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Journal of Mathematical Physics

A system theoretic representation of mechanical systems. I. Decomposition of a mechanical system into a hierarchy of orthogonal stationary linear dynamical systems

Dipak K. Saraswati, Hans Primas

Published Dec 1, 1978• 1 citation
Publisher

Abstract

A system theoretic representation of mechanical systems is proposed. It is shown that mechanical systems, classical or quantal, described by the Liouville equation, can be realized as a stationary linear input–output system. This state space representation of mechanical systems is then extended to decompose a mechanical system into a hierarchy of orthogonal stationary linear dynamical systems.

DOI
10.1063/1.523622

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A system theoretic representation of mechanical systems is proposed. It is shown that mechanical systems, classical or quantal, described by the Liouville equation, can be realized as a stationary linear input–output system. This state space representation of mechanical systems is then extended to decompose a mechanical system into a hierarchy of orthogonal stationary linear dynamical systems.
In a previous paper by the authors it was shown that mechanical systems, classical or quantal, can be realized as stationary linear dynamical systems. In this state space representation of mechanical systems, the input space, the output space, and the state space are Hilbert spaces and to make a connection with the language of a mathematical system theory, in which the corresponding spaces are topological vector spaces, it is shown here that the input, the output, and the state of the dynamical systems representing mechanical systems can be interpreted as second order stochastic processes. An orthogonal decomposition of mechanical motion into a deterministic and a purely nondeterministic motion, suggested by the stochastic interpretation, is discussed and it is pointed out that by iterating the decomposition procedure one can obtain a simple model which approximates the true mechanical motion.
Using Liouville space and superoperator formalism we consider pure stationary states of open and dissipative quantum systems. We discuss stationary states of open quantum systems, which coincide with stationary states of closed quantum systems. Open quantum systems with pure stationary states of linear oscillator are suggested. We consider stationary states for the Lindblad equation. We discuss bifurcations of pure stationary states for open quantum systems which are quantum analogs of classical dynamical bifurcations.
In the present paper we propose a method of quantization of mechanical systems that is applicable, inter alia, to systems acted on by nonconservative forces, such as dissipative systems. First, the correspondence between classical and quantum mechanics is considered and we make some important observations that can be used in defining the rules of quantization. Next, the quantization rules are presented and we give a few examples of the application of these rules to mechanical systems that are difficult to quantize within any other systemetic approach. Finally, it is shown that for any quantum-mechanical system there are stationary solutions of the evolution equations, and that the existence of these solutions is directly related to the proposed quantization rules.
The theoretical framework of the mixed quantum-classical description given by Burghardt and Parlant [J. Chem. Phys. 120, 3055 (2004)] is detailed. A representation in terms of partial hydrodynamic moments is developed, the dynamics of which is determined by a hierarchy of equations derived from the quantum Liouville equation. Exact equations of motion are obtained, whose quantum-classical approximants are associated with a fluid-dynamical trajectory representation which couples classical variables to quantum hydrodynamic variables. The latter evolve under a generalized hydrodynamic force which also depends upon the classical phase-space variables. The hydrodynamic moment description is shown to be closely connected to mixed quantum-classical phase-space methods.
Deterministic dynamical models are discussed which can be described in quantum mechanical terms. In particular, a local quantum field theory is presented which is a supersymmetric classical model. -- The Hilbert space approach of Koopman and von Neumann is used to study the evolution of an ensemble of such classical systems. With the help of the supersymmetry algebra, the corresponding Liouville operator can be decomposed into two contributions, with positive and negative spectrum, respectively. The unstable negative part is eliminated by a constraint on physical states, which is invariant under the Hamiltonian flow. In this way, choosing suitable phase space coordinates, the classical Liouville equation becomes a functional Schroedinger equation of a genuine quantum field theory. Quantization here is intimately related to the constraint, which selects the part of Hilbert space where the Hamilton operator is positive. This is interpreted as dynamical symmetry breaking in an extended model, introducing a mass scale which discriminates classical dynamics beneath from emergent quantum mechanical behaviour.