Classical Hamiltonian Dynamics

Thomas Dittrich

Massimo Tessarotto, M. Mond, Davide Batic

Eugene Oks, T. Uzer

Dirac's generalized Hamiltonian dynamics (GHD) is a purely classical formalism to describe mechanical systems with constraints, which it incorporates into the Hamiltonian. We show how quantization information can be included by choosing the integrals of motion as constraints, and demonstrate the construction of this novel classical Hamiltonian by applying GHD to electron motion in a hydrogenic atom.

Leon Bess

Joan C. Micó

This paper presents a quantum formulation for classical abstract dynamical systems (ADS), defined by coupled sets of first-order differential equations. They are referred to as “abstract” because their dynamical variables can be of different interrelated natures, not necessarily corresponding to physics, such as populations, socioeconomic variables, behavioral variables, etc. A classical linear Hamiltonian can be derived for ADS by using Dirac’s dynamics for singular Hamiltonian systems. Also, a corresponding first-order Schrödinger equation (which involves the existence of a system Planck constant particular of each system) can be derived from this Hamiltonian. However, Madelung’s reinterpretation of quantum mechanics, followed by the Bohm and Hiley work, produces no further information about the mathematical formulation of ADS. However, a classical quadratic Hamiltonian can also be derived for ADS, as well as a corresponding second-order Schrödinger equation. In this case, the Madelung reinterpretation of quantum mechanics provides a quantum Hamiltonian that does provide the quantum formulation for ADS, which provides new quantum variables interrelated dynamically with the classical variables. An application case is presented: the one-dimensional autonomous system given by the logistic dynamics. The differences between the classical and the quantum trajectories are highlighted in the context of this application case.