- #1
mma
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The canonical symplectic form on [itex] T^*M[/itex] is the exterior derivative of the tautological 1-form:
Let [itex] Y \in T_pT^*M[/itex] a vertical vector, that is [itex] d\pi(Y)=0[/itex].
It's trivial to prove using canonical coordinates that for all [itex] X \in T_pT^*M[/itex]
But how can it be proved in a coordinate-free manner?
[tex] \omega=d\alpha[/tex]
where [itex] \alpha_p(X):=p(d\pi(X))[/itex] is the tautological 1-form.Let [itex] Y \in T_pT^*M[/itex] a vertical vector, that is [itex] d\pi(Y)=0[/itex].
It's trivial to prove using canonical coordinates that for all [itex] X \in T_pT^*M[/itex]
[tex] \omega(X,Y) = y(d\pi(X))[/tex]
where [itex] y \in T_{\pi(p)}^*M[/itex] such that for any differentiable function [itex] f: T^*M \to \mathbb R[/itex] [tex] Y(f)=\left. \frac{df(p+ty)}{dt}\right|_{t=0}[/tex].But how can it be proved in a coordinate-free manner?
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