Jacobi Dynamics

In General > s.a. classical systems [metrizable]; hamiltonian dynamics.
* Jacobi Hamiltonian: One of the form

H_J(q, p) = \(1\over2\)g^ab(q) p_a p_b ,

i.e., without potential; Classical solutions are geodesics in a configuration space with (possibly curved) metric g_ab.
* Jacobi metric: Given a Hamiltonian of the general form

H = \(1\over2\)h^ab p_a p_b + V(q) ,

the dynamics in a region where E − V(x) ≠ 0, for some fixed value E for the energy, can be mapped to that of a Jacobi Hamiltonian H_J by the transformation

g_ab = 2 (E−V) h_ab ,       dt_J = 2 (E−V) dt .

@ General references: in Landau & Lifshitz v1; Glass & Scanio AJP(77)apr; in Goldstein 80; Lynch AJP(85)feb; Izquierdo et al mp/02-conf [and Morse theory]; Gryb PRD(10) [and the disappearance of time]; Maraner JMP(19)-a1912 [for a general Lagrangian system].
@ Relativistic: Kalman PR(61); Sonego PRA(91).
> Related topics: see poisson structure [Jacobi structure on a manifold]; variational principles in physics [Jacobi principle].

Special Cases, Applications > s.a. chaotic motion.
@ For fields: Faraoni & Faraoni FP(02) [Klein-Gordon field and Schrödinger equation].
@ For modified theories: Horwitz et al FP(11)-a0907-proc [with world scalar field, and TeVeS].
> In gravity: see bianchi IX and other chaotic models; formulations of general relativity; spacetime singularities.

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