For details of published calculations, see [1, 2]. Our approach follows [3]. Starting with a Hamiltonian

,

we transform to the polar coordinates of phase space

,

where

. The preceding algorithm approximately solves this implicit equation by series inversion, producing a truncated sum

.

The corresponding approximate period is then calculated using

.

Analyzing the general period function

first reported in [4], we prove that every power of the Hamiltonian energy

attaches to a function of the potential expansion coefficients

with a pair

that does not occur in the coefficient of any

with

. This fact allows order-by-order construction of isoperiodic potentials as series expansions around a stable minima.

The isoperiodic constraint between two distinct potentials with expansion coefficients

and

is

.

Fixing the

values and applying the isoperiodic constraint to the

yet leaves one continuous degree of freedom in every coefficient

. These continuous degrees of freedom are controlled by sliders in this Demonstration, which directly enables you to calculate a range of isoperiodic potentials. Methods used here have also contributed to award-winning posts on Wolfram Community [5, 6].

By direct evaluation of the period function for the expansion coefficients of the Morse potential and the Pöschl–Teller potential, it is possible to prove approximate isoperiodicity order-by-order, as in the commented code at the end of the initialization section. Comparing coefficients, we expect direct evaluation of both period integrals to yield

.

[1] M. Asorey, J. F. Cariñena, G. Marmo and A. Perelomov, "Isoperiodic Classical Systems and Their Quantum Counterparts,"

*Annals of Physics*,

**322**(6), 2007 pp. 1444–1465.

doi:10.1016/j.aop.2006.07.003.

[2] E. T. Osypowski and M. G. Olsson, "Isynchronous Motion in Classical Mechanics,"

*American Journal of Physics*,

**55**(8), 1987 pp. 720–725.

doi:10.1119/1.15063.

[4] The On-Line Encyclopedia of Integer Sequences. (Apr 4, 2017)

oeis.org/A276816.