Construction of exact invariants for classical dynamical systems in three dimensions
type: Item
instance of: scholarly article
title: Construction of exact invariants for classical dynamical systems in three dimensions
zbMATH Open document ID: 0937.70012
author: K. Appert
published in: Annals of Physics
publication date: 2000-05-25T00:00:00Z
review text: The authors calculate invariants (first integrals) \(I\) of time-dependent dynamical systems with Hamiltonians of the form \(H= \sum h_i(t)\Gamma_i(x_1,x_2,x_3, p_1,p_2,p_3)\), assuming that the family of phase of functions \(\Gamma_i\) is closed with respect to Poisson brackets (i.e., \([\Gamma_i,\Gamma_j]= \sum C^k_{ij}\Gamma_k\) with constant coefficients \(C^k_{ij}\)). Then the assumption \(I= \sum\lambda_i(t)\Gamma_i\) leads to the equations \(d\lambda_k/dt= \sum C^k_{ij}h_i\lambda_j\) which can be solved. Several examples are thoroughly discussed, e.g. \(H={1\over 2}\sum p^2_i+ \sum\alpha_i(t) x^2_i+ \beta(t)\Phi(x_1,x_2,x_3)\) for certain special classes of \(\Phi\) (Ermakov dynamical systems).
Mathematics Subject Classification ID: 70H05
Mathematics Subject Classification ID: 34A26
zbMATH DE Number: 1294618
zbMATH Keywords: first integrals
zbMATH Keywords: time-dependent Hamiltonians
zbMATH Keywords: Ermakov dynamical systems
zbMATH Keywords: invariants
zbMATH Keywords: Poisson brackets
reviewed by: Jan Chrastina
MaRDI profile type: Publication
full work available at URL: aphy.1997.5713
OpenAlex ID: W2042776007
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DOI: 10.1006/APHY.1997.5713
description: scientific article; zbMATH DE number 1294618