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I think the question is about "geometric intuition" and "geometric understanding", ill defined as those may be. Perhaps that can be answered with some comparisons. I'll be a bit lazy and just take exa …

In his classic book on classical mechanics Whittaker calls these transformations Mathieu transformations. The term appears in Wikipedia.

There is the paper ARE HAMILTONIAN FLOWS GEODESIC FLOWS? by CHRISTOPHER MCCORD, KENNETH R. MEYER, AND DANIEL OFFIN which treats this problem and gives examples of Hamiltonian flows (most cases of the …

The distribution of horizontal subspaces is always Lagrangian not only for Riemannian metrics, but also for the Ehresmann connection associated to Finsler metrics and for the Levi-Civita connection of …

Interestingly enough no one wrote about the first examples of Lagrangian manifolds in symplectic geometry. In the first papers on symplectic geometry (Theory of Systems of Rays, 1828), William R. Hami …

The answer is yes. Proposition. Let $\alpha$ be the standard contact form on the three-sphere (for which the Reeb vector field is the Hopf vector field $X$). If $f$ is a strictly positive function o …

I took the time to give a clean version of the answer and to eliminate the dependence on really hard theorems. Construction of geodesible, volume-preserving flows on $S^3$ that are not Reeb flows for …

Here is an application with which I have first hand experience. In the sixties J.J. Schaeffer conjectured that the girth of a normed space---the infimum of the lengths of all continuous curves on its …

Both limits (1) and (2) are equal to $C(1)/V(1)$ because of the homogeneity of the volume and the symplectic capacity. Namely, the symplectic form is homogeneous of degree $1$ with respect to dilation …

The intuition is that the Levi-Civita connection corresponds to the linearization of the geodesic flow plus a simple projective-geometric construction: Let $c(t)$ be an orbit of the geodesic flow (pr …

Remark. Now that Cadoi has disclosed the origin of the invariant (which was absent from the original formulation of the problem) my musings in trying to guess the symplectic geometry behind it seem a …

Dear Hapchiu, To understand the generating function construction of Lagrangian submanifolds I recommend the following: Take the few pages in "Geometric Asymptotics" by Guillemin and Sternberg, wher …

Yes, the geodesic flow on the hyperbolic plane and, in fact, on any Hadamard manifold (${\mathbb R}^n$ provided with a complete Riemannian metric with non-positive curvature) is integrable. You can …

This is a very particular case of something more general: extremals for a non-degenerate Lagrangian correspond to solutions of the corresponding Hamiltonian system. There is no need to use connections …