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American Institute of Physics, Journal of Mathematical Physics, 5(58), p. 052901

DOI: 10.1063/1.4983665

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

Journal article published in 2017 by S. G. Rajeev ORCID
This paper is available in a repository.
This paper is available in a repository.

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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 $H:Γ\toℝ$ and a Lagrangian sub-manifold $M⊂Γ$ 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]+ϕ $ 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 $∫\left[R+\frac{1}{4}F_{ik}F_{jl}g^{kl}g^{ij}-g^{ij}𝟃_{i}ϕ𝟃_{j}ϕ\right]e^{-ϕ}\sqrt{g}d^{n}q$ for the scalar, vector and tensor fields. ; Comment: 2 figs