Knowlpedia

Definition

Canonical symplectic form on a cotangent bundle

The symplectic form obtained as minus the exterior derivative of the tautological one-form on a cotangent bundle.

Definition

Let MM be a and θ\theta the on its T∗MT^*M. The canonical symplectic form is

ωcan=−dθ,\omega_{\mathrm{can}}=-d\theta,

where dd is the . It is closed because d2=0d^2=0, and it is nondegenerate, so (T∗M,ωcan)(T^*M,\omega_{\mathrm{can}}) is a . This construction is intrinsic: it uses neither a metric nor a connection on MM. The displayed sign fixes the convention used throughout this knowl.

Coordinate expression

For local coordinates q1,…,qnq^1,\ldots,q^n on MM and induced fiber coordinates p1,…,pnp_1,\ldots,p_n,

θ=∑ipi dqi,ωcan=∑idqi∧dpi.\theta=\sum_i p_i\,dq^i, \qquad \omega_{\mathrm{can}}=\sum_i dq^i\wedge dp_i.

This normal form directly shows nondegeneracy. It also explains why cotangent coordinates are the standard position-momentum coordinates of Hamiltonian mechanics.

Naturality

The of every of base manifolds preserves θ\theta, hence preserves ωcan\omega_{\mathrm{can}}. In this sense the form is canonical not only because it requires no choices, but also because it is natural under changes of variables.

Conventions and scope
References
  1. Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, 2nd ed., Benjamin/Cummings, 1978. CaltechAUTHORS record. Relevant: §3.2, canonical forms on cotangent bundles.
  2. Ana Cannas da Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics 1764, Springer, 2001. DOI record. Relevant: §1.2, tautological and canonical symplectic forms.