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.

Figure 1

Left: A cartoon of Hamilton’s principle. Dashed lines denote the virtual displacements and the solid line denotes the stationary path. Right: A cartoon of Hamilton’s principle compatible with initial data (i.e., the final state is not fixed). In both cartoons, the arrows on the paths indicate the integration direction for the line integral of the Lagrangian.