Jacobi Dynamics

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

H_J(q, p) = 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 = 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 76; Glass & Scanio AJP(77); in Goldstein 80; Lynch AJP(85); Izquierdo et al mp/02-in [and Morse theory].
@ Relativistic: Kalman PR(61); Sonego PRA(91).

Special Cases, Applications > see chaotic motion.
@ For fields: Faraoni & Faraoni FP(02) [Klein-Gordon field and Schrödinger equation].
> In gravity: see bianchi IX and other chaotic models; formulations of general relativity; singularities.

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