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 = 1 2 g i j p i p j are the geodesics. Given a symplectic manifold ( Γ , ω ) , a Hamiltonian H : Γ → ℝ , and a Lagrangian sub-manifold M ⊂ Γ , we find a generalization of the notion of curvature. The particular case H = 1 2 g i j [ p i − A i ] [ p j − A j ] + ϕ of a particle moving in gravitational, electromagnetic, and scalar fields is studied in more detail. The integral of the generalized Ricci tensor with respect to the Boltzmann weight reduces to the action principle ∫ [ R + 1 4 F i k F j l g k l g i j − g i j ∂ i ϕ ∂ j ϕ ] e − ϕ g d n q for the scalar, vector and tensor fields.