Knowlpedia

Construction

Cotangent lift

The contragredient diffeomorphism of cotangent bundles induced by a diffeomorphism of base manifolds.

Core idea

Let f:M→Nf:M\to N be a . Its cotangent lift is the map

T∗f:T∗M⟶T∗N,T∗f(αx)=(f−1)∗αx=αx∘d(f−1)f(x).T^*f:T^*M\longrightarrow T^*N,\qquad T^*f(\alpha_x)=(f^{-1})^*\alpha_x =\alpha_x\circ d(f^{-1})_{f(x)}.

Thus T∗f(αx)T^*f(\alpha_x) lies over f(x)f(x). Although is contravariant, inserting f−1f^{-1} makes the lift travel in the same direction as ff. The lift is a diffeomorphism with inverse T∗(f−1)T^*(f^{-1}), and it requires no metric, connection, or other auxiliary choice.

Functoriality

For composable diffeomorphisms ff and gg,

T∗(g∘f)=T∗g∘T∗f,T∗(id⁡M)=id⁡T∗M.T^*(g\circ f)=T^*g\circ T^*f, \qquad T^*(\operatorname{id}_M)=\operatorname{id}_{T^*M}.

These identities follow from the and the reversal built into covector pullback. They make the cotangent construction covariant on the groupoid whose morphisms are diffeomorphisms.

Preservation of canonical forms

The lift preserves the tautological one-form:

(T∗f)∗θN=θM.(T^*f)^*\theta_N=\theta_M.

Consequently it preserves the , so every cotangent lift is a . This intrinsic preservation property is a central reason cotangent lifts occur in mechanics; see Abraham and Marsden, Chapter 3.

Scope
References
  1. Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, 2nd ed., Benjamin/Cummings, 1978. CaltechAUTHORS record. Relevant: Chapter 3, cotangent lifts and canonical forms.
  2. Jerrold E. Marsden and Tudor S. Ratiu, Introduction to Mechanics and Symmetry, 2nd ed., Springer, 1999. DOI record. Relevant: §6.3, cotangent lifts.