On Tautochronic Motions

Abstract

Linear motion of a point particle influenced by two forces varying according to power laws with arbitrary exponents is considered. Exponents are found for which the governing equation is nonlinear and the oscillation period is independent of the initial data (tautochronic motion). The equations are brought to Hamiltonian form, and the Hamiltonian normal form method is used to prove that there exist only two variants of tautochronic motion, namely, when the exponents are equal to 1 and –3 (variant 1) and when the exponents are equal to 0 and –1/2 (variant 2). For the other power laws, the motion of the point particle is not tautochronic. The Hamiltonian normal form of tautochronic motion is the Hamiltonian of a linear oscillator. The canonical transformation reducing the original Hamiltonian to normal form is expressed in terms of elementary functions. Hamiltonians of tautochronic motions can be used to test computer codes for calculating Hamiltonian normal forms.