Conformally mapping 2-D SHM of Hooke to inverse-squared central force law Coulomb

We can solve for the motion of a particle in an arbitrary central force and if the force is attractive we will get orbits that circulate and precess as in picture to the right.  Of the central forces, for which there is  a Poisson potential A,  F=−dA/dr, the inverse square force laws of Newton or Coulomb,

                                             F=−k/r²

have the special property that they do not give rise to precessing orbits. The reason for this is the presence of the Runge–Lenz vector: an extra conserved quantity besides energy and angular momentum, peculiar to the inverse square force law. Such closed orbits can be mapped to the motion of a 2-d harmonic oscillator, whose force is described by Hooke's law:


                                          F=−kr

Newton also showed that the law of areas, also known as Kepler’s Second Law, must hold in any central field. This states that the line connecting an object to the centre must sweep out equal areas in equal times. baez/gravitational

Even better, in both cases the motion is in an ellipse! 

To the left is a one-dimensional oscillator with its phase space diagram that motivates the use of the complex plane geometry we will use to describe its motion. We will consider it to have freedom to move in two dimensions.

A particle that is free to move in two dimensions as per picture bottom right, subject to an attractive force that’s a constant multiple of its distance from a centre of attraction is a two-dimensional harmonic oscillator. If we choose coordinates whose origin lies at the centre of attraction, the particle’s acceleration according to will take the form:

d²r(t)/dt² = –ω² r(t)

where ω here is the angular speed and is related to the force constant and the particle’s mass. In Cartesian coordinates, the particle’s x and y coordinates will both oscillate with the frequency ω, while being free to take on arbitrary amplitudes and phases:

x(t) = C sin(ω t + φ)
y(t) = D sin(ω t + ψ)

which by a suitable rotation of the coordinates and choice of origin for t, can be put into the form (with a ≥ b)

x(t) = a cos(ωt)
y(t) = b sin(ωt),

thus tracing out an ellipse with semi-axes a and b, centred on the origin, with the major axis lying on the x-axis.

To the left are various combinations of phase differences for the two SHM oscillators.

Conformal Mapping, w(z)= z²

The motion of a point in the complex plane in the following is to be given by w (not to be confused with ω) parametrised by t. 

Using geometric analysis accessible to A level FP2 students, such as the complex conformal map transformation z →w(z)= z² acting on circles and lines

we see the function, w(z) takes circles of radius R centered at the origin to circles of radius R²  at the origin, and lines through the origin to rays from the origin. Bohlin and Sundman use precisely this map to associate trajectories of Hooke’s Law w'' = −Cw to trajectories of Newton’s Law,

 d²z/dt² = −C* z/|z|³ for C*= C(w(z),C). 

The map z → z²  thus proves that orbits of planets under Newton’s law of gravitation are conic sections, or similarly stated: a point following the trajectory z(τ(t)) = [w(t)]² where dτ/dt = |w|² moves according to Newton’s law of gravitation. Bohlin and Sundman.

Now we know that the orbital motion is an ellipse not a circle, so lets look at its deformation under the map in a little more detail. An ellipse centred on the origin in the complex plane, with its major axis lying on the x-axis. when “squared” will have every point move according to:

(x + y i)² = (x² – y²) + 2 x y i

This geometry of the conformal map z → z²

is such that an ellipse with semi-axes a and b, with any point 

(x, y) = (a cos φ, b sin φ)

and φ a parameter along the curve (not the polar angle of this complex number). Substituting for the coordinates of our squared points as X= x² – y² and Y = 2xy so

(X, Y) = (a² cos² φ – b² sin² φ, 2 a b sin φ cos φ)
    = (½(a² – b²) + ½(a² + b²) cos 2φ, a b sin 2φ)

we see an ellipse centred at (½(a² – b²), 0), with semi-axes of ½(a² + b²) along the X-axis and ab along the Y-axis.

By symmetry the new foci lie at (½(a² – b²) ± c, 0), for some quantity c. The sum of the distances of any point on the ellipse from the two foci must be equal to the full length of the major axis: a² + b². Since the vertical semi-axis is ab, the point on the top of the ellipse is (½(a² – b²), ab), and its distance from each focus is √(c² + a² b²). So we have:

c² + a² b² = [½(a² + b²)]²
c = ½(a² – b²)

That means the two foci are (0,0) and (a² – b², 0). One focus is the origin

Since squaring a complex number leaves angles between curves unchanged the angle between the original ellipse and a circle of radius r at the point where they intersect will be exactly the same as the angle between the new ellipse and a circle of radius r².

Note also that  the transformation z → z + 1/z

 (z + 1/z)² = z² + (1/z)² + 2 so that
 (w(z))² = w(z²) + 2. 

The squaring map here thus takes a line parallel to the real axis z = x₀+ iy to a parabola in the complex plane with equation 
x = x²₀ − (y/2x₀)², as shown. 

Any other line in the complex plane can now be obtained from a line parallel to the x-axis by a rotation. https://www.math.uh.edu/~josic/content/05-publications/alternative.pdf