Curvature in Hamiltonian Mechanics And The Einstein-Maxwell-Dilaton Action

Riemannian geometry is a particular case of Hamiltonian mechanics: the orbits of the hamiltonian $H=\frac{1}{2}{g}^{ij}{p}_i{p}_j$ are the geodesics. Given a symplectic manifold (\Gamma,\omega), a hamiltonian $H:\mathrm{\Gamma }\to \mathbb{R}$ and a Lagrangian sub-manifold $M\subset \mathrm{\Gamma }$ we find a generalization of the notion of curvature. The particular case $H=\frac{1}{2}{g}^{ij}[p_i-A_i][p_j-A_j]+\varphi$ of a particle moving in a gravitational, electromagnetic and scalar fields is studied in more detail. The integral of the generalized Ricci tensor w.r.t. the Boltzmann weight reduces to the action principle $\int [R+\frac{1}{4}{F}_{ik}{F}_{jl}{g}^{kl}{g}^{ij}-g^{ij}{\mathrm{\partial }}_i\varphi {\mathrm{\partial }}_j\varphi ]e^{-\varphi }\sqrt{g}{d}^{n}q$ for the scalar, vector and tensor fields.