Invariants for time-dependent potentials: Use of self-similar techniques
type: Item
instance of: scholarly article
title: Invariants for time-dependent potentials: Use of self-similar techniques
zbMATH Open document ID: 0656.70014
DOI: 10.1016/0167-2789(87)90122-9
author: S. Bouquet
author: Marc R. Feix
author: H. Ralph Lewis
published in: Physica D
publication date: 1987-01-01T00:00:00Z
review text: We seek a generalization of the derivation by the third author and \textit{P. G. L. Leach} [J. Math. Phys. 23, 2371-2374 (1982; Zbl 0515.70018)] for invariants quadratic in p. Since the general equations for invariants cubic and quadratic in p cannot be solved, we specialize to self-similar solutions of the equations. Consequently, we can recover only a part of their solutions for quadratic invariants, but new pairs of potentials and invariants are obtained, where the leading term in the invariants is \(p^ 3\) or \(p^ 4\). More precisely, all the potentials are of the form \(V(q,t)=At^{2\lambda -2}(q/t^{\lambda})^{\nu}+B(\lambda)q^ 2/t^ 2\), where \(\nu\) is prescribed and A is arbitrary. Different functional forms for B lead to different families of solutions where, for each family, the parameter \(\lambda\) is allowed to go from -\(\infty\) to \(+\infty\). Finally, a generalization to potentials with three terms is given.
Mathematics Subject Classification ID: 70H05
Mathematics Subject Classification ID: 37J99
zbMATH DE Number: 4072470
zbMATH Keywords: time-dependent nonlinear Hamiltonian systems
zbMATH Keywords: one-dimensional motion
zbMATH Keywords: self-similar solutions
MaRDI profile type: Publication
full work available at URL: 0167-2789(87)90122-9
OpenAlex ID: W2025612892
cites work: First integrals for one-dimensional particle motion in a non-linear, time-dependent potential field
cites work: A direct construction of first integrals for certain non-linear dynamical systems
cites work: A direct approach to finding exact invariants for one-dimensional time-dependent classical Hamiltonians
cites work: Nonlinear time-dependent anharmonic oscillator: Asymptotic behavior and connected invariants
cites work: Invariants for dissipative nonlinear systems by using rescaling
cites work: Similarity methods for differential equations
cites work: Exact time-dependent solutions of the Vlasov–Poisson equations
cites work: Exact, time-dependent solutions of the one-dimensional Vlasov–Maxwell equations
description: scientific article; zbMATH DE number 4072470