Hamiltonian dynamics on the symplectic extended phase space for autonomous and non-autonomous systems

Abstract

We will 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 will be 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 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, will be shown to represent particular canonical transformations in the symplectic extended phase space. Furthermore, the generalized canonical transformation approach allows to directly map explicitly time-dependent Hamiltonians into time-independent ones. An "extended" generating function that defines transformations of this kind will be presented for the time-dependent damped harmonic oscillator and for a general class of explicitly time-dependent potentials. In the appendix, we will reestablish the proper form of the extended Hamiltonian H 1 by means of a Legendre transformation of the extended Lagrangian L 1 .

One-, Two-, and Three-Dimensional Hopping Dynamics

“…The actual calculation is done in the extended phase space [50]. The time scaling function between the extended phase space and the original phase space allows relative entropy estimation between different states obtained from the same initial configuration [7,23].…”

Unified formalism for nonautonomous mechanical systems

“…(See [5,7,12,14,16] for more details). In the jet bundle description of non-autonomous dynamical systems, the configuration bundle is π : E / / R, where E is a (n + 1)-dimensional differentiable manifold endowed with local coordinates (t, q i ), and R has t as a global coordinate.…”

“…One basic idea towards a geometric discretization [8,10,13] of this type of equations is first to introduce an spatial truncation, that reduce the PDE (16) to a system of ODEs preserving many of its geometrical properties. Hence, we replace the x-derivative in the Lagrangian by a simple difference (for simplicity, we will work with a uniform grid of N + 1 points, h = K/N ) as follows:…”

Entropy and Heat Capacity Calculations by Thermodynamic Approach

“…The Nosé-Poincaré thermostat, which is the Hamiltonian version of Nosé-Hoover thermostat, utilizes the Poincaré time transformation H = ν(H − H) where ν is the time scaling factor and H is the value of H regarded as a pure function of time [9]. This permits the description of Hamiltonian dynamics in the symplectic extended phase space.…”