Geometric meaning of conjugate momentum

Question

Let's say I have a free particle moving in an $n$-dimensional Manifold $M$. There is a tangent space $TM$ associated with all possible infinitesimal motions of a particle at each point in this manifold. The trajectory of the particle $C: t \rightarrow M$, associates one tangent vector v to each point $q$ along the path. Let's say the Lagrangian is simply $\mathbf{v}^2\equiv T(\mathbf{v,v})$, where $T$ is bilinear and may change from a point to a point. The matrix/column corresponding to T/v depends on the choice of coordinates. The path of the particle in a specific coordinate system can be determined by solving Lagrange equations for each coordinate

$$\frac{d}{dt}\frac{dT}{dv_i}=\frac{dT}{dq_i}$$ or $$\frac{dp_i}{dt}=\frac{dT}{dq_i}$$

What's the geometric meaning of the conjugate momentum $p$? Can it be related to some property of the curve C? By geometric I mean in terms of intuitive properties of lines and surfaces.

PS. If possible I would like an answer to this specific question in this simplified setting. Not an abstract lecture on Lagrangian and Hamiltonian mechanics unless absolutely necessary.

Answer 1

The momentum is an element of the cotangent space $T_q^\ast M$ at point $q$ (the dual space to the tangent space), i.e. the space of linear maps $T_q M \to \mathbb{R}$. This turns out to give rise to the structure of a symplectic manifold on which Hamiltonian dynamics is naturally formulated, see this post and references therein.

I am not sure there exists an intuitive visualisation, for an illustration of dual spaces in general, see e.g. this wikipedia article.