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

Abstract

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 (Γ,ω), a hamiltonian $H:\Gamma \to \mathbb{R}$ and a Lagrangian sub-manifold $M\subset \Gamma$ we find a generalization of the notion of curvature. The particular case $H=\frac{1}{2}{g}^{ij}\left[p_i-A_i\right]\left[p_j-A_j\right]+\varphi$ 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 $\int \left[R+\frac{1}{4}{F}_{ik}{F}_{jl}{g}^{kl}{g}^{ij}-g^{ij}{\partial }_i\varphi {\partial }_j\varphi \right]e^{-\varphi }\sqrt{g}{d}^{n}q$ for the scalar, vector and tensor fields.

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References in corpus (1)

• Curvature and geodesic instabilities in a geometrical approach to the planar three-body problem