Invariants for time-dependent potentials: Use of self-similar techniques
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.
time-dependent nonlinear Hamiltonian systems
scientific article; zbMATH DE number 4072470
scientific article