Invariants for time-dependent Hamiltonian systems

Abstract

An exact invariant is derived for $n$-degree-of-freedom Hamiltonian systems with general time-dependent potentials. The invariant is worked out in two equivalent ways. In the first approach, we define a special {\it Ansatz\/} for the invariant and determine its time-dependent coefficients. In the second approach, we perform a two-step canonical transformation of the initially time-dependent Hamiltonian to a time-independent one. The invariant is found to contain a function of time $f_2(t)$, defined as a solution of a linear third-order differential equation whose coefficients depend in general on the explicitly known configuration space trajectory that follows from the system's time evolution. It is shown that the invariant can be interpreted as the time integral of an energy balance equation. Our result is applied to a one-dimensional, time-dependent, damped non-linear oscillator, and to a three-dimensional system of Coulomb-interacting particles that are confined in a time-dependent quadratic external potential. We finally show that our results can be used to assess the accuracy of numerical simulations of time-dependent Hamiltonian systems.

arXiv comment9 pages, 6 figures

Cited by in corpus (8)

• Hamiltonian dynamics on the symplectic extended phase space for autonomous and non-autonomous systems
• Noether's theorem and Lie symmetries for time-dependent Hamilton-Lagrange systems
• Reduced phase space optics for general relativity: Symplectic ray bundle transfer
• Dynamical invariants and diffusion of merger substructures in time-dependent gravitational potentials
• Integrability in time-dependent systems with one degree of freedom
• Dirac's Method for the Two-Dimensional Damped Harmonic Oscillator in the Extended Phase Space
• Critical Collapse and Solitons in Classical Conformal Field Theory
• Evolution of the wave-function's shape in a time-dependent harmonic potential