  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 non-conservative systems,
thereby filling a long-standing gap in classical mechanics. Thus dissipative
effects, for example, can be studied with new tools that may have application
in a variety of disciplines. The new formalism is demonstrated by two examples
of non-conservative systems: an object moving in a fluid with viscous drag
forces and a harmonic oscillator coupled to a dissipative environment.
