Contraction of the canonical symplectic form by vertical vectors

In summary, the canonical symplectic form on T^*M is the negative of the exterior derivative of the tautological 1-form \alpha_p(X):=p(d\pi(X)). This can be proven in a coordinate-free manner by looking at the properties of the tautological 1-form in books such as those by Anna Canna Da Silva and Libermann and Marle.
  • #1
mma
245
1
The canonical symplectic form on [itex] T^*M[/itex] is the exterior derivative of the tautological 1-form:
[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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  • #2
Erratum.

mma said:
The canonical symplectic form on [itex] T^*M[/itex] is the exterior derivative of the tautological 1-form:
[tex] \omega=d\alpha[/tex]​

should be

The canonical symplectic form on [itex] T^*M[/itex] is the negative of the exterior derivative of the tautological 1-form:
[tex] \omega=-d\alpha[/tex]​


Any idea?
 
  • #3
Your question is giving me a headache, but have you looked in the book by Anna Canna Da Silva (first or second chapter I think)? There, she proves many properties of the tautological 1-form in coordinate free form.
 
  • #4
quasar987 said:
Your question is giving me a headache, but have you looked in the book by Anna Canna Da Silva (first or second chapter I think)? There, she proves many properties of the tautological 1-form in coordinate free form.


Yes, I know (and like) da Silva's book, but I didn't find this in it. I give a better chance to the book of Libermann and Marle. Thanks anyway.
 
  • #5
I didn't know about this book until now. Thanks.
 

1. What is the canonical symplectic form?

The canonical symplectic form is a fundamental mathematical concept in symplectic geometry. It is a differential 2-form that encodes the geometric properties of a symplectic manifold, such as phase space in classical mechanics.

2. What are contractions of the canonical symplectic form?

Contractions of the canonical symplectic form refer to the operation of "contracting" or "pulling back" the form along certain vectors on the manifold. This results in a new form that captures the symplectic structure from a different perspective.

3. What do vertical vectors have to do with the contraction of the canonical symplectic form?

In symplectic geometry, vertical vectors refer to vectors that are tangent to the fibers of a symplectic manifold. These vectors play a key role in the contraction of the canonical symplectic form, as they are used to "pull back" the form along the fibers.

4. How are contractions of the canonical symplectic form useful in physics?

In physics, symplectic geometry is used to study the dynamics of physical systems. Contractions of the canonical symplectic form allow us to transform the symplectic structure of a system, which can provide insights into the behavior of the system and its underlying symplectic geometry.

5. Are there any practical applications of contractions of the canonical symplectic form?

Yes, contractions of the canonical symplectic form have applications in fields such as mechanics, fluid dynamics, and control theory. They can be used to analyze and design systems with complex dynamics, and to study the symplectic structure of physical systems in a more intuitive way.

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