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Hamiltonian dynamics on the symplectic extended phase space for autonomous and non-autonomous systems. (English) Zbl 1083.70021
Summary: We present a consistent description of Hamiltonian dynamics on the ’symplectic extended phase space’ that is analogous to that of a time-independent Hamiltonian system on the conventional symplectic phase space. The extended Hamiltonian H 1 and the pertaining extended symplectic structure that establish the proper canonical extension of a conventional Hamiltonian H are derived from a generalized formulation of Hamilton’s variational principle. The extended canonical transformation theory then naturally permits transformations that also map the time scales of the original and destination system, while preserving the extended Hamiltonian H 1 , and hence the form of the canonical equations derived from H 1 . The Lorentz transformation, as well as time scaling transformations in celestial mechanics, are shown to represent particular canonical transformations in the symplectic extended phase space. Furthermore, the generalized canonical transformation approach allows us to directly map explicitly time-dependent Hamiltonians into time-independent ones. An ’extended’ generating function that defines transformations of this kind is presented for the time-dependent damped harmonic oscillator and for a general class of explicitly time-dependent potentials. In the appendix, we re-establish the proper form of the extended Hamiltonian H 1 by means of a Legendre transformation of the extended Lagrangian L 1 .
MSC:
70H05Hamilton’s equations
37N05Dynamical systems in classical and celestial mechanics
70H15Canonical and symplectic transformations in particle mechanics