Isoperiodic Potentials via Series Expansion

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In a one-dimensional oscillation obeying conservation of energy, the potential function determines the period of motion as a function of dimensionless energy . However, the period function only uniquely determines the potential in the case of parity symmetry, where . In all other cases, it is possible to construct an uncountable infinity of potentials with the same period . Regardless of symmetry, we call any two potentials isoperiodic if they have the same period function [1]. Careful examination of inverse functions leads to a precise definition of isoperiodic potentials using power series expansion [2]. As in [3], we use a phase-space technique to write as a function of the potential expansion coefficients around a stable minima. The general form of leads to a set of linear constraints between the expansion coefficients of isoperiodic potentials (see Details).


The examples here explore energy-dependent potentials between Morse and Pöschl–Teller, and energy-independent potentials including the familiar quadratic . The validity and convergence of intermediate potentials can be examined again by numerical time evolution along the potential surface.


Contributed by: Brad Klee (April 2017)
Open content licensed under CC BY-NC-SA



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.

[3] B. Klee, "Plane Pendulum and Beyond by Phase Space Geometry."

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

[5] B. Klee, "A Period Function for Anharmonic Oscillations" from Wolfram Community—A Wolfram Web Resource. (Apr 4, 2017)

[6] B. Klee, "Plotting the Contours of Deformed Hyperspheres" from Wolfram Community—A Wolfram Web Resource. (Apr 4, 2017)

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