Construction of exact invariants for classical dynamical systems in three dimensions (Q1290562)

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).