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Canonical transformations and exact invariants for time-dependent Hamiltonian systems. (English) Zbl 1024.70009
The aim is to present a method in order to obtain exact invariants for non-relativistic Hamiltonian systems with general time-dependent potentials. The main tool is an infinitesimal canonical transformation of the extended phase space. For a specific class of Hamiltonian systems, the exact invariant is obtained equivalently performing in the extended phase space a finite canonical transformation of the initially time-dependent Hamiltonian to a time-independent one. It is furthermore shown that the invariant can be expressed as an integral of energy balance equation. More concretely, three examples of time-dependent damped and undamped oscilators are detailed.
MSC:
70H15Canonical and symplectic transformations in particle mechanics
70H05Hamilton’s equations
37J15Symmetries, invariants, invariant manifolds, momentum maps, reduction