From flows and metrics to dynamics

Recall that a vector field on an n-dimensional differentiable manifold M is a mapping X defined on M with values in the tangent bundle TM that assigns to each point $x\in M$ a vector X(x) in the tangent space $T_x M$. A vector field may be interpreted alternatively as the right-hand side of an autonomous system of first-order ordinary differential equations, i.e., a flow. Now we show that any flow can be enveloped by a conservative dynamics using a semi-Riemann metric g on M. This kind of dynamics was called {\it geometric dynamics} [7]-[9]. The given vector field, the initial semi-Riemann metric, the Levi-Civita connection, and an associated (1,1)-tensor field are used to build a new geometric structure (e.g., semi-Riemann-Jacobi, semi-Riemann-Jacobi-Lagrange, semi-Finsler-Jacobi, etc) on the manifold M ensuring that all the trajectories of a geometric dynamics are pregeodesics (Lorentz-Udri\c{s}te world-force law). Implicitly, we solved a problem rised first by Poincar\'e: find a suitable geometric structure that converts the trajectories of a given vector field into geodesics.