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].

@ __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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