a Coulomb force (9.1) and potential (9.2) of 3D point monopoles. However, 2D dipole field (10. 41a) is quite different as is 2D potential (10. 41b) whose Ö=const. and A=const. lines make a circle-net in Fig. 10.11.

                        (10.42)

(Note that complex z=x+iy is cleared from the denominator by using z^*=x-iy to give real r²= z^*z=x²+y².)

           

Fig. 10.11 Dipole F-field f(z)=1/z² and scalar potential (Ö=const.)-circles orthogonal to (A=const.)-circles.

Fig. 10.12 Stereo 3D plot of dipole ö(z)=1/z scalar potential Ö(x,y) with A-streamlines between poles.

 

Complex power series are 2D multipole expansions

A z-derivative turns 1-pole fields into 2-pole fields in (10. 41). It makes a copy of 1-pole in (10. 40) with a sign change and puts the (-)copy very near the original. What if we put a (-)copy of a 2-pole near its original? Well, the result is 4-pole or quadrupole field f^4-pole and potential ö^4-pole, each a z-derivative of f^2-pole and ö^2-pole.

(10.43a)                                  (10.43b)

Fig. 10.13 shows 4-pole structure. Two +∞-poles loom above Y-axis and two -∞-poles lurk below X-axis . The F-field vectors and their A-streamlines are shown running at 90° to Ö-equipotential lines in Fig. 10.13.

Fig. 10.13 Stereo 3D plot of quadrupole ö(z)=1/z² scalar potential Ö(x,y) with A-streamlines between poles.

 

Fig. 10.14 F-field f(z)=1/z³ of 4-pole with scalar (Ö=const.)-equipotentials normal to (A=const.)-streamlines.

 

            A field f(z) with sources only at origin (z=0) or at infinity (z=∞) may be given by power series that generalize Maclaurin series derived in (10.11) by using both positive and negative powers z^±n. Series Óa_±nz^±n is called a Laurent series or multipole expansion (10.44) of a given complex field function f(z) around z=0. All field terms a_m-1z^m-1 except 1-pole have potential term a_m-1z^m/m of a 2^m-pole at z=0 (z=∞) for m<0 (m>0).

     (10.44)

The unique 1-pole(2⁰-pole)öterm is not a constant a_-1z⁰=a_-1. (Constantö has no field:) Also a 1-pole at z=∞ gives zero field near z=0. However, a 2¹-pole at z=∞ gives a constant field f(z)=a₀ near z=0. A quadrupole (2²-pole) at z=∞ gives the linear field f(z)=a₁z shown if Fig. 10.7, but a 2²-pole at z=0 gives the field a_-3z^-3 in Fig. 10.14. Octupoles (2³-poles) at z=∞ (or z=0) give a₂z² (or a_-4z^-4), and so on for m=4,5,…

Complex 1/z gives stereographic  projection

            The potential öexpansion is most useful for revealing multi-pole structure. A negative power öterm a_-m-1z^-m/m belongs to a 2^m-pole at z=0. A positive power öterm a_m-1z^m/m belong to a 2^m-pole at z=∞. Pole field geometry involves mapping z-points onto a sphere so z=0 is its North Pole and z=∞ is its South Pole in Fig. 10.15. There a stereographic projection maps a point z=x+iy on the z-plane tangent to North Pole into a point w=1/z=u+iv in the inverse w-plane tangent to the South Pole. The map geometry uses an inscribed rectangle. A pair of red unit circles |z|=1 and |w|=1 map into each other. Any point z inside the |z|=1 circle maps into a point w outside the |w|=1 circle as shown and vice-versa outside z maps to inside w.

 Fig. 10.15 Stereographic projection of z-plane through a unit-diameter sphere to inverse 1/z=w-plane.

 

Replacing z with w=z^-1 in (10.13) switches positive multi-pole-m terms in potential ö with negative ones.

 (from (10.44))

(with z=w^-1)

(with w=z^-1)

But, the unique monopole source term stays put with only a sign change () as seen in Fig. 10.16a. Constant field f=a₀ in (10.44) appears if there is a dipole at the South Pole and, vice-versa, a dipole field at the North Pole appears to be a constant field near the South Pole as seen in Fig. 10.16b.

            Of all 2^m-pole field terms a_m-1z^m-1, only the m=0 monopole a_-1z^-1 has a non-zero loop integral (10.39).

                                                                          

This m=1-pole constant-a_-1 formula is just the first in a series of Laurent coefficient expressions.

  

Fig. 10.16 Projective sphere view of North Pole (z=0) sources. (a) monopole (b) dipole.

 

         Cauchy integrals

Source analysis starts with 1-pole loop integrals  or, with origin shifted . They hold for any loop around point-a. A continuous function f(z) is just f(a) on a tiny circle around point-a.

              (10.45a)                           (10.45b)

The f(a) result is called a Cauchy integral. Then repeated a-derivatives gives a sequence of them.

This leads to a general Taylor-Laurent power series expansion of function f(z) around point-a.

                         (10.45c)

If the function f(z) has no poles inside the contour then only positive powers n>0 are needed in its expansion and the series above reduces to a Taylor series or (if a=0) a Maclaurin series like (10.12) derived previously. There the n^th expansion coefficient a_n is given by n^th derivative of f(z) as in (10.45c) above. Otherwise, negative powers are needed with coefficients given by n^th order pole loop integrals above.

            This represents just a “tip of an iceberg” for an enormous subject of complex analysis. We shall use only tiny portions of this grand mathematical subject, and later we will consider generalizations of complex numbers to hyper-complex quaternions and spinor operators in Unit 4. This takes the analysis from a 2D framework into a 3D and 4D description that is more like the space-time we seem to live in.

 

 

 

 

Exercises

Construct dipole function geometry of Fig. 10.11.

 

 

Chapter 11. Oscillation, Rotation, and Angular Momentum

We last left the neutron starlet orbiting on an ellipse inside the Earth in Fig. 9.10 according to

x = a cos ù t     (9.13a)_repeated                             y = b sin ù t    (9.13b) _repeated

Here we show a Kepler construction for such an orbit that works for any ellipse. (It is like Fig. 3.6.) We also expose more geometry of velocity-velocity KE-ellipses used to introduce Lagrangian, Hamiltonian, action, and contact transformations in the following Chapter 12. That leads to more efficient ways to treat orbits.

Keplerian construction of elliptic oscillator orbits

To be historically correct, Kepler was concerned with elliptic orbits that lie outside of the Earth not the inside-Earth orbits in a linear force law F(r) = -kr that we plotted. As we will show in Unit. 5, outside orbits in a Coulomb force law F(r) = -kr^-2 also have elliptic orbits, albeit with origin r=0 at a focal point. That’s a little more complicated. So, we first study the easier inside-Earth orbit ellipses that have r=0 centered. This gives some properties of their country cousins who live well outside the city limits.

Elementary ellipse construction

Fig. 11.1 shows an easy 4-step construction for points on a (major-radius=a, minor radius=b)-ellipse. Note that you don’t have to draw OA first. Pick a vertical (AX) or a horizontal (BR) line first and then find the others including the OA radius that goes with your choice. Given x or y, you find t or vice versa.

The big a-circle acts like a clock dial. The x-shadow or projection of the clock dial is x = a cosù t and every mass that starts at x=a at zero-x-velocity will forever live in the shadow of the tip of the clock hand. This includes any ellipse with semi-major axis a, but arbitrary semi-minor axis b.

The ellipse in Fig. 11.1 has b=1 and a= 2.2. The speed of the orbiting mass can be estimated by the space between positions at equal time intervals. Speed is smaller as the mass rounds the long end of the ellipse than it is as it zips by the minor axis. In fact we shall show that it is exactly 2.2 times faster, a result that is attributed to Johannes Kepler and is the result of the conservation of angular momentum.

As mentioned before after Fig. 9.8, all orbits have the same period, and the mass that tunnels through the Earth center at the bottom of Fig. 11.1 has exactly the same x-equation x = a cos ù t as the ellipse-following mass above it. They differ only in their y-equation y = b sin ù t ; in the first case the tunneling mass has b=0. A circular orbit would have b=a, but its x-equation would be the same. Note how the radius vector r of the mass lags behind the ù t-clock-hand at first, but then at the b-axis low point (perigee) of the orbit it catches up and passes until the clock hand catches it again at the other a-axis high point (apogee or “up-ogee”). This leap-frog motion relates to one of Kepler’s most famous laws and the conservation of angular momentum as will be reviewed shortly.

Fig. 11.1. Harmonic force-field elliptical orbit construction.

 

Fig. 11.2 Two different systems with identical oscillator orbits. (a) Inside Earth, (b) Mass on spring.

 

Orbiting versus rotating: Centripetal versus centrifugal

Imagine an “Earthronaut” orbits inside the Earth in a linear gravity field F=-kr, as sketched in Fig. 11.2(a). (Recall “starlet” in Fig. 9.9.) Let’s compare to a kid rotating in a carnival ride at one end of a spring as the other end pivots frictionlessly about a fixed point. (See Fig. 11.2(b).) Each m does the same orbit, but there’s a big difference. You’d notice it if you were the mass m.

The Earthronaut feels weightless like astronauts in orbit. But the rotating kid feels a great outward pull, a centrifugal or center-fleeing force F=+kr. Stop the rotating “carnival kid” and the centrifugal force goes away. If the kid lets go he feels weightless in space. Stop the orbiting Earthronaut and the inward tug F=-kr by the centripetal or center-pulling force of gravity returns as the Earthronaut resumes weighing mg=kr. Earth gravity is no longer cancelled by inertial reaction force and he cannot let go of g.

An orbiting Earthronaut feels weightless because the two forces, outward centrifugal F=+kr and inward centripetal F=-kr , cancel to zero for body mass m or any part of it. On the other hand, the carnival kid feels stretched out by two equal and opposite forces, again an outward centrifugal F=+kr pulling the kid up opposes an inward centripetal F=-kr provided by the spring that the kid is holding onto.

In each case, outward centrifugal F=kr is due to rotation at angular rate ù around a circle of radius r at velocity V= ù r. The angular rate ù is the Earth or spring oscillator frequency from (9.9) or (10.23).

                                  (11.2a)             or:                                           (11.2b)

Centrifugal force formulas that result are among the most famous formulas in rotational mechanics.

                                    F_centrifugal = k r = mù² r = m V²/r         where: V= ù r                (11.3a)

Removing the mass m gives the also-famous centrifugal acceleration formulas.

                                                a_centrifugal = ù² r = V²/r              where: V= ù r                (11.3b)

Circular curvature

A geometer likes to imagine fitting a curve by circles at each point with smaller circles fitting more curvy points. These so-called circles of curvature become bigger circles as a curve straightens out. A geometer-physicist does the same, but imagines driving at a constant speed V along the curve with an accelerometer to measure transverse centrifugal acceleration a_centrifugal. By (11.3b) the accelerometer reads V²/r_curv outward from a curve if the car is rounding a circle of radius r_curv = V²/ a_centrifugal with its center that distance inside the curve. The a_centrifugal-reading is inversely proportional to radius of curvature for fixed linear velocity V, but directly proportional to it for fixed angular velocity ù.

                                     r_curv = V²/a_centrifugal= a_centrifugal /ù²         where: V= ù r_curv                        (11.3c)

It is a strange but useful view of a curve! The physicist imagines riding a carnival Merry-Go-Round whose rim speed V is constant but whose radius and center keep changing! If the road straightens to veer the other way, the Merry-Go-Round center becomes infinite and reappears on the other side.

Note that road speed V is constant in the physicist’s image. There’s no acceleration along the road, only perpendicular to it. However, in real orbits around planets or springs, velocity V holds constant only for circular orbits or, ever so briefly, at special points on elliptical ones. One special point is a low point or perigee. Another is a high point or apogee. (Think “ap” means “up” in space lingo.)

An astronaut in an elliptic orbit or a mass on an elliptic oscillator orbit like Fig. 11.1 will increase speed (accelerate) as it “falls” from the high-point apogee on the x-axis toward the low-point perigee on the y-axis. Then it will decrease speed (decelerate) as it rises back to apogee. Only at apogee or perigee is the speed momentarily constant. Then, and only then, is force and acceleration perpendicular to the flight path. In between, the F=-kr vector makes an angle è with velocity V that is not 90° so the work (dW= F•dr= |Fdr| cosè) or power (P= F•V= |FV| cosè) is non-zero so kinetic energy varies.

 

       

Fig. 11.3 Elliptic orbit force, velocity, and power variation.

More inertial forces: Coriolis and tidal forces

Carnival kid would feel even more forces on an elliptic orbit, though the Earthronaut may still be nearly weightless. Gravitational force is balanced by centrifugal force and, between apogee and perigee, by another kind of inertial force called the Coriolis force that opposes orbital velocity.

To visualize Coriolis force imagine what you would feel walking along a radial railing toward the center of a Merry-Go-Round rotating to your right as in Fig. 11.4(a). The railing pushes you left (against the rotation) to slow you down to zero speed when or if you get to the center of the Merry-Go-Round. The Coriolis force is proportional to your radial walking speed. Stop walking inward and all you feel is the usual centrifugal force pulling back out along the radial railing path. Walk back out and Coriolis pushes you to the right to get you up to the Merry-Go-Round rotation speed at each point.

Coriolis forces can make you dizzy and nauseous. Centrifugal force is steady as long as you are fixed to the Merry-Go-Round. But, if you just turn your head, the fluids in your inner ear get a kick perpendicular to the direction of motion and they’re not used to that.

Fig. 11.4(b) shows centrifugal and Coriolis forces of an inward falling orbiting mass analogous to that of the Merry-Go-Round. The Coriolis force acts oppositely to orbital velocity V on the way in and then acts with V on the way out in Fig. 11.4(c). At apogee or perigee in Fig. 11.4(d) there is centrifugal-centripetal force but no Coriolis force since the mass momentarily stops its radial motion.

The Earthronaut may not feel centrifugal or Coriolis forces if every atom almost perfectly balances inertial force by equal and opposite gravitational force to make a feel-force-free orbit. But, “almost” is not zero! Suppose our astronaut is on a 1 kHz neutron star orbit. (That’s ù=2000π.) He (or she) is toast and jelly due to what is called tidal force. Only the astronaut’s center-of-gravity is right on a feel-force-free elliptic orbit. For the rest that’s the wrong ellipse! The poor astronaut’s left and right hands (and ears and other bilateral pieces of anatomy) try to change places 2000 times per second as disparate free-fall orbits crisscross back-and-forth twice each period. Really barf!

The k-constant or spring constant for an oscillator tidal force felt by a neutron-astronaut (who will be reduced to a “neuternaut” by one orbit) is given by (11.2b).

 k=mù ²= m6.28² E6  (N/m)     for: ù=2000π                           

That is almost 40 million Newtons (10 million lbs. or 5 thousand tons) for each kilogram of mass a meter off-center or 50 tons of pressure just on a 1 cm-sized fingertip.

Quite a number of astrophysical effects are due to tidal forces like ocean tides. The Moon presents one face because its tides due to Earth have wasted as much of its rotational energy as possible. So it’s locked relative to Earth with only slight (but interesting) nutational wobbling due partly to its eccentric orbit.

Fig. 11.4 Centrifugal and Coriolis forces. (a) Simple Merry-Go-Round. (b-d) Various orbital phases.

Vector analysis and geometry of elliptic oscillator orbit

An ellipse orbit is characterized using vectors r, v, and F= ma that are arrows in Fig. 11.4. First, there is the location, position, or radius vector r that we found in (9.13).

                                                                                                          (11.5a)

Second, there is the rate, speed, or velocity vector v that is a 1^st time derivative.

                                                                                                 (11.5b)

Unit-m force F is proportional to 2^nd derivative (Change-of-velocity is acceleration a) and is just -ù²r.

                                                                          (11.5b)

Then the 3^rd derivative (Change-of-acceleration is jerk j), is just -ù²v,

                                                                                        (11.5c)

and finally the 4^th derivative (Change-of-jerk is inauguration i), equals the r-vector with a scale factor ù⁴.

                                                                                   (11.5d)Linear Hooke force F=-kr gives F=ma=-kr then a=-ù²r then j= -ù²v, and so on. The 5^th derivative (Change-of-inauguration is revolution) is just ù⁵v. As plotted in Fig. 11.5, the four vectors r, v/ù, a/ù², and j/ù³ follow each other on one ellipse orbit of Fig. 11.1 taking turns to slow down then to speed up.

Fig. 11.5 Harmonic oscillator orbit ellipse (a) All derivatives follow same orbit. (b) Related tangents.

 

Each of the four vectors r, v/ù, a/ù², and j/ù³ in Fig. 11.5 has a time-phase angle or mean anomaly value ö=ùt that is spaced at π/2 intervals ùt, ùt+ð/2, ùt+ð, and ùt+3ð/2, respectively, as listed below.

 (11.6a)    (11.6b)    (11.6c)      (11.6d)

Matrix operations and dual quadratic forms

Ellipse equation  may be written using a matrix  on position vector .

                             (11.7)

Function r•Q•r is a quadratic form QF. QF’s are useful to mechanics and their powerful geometry will be demonstrated for orbit ellipses and later for KE ellipses. First note that if a matrix  acts on a radial position vector  it gives a vector p perpendicular to ellipse tangent  at r.

           (11.8a)

p is perpendicular, that is, orthogonal to the velocity vector v= (11.5b) as seen here and in Fig. 11.6.

(11.8b)

These p-vectors define their own ellipse r•Q^-1•r=1 of an inverse quadratic formQ^-1F. Its radii are inverse (1/a,1/b) of the original Q-ellipse radii(a,b) in (11.7). The Q^-1F-ellipse is the dashed oval in Fig. 11.6.

                    (11.9)

Inverse operation Q^-1•p on perpendicular p returns the radial position vector r on the Q-ellipse.

                                 (11.10a)

r is orthogonal to the Q^-1F-ellipse tangent, just as p is orthogonal to the QF-ellipse tangent in (11.8b).

(11.10b)

Vectors p and r maintain a unit mutual projection, that is, dot-products p•r and always equal 1.

                          (11.10c)

Fig. 11.6b shows a geometric-algebraic symmetry where ellipse plots are scaled by geometric mean S=√(ab) so that each scaled major radius a_S=a/S is the inverse of its minor radius b_S=b/S and a_S b_S=1.

            a_S=a/S=√(a/b)=1/b_S   (11.11a)                                   b_S=b/S=√(b/a) =1/a_S             (11.11b)

Then inverse ellipse r•Q^-1•r=1 is an axis switch () or 90° rotation of ellipse r•Q•r=1 by symmetry.

Fig. 11.6 Ellipse vectors and tangents for quadratic forms. (a) Ellipse vectors. (b) Tangent geometry.

Slope multiplication and eigenvectors

Matrixor  acting on a vector  multiplies its slope by a/b or b/a respectively.

             (11.12a)                           (11.12b)

Matrixor  multiplies slope by a²/b² as in (11.8a) or b²/a² as in (11.10a).

                    (11.13a)                              (11.13b)

Only vectors of slope zero or infinity, such as or , are immune to slope-change by R or R^-1.

             (11.14a)                    (11.14b)

They are called eigenvectors of R^-1 or any power of R or R^-1.

              (11.15a)                                    (11.15b)

           (11.15c)                                 (11.15d)

These special vectors are operator R^p’s own base vectors for any power p. Eigenvector is German for “own-vector.” Base vectors anddefine a Q^p-and-R^p-ellipse’s own major and minor axial directions. The axial radii a and b are the eigenvalues of R^-1 in (11.14). Powers a^p and b^p are eigenvalues of R^-p.

Geometric slope series

Each action of R (or Q) on vector r grows its slope by a/b (or a²/b²) so it approaches eigenvector while R^-1 and Q^-1 make it approach eigenvector . Each slope polar angle ö_k plotted in Fig. 11.7 is obtained from its neighbors ö_k-1 and ö_k+1 by inscribing a rectangle between r=a and r=b with its main diagonal on the  ö_k line. Lower and upper corners on the cross-diagonal give radial position r(ö_k-1) on the Q-ellipse and perpendicular p(ö_k+1) on the Q^-1-ellipse, respectively, following Fig. 11.1 and (11.13).

                                                p(ö_k+1)=Q•r(ö_k-1)         where: tan(ö_k+1)= (a/b) ² tan(ö_k-1)                     (11.16)

For the k^th triad, angle ö_k =ùt_k is the “timer” angle and polar angle of main diagonal while ö_k-1 is the polar angle of radial position r(ö_k-1) and ö_k+1 is the polar angle of perpendicular p(ö_k+1) to velocity (ö_k-1)= v(t_k).

            p(ö_k+1)•(ö_k-1)=0= (ö_k+1)•(ö_k-1)    (11.17a)                                   p(ö_k+1)•r(ö_k-1)=1          (11.17b)

This restates the duality relations (11.10) for an entire sequence, part of which is shown by Fig. 11.7b.

            A {ö_k} sequence may start on any angle but a choice ö₀=ð/4 in Fig. 11.7 gives symmetric results. Also, we may let: ab=1 and: below by assuming unit scale S=1 in (11.11).

               

    (11.18a)               (11.18b)                    (11.18c)

              

Triads {r_k+1,r_k,r_k-1} and {p_k+1,p_k,p_k-1} of vectors r_k=r(ö_k) and p_k=p(ö_k) are given for ö₀=ð/4.

                                             (11.19)

Each triad {r_k+1,r_k,r_k-1} easily gives the tangent v_k-1=v(ö_k-1)= that contacts the Q-ellipse at r_k-1. An arc by r_k intersects r_k+1 where v_k-1 is perpendicular to r_k+1 or p_k+1=p(ö_k+1). (See Fig. 11.7a and exercises.)

            So far the ellipse axes line up with the Cartesian coordinate axes of a standard page. Ellipses in other bases may be rotated, and certainly an orbit of an isotropic oscillator may choose any direction for its axes. The following general 2D quadratic form gives a rotated conic section (ellipse or hyperbola).

                      (11.20)

Fig. 11.7 Triad sequence geometry of radial position vector r(ö_n-1) and tangent-perpendicular p(ö_n+1).

 

However, all the relative geometric properties such as their tangent geometry are the same in all bases. It’s abstract vector equation 1=r•Q•r looks the same in any coordinate base system, but the matrix components may include non-zero off-diagonal elements B≠0 that indicate it is a rotated ellipse.

Angular momentum and Kepler’s law

The shape and rotational orientation of an isotropic oscillator orbit ellipse is constant with time. The cross product of rxv of position and velocity is also a constant of the motion by (11.5). (See App. 1.A.)

                               (11.21)

The quantity L=m rxv is called orbital angular momentum. It’s conserved as mass m orbits.

                                                                             (11.22)

It means the area of r+v or r-v triangles, as discussed in Appendix 1.A, are constant on an orbit as indicated in Fig. 11.8 below. Area enclosed by r and v is proportional to the area π ab of the whole orbit.

 

Fig. 11.8 Vector r+v and r-v parallelogram and triangle areas are constant all during orbit.

 

By (11.22), velocity at perigee (x=0,y=b) is v_b=L/mb=aù. At apogee it slows to v_a=L/ma=bù. This is consistent with velocity formula (11.5b). Constant momentum relates to Kepler’s Law: the radius r-vector sweeps the same area every second or every hour and equal time means equal area.

This is true since the triangle made of r and dr=v dt has the same area ¹/₂rxv dt = (L/m)dt for the same time interval dt. This law applies to any central force that is a function of radius r alone, not just the oscillator force F=-k·r. This includes the Coulomb force F=-k/r², which is the only other force to have elliptical orbits that maintain their orientation.

The oscillator and Coulomb forces each have hidden symmetry beyond their Keplerian rotational isotropy that conserves angular momentum and this makes their orbits have simple geometric properties. This extra symmetry will be analyzed in units 4 and 5.

Flight of a stick: Introducing geometry of cycloids

If linear momentum and angular momentum are conserved they often do so together. As an example, we consider the flight of a rigid rod or stick in free space. Flying rods are treated in Sec. 6.4 of the rigid body unit (Unit 6) but elementary aspects of rigid body motion are easy to derive and they display cycloid geometry that is useful for several classical mechanical phenomena. A mass m rotating on a circle of radius r with angular velocity ù has a linear tangential velocity V=ù·r and kinetic energy . (Recall (9.10) for orbiting starlet.) An angular form is derived here again.

                                                                                                 (11.23)

Circular orbiter angular momentum (from (11.22) above) takes an angular form, too.

                                                                                                          (11.24)

In each the point mass rotational inertia  replaces linear mass m while angular velocity ù replaces V.

            A rod or lever of length l rotating about one end is viewed as an integral from r=0 to r=l of its mass points each of infinitesimal inertia . Density ñ is the rod’s total mass M per length l, and that is assumed to be uniform. Total inertia follows from taking the integral over its length.

                                                                                       (11.25)

            This inertial formula is true for two identical rods of length l welded end-to-end and rotating about that point. Now mass M is that of the total system. In free space the straight welded rod will rotate naturally with constant angular velocity ù about the welded point at its center of mass and that center travels at a constant velocity  until hit by an outside force.

            Then free-space paths of each point on the rod, including its center-of-mass, are generalized cycloids such as are shown in Fig. 11.9. There are two dots on the rod (a red dot • and a green dot •) that follow normal cycloids. Each comes to a complete stop at a cycloid cusp (as green dot • is in the lower center of the figure) while the opposite dot is just reaching its maximum velocity (as red dot • is in the upper center of the figure). On left and (later) right sides of Fig. 11.9 is similar with rod flipped.

Imagine dot • and dot • are pieces of gum stuck to opposite sides of a tire (green circle of radius p in Fig. 11.9) rolling left-to-right along a line (“road”) with rod CM point attached at tire center. A blue dot • on one end of the rod is initially above the green dot • (upper left of Fig. 11.9) while the rod’s lower end has a violet dot • attached below the red dot • where the tire initially meets the road. Rod points outside tire radius (blue dot • and violet dot •) trace curlate cycloids. Inside points (CM and yellow dot •) trace prolate cycloids. The CM just follows a straight line at constant speed. Tire radius p depends on hit-height h above CM point where momentum impulse Ð is delivered in Fig. 11.10. However, regardless of h, the CM point will travel at constant linear velocity V=Ð/M while the rod conserves linear momentum Ð=MV.

Center of percussion, radius of gyration, and “sweet-spot”

Angular velocity ù relates to hit-height h factor in angular momentum Iù=Ë=Ðh and to tire radius p where red dot • on tire comes to rest on road, and velocity ù·p=Ðh·p/I due to rotation cancels velocity V=Ð/M due to translation. This gives relation h·p=I/M. Rod inertia then gives relation  between hit-height h and radius p of percussion for rod of radius l. We call hit point h a “sweet-spot” for point P and vice-versa.

 

   

            Fig. 11.9 Free flying rod of length L=2l “rolls” left-to-right on “tire” of radius p.

           

            Fig. 11.10 Impulse hit-height h relates to rod radius l and percussion radius p of rolling “tire.”

 

            If the hit-height h is zero then percussion radius p is infinite and all points of the rod follow straight parallel paths since there will then be zero rod rotational velocity ù and zero angular momentum Ë=Ðh. Only linear velocity V=Ð/M would be nonzero then. If the hit-height h is equal to l (the maximum practical value of h is the radius l of rod) then the percussion radius is p=l/3, its minimum practical value.

Reducing h increases p proportionally. The two are equal at the value p=l/√3=h=0.866l which is called the radius of gyration of the rod. Reducing hit-height h further to l/2 and l/3 increases the percussion radius p to p=2l/3 and p=l, respectively. The percussion point is where you can hold the lever and feel the very least recoil during the hit. In fact, the dynamics at a normal cycloid cusp point at radius p in Fig. 11.9 amounts to a gentle tug along the lever but no force perpendicular to it.

            Baseball bats are made thicker at the hitting end to accommodate h and p points further from the ends than allowed by p=h=0.866l. Cricket bats, on the other hand, seem to be more like sticks.

 

Exercise 1.11.1

Quadratic form matrices are generally non-diagonal . If Q has positive eigenvalues 1/a² and 1/b² its form is called positive definite and r•Q•r =1 gives an ellipse rotated by angle è with radii a and b. So does r•Q^-1•r =1.

(a) Derive relations between matrix parameters {A,B,D} and ellipse parameters {a,b, è }. (Hint: Start with diagonal matrix and rotate it to Q´=R·Q·R^-1 by applying rotation transformation matrices  and .)

(b) Show TraceQ and detQ (or TraceQ^-1 and detQ^-1) are invariant to è and relate them to conserved quantities such as total energy  and orbital momentum  of isotropic 2D harmonic oscillator orbit elliptic orbit r(t) in (11.5). Could the energy or momentum of an isotropic 2D HO orbit depend on orientation angle è ? How or why not?

(c) Suppose Q’s eigenvalues were 1/a² and -1/b². What curve is r•Q•r =1? Plot for a=1=b and è = 45°.

 

Exercise1.11.2

Recall Fig. 11.7 geometric sequence {r_-3, r_-2, r_-1, r₀, r₁, r₂, r₃,…} of ellipse radii r_k=r(ö_k) and perpendicular-to-tangents p_k+2 =Q•r_k defined by quadratic forms r_k•Q•r_k =1= p_k•Q^-1•p_k by . (Ellipse radii had ab=1 and 0^th sequence slope was tanö₀=1 (ö₀=ð/4) but other values work as well.)

(a) Construct super-imposed r-ellipse and p-ellipse with at least seven vectors each for a/b=2 (done in class) and a/b=5/4. Give the vectors r_k you draw algebraically (in terms of a and b) for  and check k=2 cases numerically with geometry.

(b) Verify duality relations:  and  . (For time derivatives let ù=1.)

(c) The text noted that a ellipse tangent v_k-1=v(ö_k-1)=at r_k-1 is also tangent to a circular arc C_{k to k+1} swept by radius |r_k| from the tip of r_k to where C_{k to k+1} is perpendicular the r_k+1-line. Show this implies a relation

                                                .            (Notation  denotes unit vector: )

Verify this algebraically and geometrically for the case k=-1 and k=0 using vectors you have derived. Use the result to construct tangents v_k contacting each radius {r_-3, r_-2, r_-1, r₀, r₁, r₂, r₃,…} on r-ellipse. Place the tangent vectors in r-ellipse.

 

 

Chapter 12. Velocity vs momentum functions: Lagrange vs Hamilton

Relating energy ellipses in velocity and momentum space

The ellipse in Fig. 5.1 is skinny and difficult to see. To better view multiple collisions, the v₁-v₂ axes are rescaled into “quasi-velocities” V_k=v_k √m_k so the ellipse forms into a nice circle in Fig. 5.2.

             where:  and:          (12.1)

The half-power mass scale is helpful. A full power m_k-scale converts velocity v_k to momentum p_k.

             where:  and:                       (12.2)

Geometry of a p-ellipse is just a flip of the v-ellipse, but there are compelling algebraic reasons for dealing with such alternative functions. In fact two of these functions have famous names attached.

Lagrangian, Estrangian, and Hamiltonian functions

An energy that is an explicit function of velocities is called a Lagrangian function L=L(v_k..).

                                                                                 (12.3)

An energy that is an explicit function of momenta is called a Hamiltonian function H=H(p_k..).

                                                                                    (12.4)

A compromising function like (12.1) has no famous name so we’ll call it an Estrangian E=E(V_k..).

                                                                                     (12.5)

            While all these functions may have the same numerical value for a given situation, the have quite different functional dependence. To emphasize this let us write our first equations of (non)-motion.

             (12.6a)            (12.6b)             (12.6a)

The first two for L and H say that L has no explicit p-dependence and H has no explicit v-dependence. L may still vary if p varies but L is not defined by p and the same for H and v. Calculus distinguishes total derivatives or  from partial derivatives  orand begins by defining differential chain rule sums.

                        (12.7a)                 (12.7a)   

Then L (or H) varies with any variable z such as v_k, p_k, or time t according to derivative chain rule sums.

                         (12.7a)              (12.7a)

(Imagine L(v...) is “married” to v and H (p...) to p. Dots denote coordinates and time discussed later.)

 Neither may use another’s dependents without legal difficulty! Geometry helps clarify this below. 

L, E, and H ellipse geometry

A Lagrangian ellipse plot const. = L(v) in Fig. 12.1a is similar to the superball collision diagram in Fig. 5.1. It is to be compared with the corresponding Estrangian ellipse (circle) plot const. = E(V) in Fig. 12.1b and the Hamiltonian ellipse plot const. = H(p) in Fig. 12.1c. COM and collision line slopes are compared.

 

Fig. 12.1 KE ellipse functions related by scale. (a) L in velocity v_k space. (b) E in V_k. (c) H in p_k.                                                                

Functions L, E, and H are quadratic forms of vectors v, V=R•v, and p=M•v=R²•v, respectively.

  (12.8a)            (12.8b)           (12.8c)

The corresponding scaling matrices are powers of the root-mass matrix:  R=

                               (12.9)

The 2^nd power rescaling M=R² mass matrix maps L(v) space (a) into Hamiltonian H(p) space (c).

            The p-to-v tangent-normal mapping is analogous to r-to-p mapping in Fig. 11.6 as displayed in a Fig. 12.2c that overlaps parts (a) and (c) of Fig. 12.1. Velocity v is a p-space gradient operation H= v and thus normal to H-ellipse, and vice-versa for the normal p=L to the L-ellipse. Matrix notation is given, too.  

                        (12.10a)                               (12.10b)

 (12.10c)         (12.10d)

Fig. 12.2 Tangent-normal mapping between (a) Lagrangian L(v) space and (b) H(p) space.

If mass increases s-times then L-ellipse radii become s times H-ellipse radii. (d) s=1/2, (e) s=2.

 

Fig. 12.2 is a top view of L(v)-vs-v and H(p)-vs-p plots. Side views are shown and discussed below.

Legendre contact transformations

Given mapping  or inverse , it might appear that either quadratic form or  may be written simply as  or  . This is correct numerically but its calculus is not. Instead, it is  or else  that gives correct derivatives. This results in the Legendre contact transformation between H(p) and L(v) expressed by the following identical equations.

                           (12.11a)                              (12.11b)

 They give correct partial derivatives with zero for  and  according to definitions in (12.6), as follows.

   (12.12a)                                     (12.12b)

The results are the first Hamilton equation and the first Lagrange definition (Recall (12.10). Reversing p and v derivatives gives them again in reverse order but quite consistently.

   (12.12c)                                      (12.12d)

Side-view sketches of eqs. (12.11) and (12.12) are given in Fig. 12.3a-b and Fig. 12.4a-b below.

 

  

   Fig 12.3 Geometry of Legendre contact transformation relating (a) L(v)-vs-v and (b) H(p)-vs-p plots.

 

Extreme geometry of contact transformations

Contact transformations are among the most enduring and fundamental ideas in either classical or modern physics. Yet few texts on these subjects explain them adequately, if at all. Most can’t even tell why “contact” appears in their name. Our explanation revolves around explicit-function issues involving equations (2.12):

            “L(v...) is not function of p, H(p...) is not function of v, yet L(v...), H(p...), p, and v are all related!”

There is method in this madness! One should learn how these ideas “contact” so much of physics.

            The term “contact” refers to a line or curve touching or being tangent-to another curve. It is opposite to more common cases of crossing or being secant-to another curve (swordlike). Examples in Fig. 12.4 are based on Fig. 12.3.  Lagrangian side (a) shows secant lines  all of slope  but decreasing intercept  tied to increasing velocity points  leading to a unique tangent to the curve at tangent contact point  that has a max-value  of . At that point the Hamiltonian  has no 1^st order variation with respect to velocity, that is, has zero 1^st v-derivative.

                                                                     (12.13a)

Thus loses its explicit v-dependence at each tangent point but does depend on its slope. So also does lose its explicit -dependence at each tangent point but does depend on that tangent slope .

                                                                    (12.13b)

Fig 12.4 Geometry of explicit dependence. (a) loses dependence.  (b)loses dependence.

 

Let’s examine more general examples of contact mapping that help clarify its beautiful structure and utility.

 

General contact transformation geometry

Consider now a contact transformation that relates to some classical physics or to some modern physics while reviewing some sophomore physics. It involves the geometry of volcanic plumes on Io or atomic clouds rising and falling in Earth gravity as sketched in Fig. 12.5a or b, respectively. Each is modeled by a parabolic trajectory fountain in Fig. 12.5c that you may have studied in sophomore physics.

Fig. 12.5 Modeling (a) Io volcano and (b) Atomic clock by (c) trajectories of initial velocity v0 and angle á.

       

Initial position x(0)=0=y(0) and velocity below lead to a fixed-g trajectory x(t) = (x(t), y(t)).

                              (12.14a)                             (12.14b)

                                              .                               

The x(t)-solution has time t=x/(v0 cos á) to put in y(t) and get each trajectory y(x) plotted in Fig. 12.5c.

                                    .                  (12.14c)

            Each trajectory is a zero value of a Contact Generating Solution S(v0, á : x, y)given by

                                    .                                 (12.15)

 S(v0, á : x, y) maps initial value point (v0=1, á=45°) in Fig. 12.6 onto red trajectory curve y(x) in Fig. 12.5c. A horizontal line of points (same v0 but á varies) fills a region of y(x) space with the v0-trajectory family.

Fig. 12.6 Contact transformation maps point (v0, á) to trajectory of initial velocity v0 and angle á.

 

Envelopes of the v0-trajectory region contain extremal contact points with each trajectory. Varying á at such a point does not change S so 1^st á-derivative of S is zero, quite analogous to zero derivatives in (12.13).

                                                                                                                             (12.16a)

                                                                    (12.16b)

Solving this equation relates the x-value and the á-value of each contact point for a given fixed v0.

                                                                                             (12.16c)

If you put this relation into generating function (12.15), it gives the contact envelope function y_env(x).

            .            (12.17)

It is the dashed parabolic curve in Fig. 12.5-6 contacting each and (almost) every parabolic trajectory from above. The envelope happens to share the shape of the (á=0)-trajectory hilited in green in Fig. 12.5. That is the single trajectory that never contacts the envelope. Do exercises to see more of this lovely geometry!

            A generic general contact transformation S(x,y:X,Y) shown in Fig. 12.7 maps points (xk, yk) in (x, y)-space into points (Xk, Yk) in (X, Y)-space so function y(x) is contact-transformed to function Y(X) there. Such a transformation can occur from one curved set of points to another in the same space as in Huygen’s wavelet view of wave propagation: each wavefront curve at one instant is a contact transform of a wavefront at another time. (Presumably, it’s an earlier time but quantum waves are time reversible.) Contact transform geometry plays such a big role in connecting (contacting) classical and modern physics as we’ll see. 

            Contact transforms are key to classical thermodynamics. For example, internal energy U(S,V) is defined as a function of entropy S and volume V. A new function enthalpy H(S,P) depends on entropy and pressure P. It is a Legendre transform H(S,P)=P·V+U of energy U(S,V) to new variable  . Except for ±signs, it’s our Hamiltonian H(p)=p·v-L(v) from Lagrangian L(v) to new variable momentum.

Fig. 12.7 General contact transformation S(x,y:X,Y) maps each point (xk, yk) to a contact point (Xk,Yk).

 

Let us return briefly to the Legendre-Lagrangian-Hamiltonian relation (12.11) by comparing it’s geometry in Fig. 12.8 to the generic case in Fig. 12.7. The general case makes contacts using curved tangents. Legendre uses straight line tangents and is thus easily invertible as shown below in Fig. 12.9 and before in Fig. 12.3.

Fig. 12.8 Legendre contact transform S(x,y:X,Y) maps each point (xk, yk) to a contact point (Xk,Yk).

 

Fig. 12.9 Summary sketch of Legendre-Lagrangian-Hamiltonian geometry of Fig. 12.3 and 12.4.

The Equations of the Classical Universe (Lagrange, Hamilton, and others)

While string theorists search for an “Equation of the Universe”(EOTU) or a “Theory of Everything” (TOE) it should be noted that virtually all our modern physics, however avant-garde, has a classical foundation in Lagrangian or Hamiltonian equations. So a derivation of these all-important equations is in order. We already have deduced, mostly by symmetry and functional trickery, half of the Lagrange equations ( in 12.12c) and half of the Hamilton equations ( in 12.12d) for purely kinetic quadratic form Lagrangian or Hamiltonian. Needed now are the other half that add a potential energy U(x) with spatial coordinate-x dependence to  and . That wrecks our nice translational symmetry and momentum conservation. Fortunately, the halves we have so far still apply. (Nature is so kind!) Also, the to-be-derived EOTCU (Equations of the Classical Universe) translate easily to “quite queer” coordinates {q¹, q²,...} such as polar, parabolic, hyperbolic, etc. that were introduced in  Ch. 10 in connection with complex potential fields.

            Generalized curvilinear coordinate (GCC) q^m systems are a big deal with a long history coming after (de)Cartesian coordinates x^j (or x_j) introduced in Chapter 1. GCC superscript q^m convention that puts the index m up (where exponents usually go) and its notation by letter-q may seem, well, quite queer. But, such oddity can be forgiven if the q^m let us write all EOTCU in a single compact translatable form! It is based on an N-dimension differential chain relation between Cartesian coordinate (CC) differential dx^j  and GCC dq^m.

                                                                 (12.18)

An N-dimension sum is implied over any index repeated (like m above) to avoid writing “Sigma”-sums. 

             An identical linear relation exists between CC velocity and GCC velocity.

                                                                                                         (12.19)

Total time-derivatives of CC (Cartesian velocity) and GCC (Generalized velocity) are denoted as follows.

                                                                                      

In (12.19) Jacobian J_m^j matrix gives each CCC differential or velocity in terms of GCC or .

                                                                                             (12.20a)

Inverse (so-called) Kajobian K_j^m matrix is defined by a partial derivative that is a flip of the one for J_m^j.

                                                                                      (12.20b)

Product of matrix J_m^j and K_j^m is a j-sum or that by definition of partial derivatives, gives unit matrix.

                                                                 

So a K_j^m matrix gives GCC or in terms of or , respectively, the reverse of (12.18) or (12.19).

                                                                                 

 GCC acceleration or 2^nd time-derivatives are a bit more complicated. We first apply  to velocity (12.19).

                                               

Then a differential chain sum is applied to the Jacobian. Partial derivatives  and  are reversible.

                       

Velocity eq. (12.19) then equates total t-derivative of Jacobian to a partial q^m-derivative of CC velocity.

                                                                                                                                (12.21)

This and (12.20) are two main keys to converting the Newton Second Law (Newt II) to GCC forms that apply to all classical mechanical phenomena due to any number of particles in one, two, or three dimensions. These then serve as mathematical analogies or analogs to modern physics of relativity and quantum theory that describe optical, nuclear, atomic, and molecular phenomena that involve bizarre wavelike behavior in unimaginably countless dimensions. Even LHC physics is glimpsed but that’s still a work in progress. 

 Lagrange’s version of Newt-II (f=Ma)

            Lagrange’s derivation starts with the following multidimensional CC version of Newt-II (f=Ma).

                                                                                                        (12.22)

It is based upon a multidimensional CC version of kinetic energy that generalizes in (12.8).

                           (12.23)

Into the CC expression for differential work  is put the 1^st GCC differential (12.19).

                                                          (12.24)

The dq^m-sum is true term-by-term since dq^m are independent. (Sum still holds if all dq^m are zero but one.)  This gives an expression for each generalized GCC force component F_m defined as follows.

                                                  (12.24a)

                                                                                 (12.24b)

Now for Lagrange’s clever end game.  First setand with  to get:

                       

Then convert  to  by (12.20a) on 1^st term and (12.21) on 2^nd term. Simplify by: 

                      

The result is Lagrange’s GCC force equation using kinetic energy (12.23).

                                                where:           (12.25a)

            But, Lagrange isn’t done yet! If the force is conservative it’s a gradient or in GCC:.

                                                                                 

This gives Lagrange’s GCC potential equation with a new definition for the Lagrangian: L=T-U.

                                    where:         (12.25b)

 Note that the potential function U cannot have any velocity dependence or else the first term of (12.25b) would be wrong. Lagrange’s equations have simple forms that use GCC momentum p_m.

                                                                                                          (12.25c)

             (12.25d)                                                       (12.25e)

The first one (12.25d) is what we did not have when we found in (12.12c) the form . The latter is the CC version of (12.25e) above. Pretty nice EOTCU (Equations of the Classical Universe) here! Let’s try them out.

            Consider an example of Lagrange equations in polar coordinates (q1 = , q2 = ö) defined as follows.

                                                            x =x¹=  cos ö ,       y=x² =  sin ö                                                     

First we find Jacobian and Kajobian matrices. <J> is easy to get by (12.20a). Inverting <J> gives <K>.

               (12.26)

Two kinds of quasi-unit vectors show up. Columns of <J> are covariant vectors and  Rows of <K> are contravariant vectors and . They are plotted and sketched generically in Fig. 12.10.

          

Fig. 12.10  Covariant force vector components in a polar space ( Eñ,Eö ).

 

            Covariant vector  is tangent to the q^m coordinate line of a cell wall.  grows as its cell grows according to the 1^st differential relation (12.18) sketched in Fig. 12.10b and rewritten here in vector notation.

                                                                                            (12.27)

Note grows in Fig. 12.10a. are convenient bases for extensive quantities like distance and velocity.

            Contravariant vector  is normal to the q^m=const. surface or coordinate line of a cell wall.  shrinks as its cell side grows according to gradient relation (12.28) sketched in Fig. 12.10c.

                                                                           (12.28)

are convenient bases for intensive quantities like force and momentum.

            Polar coordinates are orthogonal, but GCC and are at home in more exotic non-orthogonal manifolds and provide mutually orthonormal dual bases with GCC orthogonality relations. (Prove this!)

                                                                                                                (12.30a)

Scalar products • give covariant metric g_mn-tensor matrix and similarly • gives contra-g^mn.

                           (12.30b)                                                                         (12.30c)

GCC version for 1-free-particle Lagrangian involves g_mn factors using (12.27) and (12.30b).

                                                 (12.31)

Polar coordinate metric tensors follow from (12.26). g_mn and g^mn are diagonal (orthogonal) mutual inverses.

                     

The resulting polar-coordinate Lagrangian and its covariant momentum p_m expressions (12.25e) follow.

                                                                               (12.32)

is radial linear momentum. is angular momentum complete with moment of inertia. A potential  turns the Lagrangian into of (12.25b). Momentum t-derivatives (12.25d) follow.

                                  (12.33)

Note how centrifugal force  and Coriolis force are derived so easily by Lagrange GCC. Also, if potential is radial-isotropic (no -dependence) then angular momentum is a conserved constant.

                                                                                                                       (12.35)

Lagrange GCC is elegant and powerful. So is Hamiltonian GCC and, as we’ll see, it’s more conservative!

Hamilton’s version of Newt-II (f=Ma)

            A GCC Hamiltonian H(p,q) uses momenta and coordinates as independent variables rather than generalized velocities and coordinates employed by Lagrangian. The total time derivative of L is

                                                .                                  

Here we leave open the possibility that the Lagrangian may have explicit time dependence and a non-zero partial t-derivative. (Think of a mad scientist dialing up the force field as the experiment progresses.)

            The Lagrange equations (12.25) let us insert momentum and its time derivative into (12.36).

                                                                                                    

A derivative identity lets us collect the two pq terms. Reordering gives a total t-derivative of a GCC version of a form () first encountered in (12.11b). That’s the GCC Hamiltonian H(p,q).

                                                                     (12.36a)

The momentum side (12.12) of Hamilton’s equations follows. So does the other (coordinate) side (12.36c).

                               (12.36b)                                                           (12.36c)

To get (12.36b) we apply to H. (Recall =0.) To get (12.36c) we apply to H. (Recall (12.25d).)

            The Hamiltonian function has an unusual property. Its total time derivative equals its partial time derivative. If H lacks explicit time dependence it is a conserved constant. (Imagine our mad scientist asleep at the dial!)  That constant is energy. To show this, apply metric definition (12.31) of T in L=T-U.

                                                     (12.37a)

                                                                            (12.37b)

                                                                          (12.37c)

So, the Hamiltonian is the sum of kinetic energy T and potential U which is the total energy E=T+U. Equations (3.12.5) amount to the conservation of total energy if L and H are not explicit functions of time.

            Note that we use the covariant metric g_mn for the velocity v-dependent Lagrangian, but the inverse contravariant metric g^mn comes into play for the momentum p-dependent Hamiltonian. Seem like so much formalistic foo-foo? Perhaps. But, just wait until Unit 2 where we develop the relativistic and quantum mechanical versions of this. Then this “formalism” will morph into the most sublimely gorgeous geometry that you have ever imagined!
 

Variational calculus of Lagrangian mechanics

Here is a funny way to derive Lagrange’s equations. It involves something called variational calculus.

            Variational calculus finds extreme (minimum or maximum) values to entire integrals such as

                                                                                                    (12.38)

Here the curve q(t) can vary at each time t. If S(q) was a simple function like S(q)=q2 -4q we would find zeros of its derivative dS/dq =(2q-4)=0 at q=2 and be done. However, here S(q) is a functional, that is, a function  of entire functions q(t) and t-derivative  either of which can be varied arbitrarily at any point between limits t0 and t1 of the dependent time integration variable as shown in Fig. 12.11. (Again, we allow the possibility that  may have explicit t-dependence, too.)

           

            Fig. 12.11 Variation of functional curve or trajectory path from q(t) to

 

            An arbitrary but small variation function is allowed at every point t in the figure along the curve except at the end points t0 and t1. There we demand it not vary at all.

                                                                                                                 (12.39)

The variant changes integral (12.38) according to a first order Taylor series.

                                     (12.40)

Replacing  with  gives a sum of two and then three integrals.

                                  

The third term vanishes according to (12.40). This leaves the following first order variation äS.

                                                                  

If integral S is an extremum its first order variation äS must be zero for all äq(t) even the case where äq(t) is only non-zero at only one tiny t-interval. Thus the äS integrand must be zero everywhere.

                                                                                                  (12.41a)

The result is an Euler-Lagrange equation. It is a 1-dimensional version of GCC Lagrange equation (12.25c)

                                                                                                                       (12-41b)

We may just as well demand extreme (actually minimum) values for multi-dimensional Lagrange integrals.

                                                                                               (12.38)_N-dim

The result is the N-dimensional Lagrange equations (12.25) derived from Newt-II. But, why should Newt-II make the integral of Lagrangian minimum? Something weird underlies classical laws! Henri Poincare recognized early on that a new physics (modern physics) must be hiding down there.

Poincare’s invariant, quantum phase, and action

The Legendre relation (12.11a) becomes Poincare’s invariant differential if has dt cleared.

                                                                                     (12.42a)

It is also the time differential dS of action whose time derivative is rate L of quantum phase.

                                                                          (12.42b)

In Unit 2 we find DeBroglie law and Planck lawthat, if conserved, give a quantum plane wave:

                                                                                     (12.42c)

Time-independent or Hamilton’s reduced action is the spatial integral . Classical trajectories minimize action integrals S and S_H according to Least Action Principles like (12.41).

Huygen's principle: "Proof" of classical axioms and path integrals

            Enveloping curves generated by contact transformations like (12.15) or Fig. 12.7 are closely related to Huygen's principle of wave optics. This also applies to quantum waves of matter. Suppose a hypothetical action function SH(r0 : r) generates the curves SH(r0 : r)=10, 20, and 30 as sketched in Fig. 12.12.

            Now imagine the same generator acts starting from two points r10 and r'10 on the SH(r0 : r)=10 wave front thereby generating two sets of intermediate wave fronts: SH(r10 : r)=10 and SH(r'10 : r)=10 around each of these two points. All points on these curves represent a total accumulation of 20 J.s of action since leaving r0, but only for select points like r20 and r'20 is 20 J. the least action that can be accumulated after leaving r0. All other points have a more direct route from r0 that is cheaper than 20 J.s.

   

      Fig. 12.12 Comparison of paths and wave fronts for discussion of Huygen's principle.

 

            These special points r=r20 and r=r'20  of least action are just the contacting ones that lie on the envelope curve SH(r0 : r)=20. They also lie on optimal (least action) trajectory paths from r0 which have never failed to follow the undeviating "straight-and-narrow" paths determined by Lagrange equations.      What makes these paths appear to follow the classical Lagrange equations? Why do they appear to optimize their action so faithfully? Huygens knew the answer in the 1600's, at least for rays of light. The key word here is "appear" since neither light waves nor matter waves originally have any intention of following a straight and narrow path!

            Quite the contrary, every point on a Huygen's wave front broadcasts a continuum of deviant wave fronts in the form of the intermediate "wavelet" ovals such as SH(r10 : r)=10 and SH(r'10 : r)=10 in Fig. 12.12. But, for each of these non-optimal deviant "rascals" there are thousands more neighboring "rascals" whose actions vary linearly with deviation so that non-extreme action paths end up canceling each other by destructive interference of the varying phases due to deviant actions. No honor amongst "rascals" here!

            Only for those optimal paths of stationary action (and therefore, stationary phase) do the phases add constructively, and it is only for these that quantum wave intensity or classical presence appears to exist most of the time in a classical world of enormous action. All paths are possible to varying degrees and exist in some sense, but only the optimal ones make their presence known and generally do so while obeying quite precisely the classical equations of motion.

            In a sense, this constitutes an evolutionary proof of Newton's "laws" or at least justification of Newton's axioms in the case of high action or the classical limit. The classical world appears to be a result of a continual process of natural selection!

            However, the situation is different for systems with discrete or limited number of paths as in the case of low action or when wavelength is comparable to the size of a system. Then the classical myth is likely to disintegrate like Dracula out of his coffin at dawn! Now matter how dearly we believe in our precisely machined gears and fine particles there comes a time and place where the classical equations part company with new reality that appears with increasingly clever and precise experimental evidence.

            Nevertheless, the classical apparatus is far too well developed to die forever, and it rises to assist the newly appointed quantum paradigm in what is called semi-classical approximation theory. The role of generating action functions S(r0,t0 : r,t) and SH(r0 : r) is taken over in quantum theory by amplitudes, wavefunctions, or matrix elements such as the amplitude 〈r,t| r0,t0〉 of time-evolution and or the transition-overlap amplitude 〈r |  r0 〉. Here, |〈 B |  A 〉|2 is the probability for a state-A to become state-B if forced to make a choice. Bracket 〈 B |  A 〉 is called a probability amplitude; past-to-future is read right-to-left like Hebrew. Probability amplitudes may be approximated by semi-classical relations similar to (12.42c).

               (12.43a)                                            (12.43b)

Restating Huygen's principle with semiclassical amplitudes gives a completeness or closure relation.

                                 (12.44)

Intermediate r'-path sums, as in Fig. 12.12, cancel by phase variation except on the optimal stationary-action path r1←r0. The sum over phase factors from r'-paths is well approximated by the amplitude for the stationary optimal path. Methods for summing over all paths of significant importance (including deviant ones) are called Feynman path integration techniques. This is a difficult chore since “all paths...” are a tangled uncountably infinite mess. Often, the extra effort needed to count them is not needed.

           

           

Bohr quantization

Bohr quantization requires quantum phase in amplitude (12.44) to be an integral multiple õ of 2π after a closed loop integral . The integer õ (õ = 0, 1, 2,...) is a quantum number.

                                            (12.45)

A colorful way to display action and its Bohr quantization is to numerically integrate Hamilton's equations and Lagrangian L and color the trajectory according to the current accumulated value of action.

                                        SH(0 : r) = Sp(0, 0 : r, t ) + Ht =+ Ht .                               (12.46)

The hue should represent the phase angle SH(0 : r)/h modulo 2π as, for example, 0=red, π/4=orange,

π/2=yellow, 3π/4=green, π=cyan (opposite of red), 5π/4=indigo, 3π/2=blue, 7π/4=purple, and 2π=red (full color circle). Interpolating action on a palette of 32 colors is enough precision for low quanta.

            The colored paths display a confused gray mess if phases fail to interfere constructively. But, for select quantizing values of energy, there appear striking patterns of colors when Bohr quantization makes phases interfere constructively. Patterns are outlines of spatial quantum wave amplitudes (12.43b).

                                                                                                (12.43)_repeated

 This color-quantization technique was done on a CRAY-Dicomed film system by Heller and Davis in 1983.

            A quantizing example for a 2-dimensional oscillator using the ColorU(2) program is shown in Fig. 11.13. Viewing this in gray-scale is possible since only two hues actually survive: red, representing a phase of 0, and cyan, representing a phase of π. The example is a standing wave mode in (x,y)-coordinate space, so the only possible wave amplitude is ±1, that is, complimentary hues red and cyan which appear as light and dark gray in a gray scale portrait. Remaining colors pile up on nodal lines with so many phases interfering to destroy the amplitude. A time-dependent action S(0, 0 : r, t ) (12.43a) gives time-dependent moving waves such as the snapshot in Fig. 11.14 of a quantum fountain discussed after Fig. 12.5. Wave animation is done by shift the computer color wheel by the time-dependent phase angle in (12.46).  

                         (12.46)

A moving wave has a quantum phase velocity found by setting S=const. or dS(0,0:r,t)=0=p•dr-Hdt .

                                                                                                                       (12.47)

This is quite the opposite of classical particle velocity which is quantum group velocity.

                                                                                                                  (12.48)

 By making two of the entries in the phase-color palette to be black-and-white it is possible to display a wave front line which will march in step with the other hues in the palette.

                       

Fig. 12.13 Phase-color 2-dimensional harmonic oscillator paths showing (2,2) quantum wave function.

           

Fig. 12.14 Phase-color trajectory  paths showing quantum wave fronts of moving wave.

 

            A sequence of quantum wavefronts underlying the quantum fountain are drawn in Fig. 12.15 for three different values of reduced action S_H. This is equivalent to taking snapshots at different times. As classical momentum approaches zero at the top of Fig. 12.15b, the S wave phase speed diverges to infinity. Then two "cat ears" are created and race out along the top of the classical envelope in Fig. 12.15c and then slow down as the classical momentum p again picks up. High p in Fig. 12.15 means high gradient ∇SH = p so the SH  contours are closer together. S fronts move from one SH =n2π contour to the next SH =(n+1)2π contour at frequency ù=H/h so large p means slow going. Note that the lower regions of each contour in Fig. 12.15 are slower than upper regions; quite the opposite of the classical particle “ball” in Fig. 12.5. The two "cat ears" move out and down rapidly until, like Lewis Carroll's Cheshire cat, nothing remains but its smile!

           

Fig. 12.15 Constant SH contours for iso-energetic trajectory family are normal to trajectory paths.

 

Dynamics of phase and group velocity are the key to relativity and quantum theory as Unit 2 will show.

 

 

 The volcanoes of Io and NIST atomic fountain

1.12.1. A fountain spraying water in many directions at equal speed appears to form a parabola of revolution whose cross-section is plotted above. You see this also in photos of volcanoes on Jupter’s moon Io. Let’s model this with a Bang! of equal-initial-speed v₀ particles ejected from origin in uniform gravity g vacuum and develop geometry of parabolic trajectories and their envelope. Key questions: “What curve or “blast wave” do the particles form at each moment of time?”  and “Is the envelope, in fact, parabolic?”

(a) First describe how observers see the trajectories behave if they are riding in a well-shielded free-falling elevator frame-(x´,y´) that passes the origin point at the moment of ejection.

(b) Meanwhile, in the lab (x,y)-frame, describe the parabolic fragment trajectories as a function of  the initial elevation angle á for each fragment. Give equations for the focal point of each parabola and describe what curve these foci form (the focus-locus). (See if you can do it without peeking at Ch. 12.)

(c) Construct examples of parabolas with their focus-locus and find a relation between the focal point and the contact point for each parabolic path with the envelope. Is the envelope parabolic? and, if so, where is its focus? Construct drawings of á =30° and á =45° paths and “blast wave” at the moment each path contacts the envelope.

 

Problem for a volcanolgist (or baseball outfielder)

1.12.2. Suppose you are on the x-axis some distance from the origin where a baseball (or volcano rock) is thrown up and toward you so it appears to move straight up the y-axis like an imaginary elevator that is at the line-of-sight projection of baseball (or rock) onto y-axis. Elevator motion indicates if you are in position to catch the ball (or be clobbered by the rock) or whether the object will fly over or fall short. Describe apparent elevator velocity and/or acceleration that distinguishes the three cases, particularly the middle one.

 

1.12.3. Suppose a neutron starlet enters a pocket of U²³⁵ at position r(0) and goes Bang! The U²³⁵ detonates and blasts off pieces of starlet that each fly away with the same initial speed |v(0)|=1 (we assume) but various velocities v(0) =(0,1), (), (0,1), (), etc. (Units use the usual geometric ù=1 scaling.)

The short answer problems concern the resulting orbits of equal mass starlet pieces inside the Earth.

Do the starlet blast orbits conserve values for any physical quantities such as (Yes or No) initial

speed |v|?__, momentum |p|?__, angular momentum l?__, KE?__, PE?__, total Energy E?__ .

Do the starlet blast orbits conserve values for some geometric quantities such as (Yes or No)

a?__, b?__ , r?__ , H?__, ö?__, ϑ?__, á?__. (See sketch of general ellipse orbit above.)

For whichever it is possible, give |v|, |p|, l, KE, PE, or E in terms of a, b, H, ϑ or á. Note if any correspond to particular geometrical length or area that characterizes an orbit.

Do the starlet blast orbits share equal values for any physical quantities such as (Yes or No)

Average momentum p?__, angular momentum l?__, average KE?__, average PE?__, total E?__ .

Do the starlet blast orbits share equal values for some geometric quantities such as (Yes or No)

a?__, b?__ , H?__, ö?__, ϑ?__, á?__. (See sketch of general ellipse orbit above.)

Geometric orbit and envelope constructions

Using the graph on the attached page, do the following for initial position vector r(0) =(1,0):

(a) Construct the orbit for initial v(0) =(0,1) in part (a) of the graph.

(b) Construct the orbit for initial v(0) =(1,0) in part (b) of the graph.

(c) Construct the orbit for initial v(0) =() in part (c) of the graph.

(d) Construct the orbit for initial v(0) =() in part (c) of the graph.

(e) There is a “blast wavefront” or locus of starlet pieces that expands with each instant of time. Using whatever means you like, plot points of this curve for t/ô_period= 1/12, 2/12, 3/12, and 4/12.

(f) There is a constant contacting curve that envelops all orbits with r(0)=(1,0) and |v(0)|=1.

Construct this enveloping boundary for |v(0)|=1. How does the envelope vary with initial speed |v(0)| ?

(g) See if you can deduce a relation between the focal point(s) and the contact point(s) for each elliptic path with the envelope. Is the envelope also elliptic? If so, where are its foci?