Symplectic geometry and Hamiltonian flow

At each point, a differential two-form is an alternating bilinear function of two tangent vectors. A symplectic form is both closed, meaning its exterior derivative vanishes, and nondegenerate, meaning no nonzero vector pairs to zero with every vector. Nondegeneracy forces even dimension and lets a covector determine a unique vector. Closedness is an additional differential condition, not a consequence of invertibility of a matrix at each point. On a two-dimensional manifold every two-form is automatically closed, but that dimensional shortcut does not extend to arbitrary even dimensions.

In canonical coordinates (q,p), take ω=dq∧dp. It assigns the signed area ad−bc to tangent vectors (a,b) and (c,d). By contrast, a Riemannian metric is symmetric and positive definite and measures squared lengths. Confusing these structures hides why Hamiltonian evolution can stretch one direction while compressing another. The local Darboux theorem provides coordinates with a canonical symplectic form, but it does not make every Hamiltonian simple or erase global topology. A standard-looking local form is a coordinate statement, not a complete solution of the dynamics.

We define the Hamiltonian vector field X_H by contracting it into ω to obtain dH. For X_H=(q̇,ṗ), contraction into dq∧dp gives q̇dp−ṗdq. Matching coefficients with H_qdq+H_pdp yields q̇=H_p and ṗ=−H_q. Some texts define contraction with the opposite sign and adjust conventions elsewhere. A calculation should state its convention rather than mixing formulas from incompatible choices. The existence and uniqueness of the vector field follow from nondegeneracy; existence of its flow for all time requires further conditions on the vector field and manifold.

For a time-independent H, the derivative of H along its own vector field is dH(X_H)=ω(X_H,X_H)=0 by alternation. Closedness additionally makes the Lie derivative of ω vanish through Cartan's formula. Therefore the flow preserves the symplectic form wherever it exists, and hence preserves the associated phase-space volume. These conclusions do not imply that individual trajectories fill an energy surface or that the system is ergodic. Integrable and chaotic Hamiltonian systems share symplectic preservation while having very different long-time orbit structures.

For H=(q²+p²)/2, the flow is a rotation: q̇=p and ṗ=−q. Circles of constant energy remain circles and signed area is preserved. Add damping ṗ=−q−2ζp with ζ>0 and the energy derivative becomes −2ζp². The planar divergence is −2ζ, so phase-space area contracts. This damped system is useful physics but is not Hamiltonian for that same canonical form and time-independent oscillator energy. Enlarging a model with reservoirs is another construction; it does not make the reduced damped plane area-preserving.

A numerical method can approximate the trajectory accurately over a short interval yet systematically distort its geometric invariants over long intervals. Symplectic integrators preserve an appropriate discrete symplectic map, usually without exactly preserving the original energy at every step. Exact energy preservation and symplecticity are different design goals. The oscillator lab uses analytic damped solutions with fixed initial conditions; set damping to zero to inspect the nondissipative case, then increase it to compare contraction in the underlying equations. It does not implement a numerical integrator or test a general manifold's symplectic structure.