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Dual Laws, Central Forces and Tessellations

We have already discussed dual laws: undoubtedly the most famous pair is Newton's law of attraction and Hooke's harmonic oscillator. Their potentials are power laws and we can write the pair as {-2,1}, the first term is the square in the harmonic oscillator potential, and the second is the inverse in the Newtonian potential. Other integer power laws are known, specifically 3, 4, and 6, for the inverse quadratic, quartic and inverse sixth. Moreover, 3 pairs with 6 and 4 pairs with itself: {3,6}, {6,3} and {4,4}. The pairs {3,6} and {6,3} are not the same. If a is the first integer, and A the second then they satisfy the condition

(a+2)(A+2)=4.

But, where on earth does this come from?


Tristan Needham writes that the integer power laws, apart from {-2,1}, "are rather mysterious. Their physical significance is unknown to me, and that itself is strange, for the music of mathematic is seldom played without an accompanying echo being heard in Nature." He's right, as we shall now see.


A regular tessellation, like floor tiling, is a covering of the plane by polygonal figures so that the same number of polygons meet at each and every vertex. In the Euclidean plane, there are only three which satisfy the condition

1/n + 1/k= 1/2

for any pair {n,k}. The beautiful realization that the above condition on the integer exponents of dual power laws is identical to this condition. The three pairs are {3,6} for 6 equilateral triangles having sides 3 meet at each vertex; {4,4} in which squares having 4 sides meet 4 other squares at each vertex, and {6,3} in which hexagons having 6 sides meet 3 other hexagons at each vertex. The pair {-2,1} also satisfies the above law, but now it is unclear what type of tessellation it can be associated with.


Goldstein in his famed Classical Mechanics claims that only the power law potentials -3,-4, and -6 have polar equations for their orbits expressible in terms of elliptic functions. Whereas elliptic functions have two periods, and so can be tessellated in the plane, the ellipse has a single period and thus does not fit into the above classification, although the pair of exponents satisfy: -1/2+1=1/2.


For an {n,k} tessellation there are k polygons at each vertex. The angle at each vertex is 2 pi/k. And since the polygon has n equal angles, the sum is 2 pi n/k. For a triangle, n=3 and k=6 so that the angle sum is pi. This means that we are in the Euclidean plane, as we knew when we started. Angle sums less than pi lie in the hyperbolic plane while those more than pi lie in the elliptic plane. For a square, then, the angle sum is 2 pi, and for a hexagon it is 6 pi. These polygons can be rolled up and glued together to form a torus of genus 1 (one hole).


The power law that tessellates as a triangle give rise to the orbit of a limacon, or cardioid. That which tessellates as a square corresponds to a circular orbit through the origin, and the hexagon has a orbit that is described by a lemniscate of Bernoulli, or, in more general terms a Cassini oval.


Now, things get interesting. Instead of concentrating on the orbit in configuration space, we consider velocity space. The hodograph of an orbit described by Newton's law is a circle, and Hamilton, way back in 1847 designated the law as the "Law of a Circular Hodograph".

Whereas the arc length of an ellipse cannot be expressed as an elementary function, but rather as an elliptic function, in velocity space things become much simpler.


However, things can become very confusing if we are not careful. An ellipse and an elliptic curve are two different things. Elliptic curves pertain to elliptic functions of the Weierstrass type, being of the form y^2=x^3+ax^2+bx+c. Yet, by inversion in a circle, the mean eccentricity of an ellipse can be related to the Riemann metric of elliptic geometry (c.f. Seeing Gravity, Sec 4.4 and references therein). So what is Euclidean in configuration space, becomes non-Euclidean in velocity space!


The transformation from a harmonic oscillator to Newton's law can be carried out in the following way. Following Levi-Civita we consider two times, a fictitious time s and a real time t. They are related by

ds/dt=1/q=u^2/q^2.

for the harmonic oscillator Newton ellipse pair where q is the coordinate and p is momentum.

There is another set of coordinates u, and v, and the second equality expresses Kepler's equal area law. The second equality gives us the transformation q=u^2, both q and u are real coordinates.

Consider the hamiltonian (we will not worry about coefficients)

H=v^2+u^2,

where

v=du/ds=1/2 sqrt(q)*dq/dt*dt/ds=1/2 sqrt(q)*p,


where p=dq/dt. Introducing this into the above hamiltonian we get

H'=q{p^2/4+1} - 1,

where we added on a constant term. This is related to a new Hamiltonian

H"=ds/dt*H' - 1

which is easy to see is none other than

H"=p^2/4-1/q.

This gives the pair {-2,1} which satisfies 1/n+1/k=1/2.


We can generalize Levi-Civita's fictitious time relation to

ds/dt=V(q)

where V(q) is any power law, 1/q^n. But only the powers n=-3,-4, and -6 will turn out to give elliptic integrals and be of physical interest.


The metric corresponding to the above can easily be derived.

ds=dt/q and dp=dt/q^2

so

ds=q*dp.

Introducing the conservation of energy, 1/q=p^2/4-H" gives

ds=4dp/(p^2-4H"),

which can be brought into Riemann's form by inversion w*p=1 so that

ds=4dw/(1+Kw^2),

where K is related to the radius of curvature, which is positive for orbits of negative total energy H"<0.


What new metrics expect us when we generalized to elliptic functions is a matter for the next blog.



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