Covariant Symplectic Geometry of Classical Particles

Abstract

We investigate the tension between symplecticity and gauge covariance in classical Hamiltonian mechanics. The pursuit of manifest covariance over manifest symplecticity results in a unique geometric formulation. Firstly, covariant yet non-canonical coordinates are employed by adopting Souriau's approach to minimal coupling. Secondly, covariant yet non-coordinate frames arise from Ehresmann connections in phase space. Thirdly, the concept of covariant Poisson bracket is introduced, facilitating direct derivations of covariant equations of motion. In this way, we establish manifestly covariant Hamiltonian formulations of particles coupled to background gauge and gravitational fields, with or without spin. The variational principle and path integral origins of our framework are also explicated.

Figures (5)

• Figure 1: Three principles in tension.
• Figure 2: Anharmonic oscillator in canonical and non-canonical coordinates. In a canonical coordinate system (left), the apparent area element in phase space is preserved under time evolution. In a non-canonical coordinate system (right), the apparent area element shrinks or expands by 60% or 120% of the original scale. Such a non-canonical coordinate system arises by squishing the phase space by a generic diffeomorphism.
• Figure 3: The scaffolding of covariant symplectic geometry. The first two levels are sufficient for characterizing classical geometry, while the third level characterizes quantum geometry.
• Figure 4: A non-commutative diagram arises from covariantization and exterior derivative.
• Figure 5: The operator product from path integral.