Definition
Classical phase space
A symplectic manifold whose points represent the instantaneous states of a finite-dimensional classical system.
Definition
In finite-dimensional Hamiltonian mechanics, a classical phase space is a symplectic manifold whose points represent the complete instantaneous states of a classical system. A real-valued smooth function , called a Hamiltonian, specifies a particular dynamics through the unique vector field determined by
Thus the symplectic form belongs to the kinematic structure of phase space, whereas the choice of supplies the evolution law. Sign conventions may instead use .
Cotangent-bundle model
For a configuration manifold , the standard phase space is the cotangent bundle . Its points record position and momentum, and its canonical symplectic form is obtained from the tautological -form. This construction is intrinsic and does not require a metric on ; see Arnol'd, chapters on Hamiltonian mechanics.
Observables and evolution
Smooth functions on are classical observables. The symplectic form defines their Poisson bracket, and Hamiltonian flow preserves and the Hamiltonian . A Hamiltonian system is therefore more data than a phase space: it is usually a triple .
Scope and generalizations
The definition describes unconstrained finite-dimensional Hamiltonian mechanics. Systems with constraints may first produce a degenerate presymplectic form and require reduction. Infinite-dimensional field theories need functional-analytic hypotheses, and statistical or quantum state spaces use different structures. In Poisson mechanics, a Poisson manifold is sometimes also called a phase space even when its Poisson tensor is degenerate.
References
- V. I. Arnol'd, Mathematical Methods of Classical Mechanics, Springer, 1978. DOI record. Relevant: Hamiltonian mechanics, differential forms, and symplectic manifolds.
- Ana Cannas da Silva, Lectures on Symplectic Geometry, Springer, 2008. DOI record. Relevant: canonical symplectic form on a cotangent bundle and Hamiltonian mechanics.