A Sort of Relation among a Dissipative Mechanical System and Conservative Ones

TL;DR

This paper establishes a theoretical link between dissipative and conservative mechanical systems, showing that each dissipative system corresponds to a conservative one sharing a unique phase curve, enabling the application of Hamiltonian and Lagrangian mechanics without redefining canonical momentum.

Contribution

It introduces a novel proposition that any nonconservative system can be associated with a conservative system sharing a phase curve, facilitating the use of classical mechanics methods for dissipative systems.

Findings

Dissipative systems can be reformulated as conservative systems along a phase curve.

The Hamiltonian of the conservative system relates to the total energy of the dissipative system.

This approach allows applying Hamiltonian and Lagrangian mechanics without redefining canonical momentum.

Abstract

In this paper we proposed a proposition: for any nonconservative classical mechanical system and any initial condition, there exists a conservative one; the two systems share one and only one common phase curve; the Hamiltonian of the conservative system is the sum of the total energy of the nonconservative system on the aforementioned phase curve and a constant depending on the initial condition. Hence, this approach entails substituting an infinite number of conservative systems for a dissipative mechanical system corresponding to varied initial conditions. One key way we use to demonstrate these viewpoints is that by the Newton-Laplace principle the nonconservative force can be reasonably assumed to be equal to a function of a component of generalized coordinates $q_i$ along a phase curve, such that a nonconservative mechanical system can be reformulated as countless conservative systems. Utilizing the proposition, one can apply the method of Hamiltonian mechanics or Lagrangian mechanics to dissipative mechanical system. The advantage of this approach is that there is no need to change the definition of canonical momentum and the motion is identical to that of the original system.