Cotangent bundles and the canonical form

Let π:T*Q→Q be the bundle projection. A point α_q in T*Q is itself a covector at q. For a tangent vector V based at α_q, define θ(V) by applying α_q to dπ(V). This separates two levels of tangent information: V moves through phase space, while dπ(V) records only its configuration-space component. In local configuration coordinates q_j with momentum coefficients p_j, the definition gives θ=Σp_jdq_j. Although the coefficients change between charts, the intrinsic evaluation rule defines one global one-form.

Taking minus its exterior derivative yields Σdq_j∧dp_j, a nondegenerate closed form on every cotangent chart and therefore globally. It is exact even when Q has complicated topology, because the global θ provides a primitive for −ω. This does not conflict with the impossibility of exact symplectic forms on compact boundaryless manifolds: a positive-dimensional cotangent bundle has noncompact vector-space fibers. A metric may identify momenta with velocities for a mechanical Hamiltonian, but that extra identification is not part of the canonical form itself.

If a configuration diffeomorphism sends q to Q=F(q), the same covector must have equal evaluations in both descriptions. Thus PᵀdQ=pᵀdq, giving P=(DF(q))^{-T}p. The corresponding cotangent lift preserves θ exactly, and hence preserves ω. Transforming position while leaving momentum numerically unchanged generally fails this rule. In particular, nonlinear coordinate changes introduce momentum couplings through the Jacobian. This is why canonical momentum is naturally a covector rather than simply a second copy of the position coordinates.

A different family of transformations translates momenta by a one-form β on Q. Pulling back θ adds π*β, so pulling back ω subtracts π*dβ. Consequently the translation is symplectic exactly when β is closed. If β is exact, its graph comes from a global generating potential; if it is closed but not exact, a symplectic translation can still exist without a single-valued global potential. This distinction keeps local calculus and global topology separate, just as closed contraction forms did for Hamiltonian vector fields.

Choosing H(q,p)=one half the squared momentum norm requires an inverse metric on Q. Adding a potential changes trajectories without changing the canonical form. Likewise, external forcing can alter the vector field while leaving the underlying phase space available as a mathematical description. The canonical structure organizes the possible Hamiltonian equations; it does not assert that every physical process on the bundle is conservative. Constraints may require restriction and reduction rather than simply deleting momentum coordinates from the form.

The zero section and individual cotangent fibers have half the phase-space dimension and zero restricted symplectic form. They are basic Lagrangian submanifolds, developed next. Graphs of covector fields offer another bridge: their pullback of θ is the covector field itself, so their restricted ω is minus its exterior derivative. This gives a direct calculation of when such graphs are Lagrangian. The same intrinsic evaluation formula will later construct momentum maps for lifted symmetries, linking configuration-space actions to conserved phase-space quantities.