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I'm looking for examples of completely integrable systems and specifically geodesic flows. We remember that when we have a symplectic manifold $(M,\omega)$ (with $M$ of dimension $2n$) and $H:M\...

Here is my precise question. Let $M, \omega$ be a symplectic manifold and let $H: M \to \mathbb{R}$ be any smooth function. The symplectic form gives rise to an isomorphism between the tangent ...

The most concrete version of the question is : A (necessarily) invariant Lagrangian torus $L$ on the unit cotangent of a Riemannian metric on the two-torus carries a periodic orbit with period $T$. ...

Motivated by classical formulas $L_{X}=d\circ i_{X}+i_{X}\circ d$ and $L_{X} \Omega=Div(X) \Omega$ and the essential role of the diff operator $d$ in definition of divergence, we define some ...

Edit: According to the comments to the previous version of this question, I remove my essential errors in the question. I thank the commenters very much. Let $M$ be a n dimensional manifold. ...