# Geometric meaning of conjugate momentum

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.

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.