I was thinking about the foundations of geometric mechanics and its precursors. I wondered who was the first to realized the equivalence between Riemannian geometry and Lagrangian mechanics. In particular:

- Solutions (trajectories) of Lagrange equations are geodesics of Levi-Civita connection.
- The moment of inertia tensor can be viewed as a Riemannian metric tensor.

I would be grateful for any answers and relevant references to this topic.