Classical Mechanics of Nonconservative Systems

Hamilton’s principle of stationary action lies at the foundation of theoretical physics and is applied in many other disciplines from pure mathematics to economics. Despite its utility, Hamilton’s principle has a subtle pitfall that often goes unnoticed in physics: it is formulated as a boundary value problem in time but is used to derive equations of motion that are solved with initial data. This subtlety can have undesirable effects. I present a formulation of Hamilton’s principle that is compatible with initial value problems. Remarkably, this leads to a natural formulation for the Lagrangian and Hamiltonian dynamics of generic nonconservative systems, thereby filling a long-standing gap in classical mechanics. Thus, dissipative effects, for example, can be studied with new tools that may have applications in a variety of disciplines. The new formalism is demonstrated by two examples of nonconservative systems: an object moving in a fluid with viscous drag forces and a harmonic oscillator coupled to a dissipative environment.

Note:

5 pages, 1 figure. Updated to incorporate referees' comments. Matches published version

47.10.-g
05.20.-y
02.30.Xx
45.20.-d
mechanics: classical
oscillator: harmonic
oscillator: coupling
boundary condition
dissipation
field equations

References(18)

Figures(1)

[1]

Classical Mechanics, 2nd edition ed Massachusetts, 1980)

H. Goldstein

[2]

Classical electrodynamics in terms of direct interparticle action

- Rev.Mod.Phys. 21 (1949) 425-433,
•
DOI:
- 10.1103/RevModPhys.21.425

[3]

Mathematical Methods for Physicists, 2nd edition ed Press, New York, 1970)

G. Arfken

[4]

Doubled variables appear in specific applications as early as [16] and more recently in [17, 18]. However, this paper constructs a general framework relating the doubled variables, Hamilton’s principle, initial conditions, and nonconservative dynamics for general dynamical systems

[5]

The same is true in the general case if K = V1(q1, ˙q1) - V2(q2, ˙q2) for V1 = V2. The resulting equations of motion are the same as if (V1 + V2)/2 is absorbed into each Lagrangian

[6]

Using Legendre transforms with respect to pa or πa leads to different but related Hamiltonian formulations that will be detailed elsewhere

[7]

An Introduction to Fluid Dynamics (Cambridge University Press, Cambridge, 1967)