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} ,
d*t*_{J} = 2 (*E*−*V*) d*t* .

@ __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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