Jacobi Dynamics  

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

HJ(q, p) = \(1\over2\)gab(q) pa pb ,

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

H = \(1\over2\)hab pa pb + V(q) ,

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

gab = 2 (EV) hab ,      dtJ = 2 (EV) 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].
@ 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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