Knowlpedia

Definition

Tautological one-form on a cotangent bundle

The canonical one-form that evaluates a cotangent vector on the projection of a tangent vector to the base.

Definition

Let MM be a and π:T∗M→M\pi:T^*M\to M its . The tautological one-form θ\theta is the on T∗MT^*M defined at a covector αx∈Tx∗M\alpha_x\in T_x^*M by

θαx(v)=αx ⁣(dπαx(v)),v∈Tαx(T∗M).\theta_{\alpha_x}(v)=\alpha_x\!\left(d\pi_{\alpha_x}(v)\right), \qquad v\in T_{\alpha_x}(T^*M).

Thus θ\theta evaluates the covector represented by the base point of T∗MT^*M on the component of vv projected to MM. The construction uses only the cotangent projection and therefore requires no coordinates, metric, or connection.

Coordinate expression

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

θ=∑i=1npi dqi.\theta=\sum_{i=1}^n p_i\,dq^i.

Although this formula is coordinate-dependent in appearance, the defining evaluation formula proves that the one-form is intrinsic. The construction and its coordinate calculation are given in Cannas da Silva, §1.2.

Canonical symplectic form

With the convention used here, the canonical symplectic form on T∗MT^*M is

ωcan=−dθ=∑i=1ndqi∧dpi.\omega_{\mathrm{can}}=-d\theta=\sum_{i=1}^n dq^i\wedge dp_i.

It is closed because d2=0d^2=0, and the coordinate expression shows that it is nondegenerate. Hence every cotangent bundle carries canonical symplectic geometry without any auxiliary choice.

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