Hamilton-Jacobi Theory

In General > s.a. quantization of constrained systems [HJ approach]; covariant symplectic structure.
* Idea: A method for solving dynamical equations in classical mechanics, or obtain frequencies of periodic motion without solving them.
* Hamilton-Jacobi equation: The equation for S = S(q, P; t) solved by the action along a classical trajectory,

H(q, S/q; t) + S/t = 0 .

Specific Types of Theories > s.a. classical particles.
@ Constrained systems: Dominici et al JMP(84); Rothe & Scholtz AP(03) [second-class]; Nawafleh et al IJMPA(04).
@ Non-holonomic: Pavon JMP(05)mp/04 [linear in velocity]; Iglesias et al PRD(07)-a0705.
@ For general relativity: Bergmann et al IJTP(70); Salopek & Stewart CQG(92); Parry et al PRD(94)gq/93; Salopek PRD(95)ap, ap/95-in, ap/95-in [cosmic time and matter], PRD(97)ap/98; Darian CQG(98) [+ electromagnetism + scalar]; Parentani CQG(00)gq/98 [background field approximation of quantum gravity]; > s.a. time in gravity.
@ Other theories and topics: Martínez-Merino & Montesinos AP(06)gq [covariant symplectic structure]; Bertin et al AP(08)ht/07 [first-order actions for theories with higher derivatives]; Bruno JMP(07) [field theory, solutions]; Rajeev a0711 [thermodynamics]; Rabei et al PS(08) [with fractional derivatives]; de León et al a0801 [for field theories, geometrical]; Marciniak & Blazsak JGP(08) [non-Hamiltonian systems].
> For quantum gravity: see 2-dimensional quantum gravity, time in quantum gravity.
> Other quantum systems: see Bloch Theory; quantum systems; first class and second class constraints [Hamilton-Jacobi approach].

References
@ General: Marmo, Morandi & Mukunda RNC(90); Stoyanovsky mp/02/FAA [for field theory]; Butterfield qp/02-in [geometry]; Cariñena et al IJGMP(06)mp [geometric, possibly bi-Hamiltonian and constrained].
@ And quantum mechanics: Kyprianidis PLA(88); Ferraro qp/96, JPA(99)qp/96; Bhalla et al AJP(97) [and bound state spectrum]; Periwal PRL(98); Kim & Lee CJP(99)qp [canonical transformations]; Makowski PRA(02) [V(r) with no quantum correction]; Jurisch qp/06, JPA(07); Roncadelli & Schulman PRL(07)-a0712 [prescription for solving the quantum HJ equation].
@ Separation of variables: Benenti JMP(97), et al JMP(02), JMP(02), JMP(05).
@ Other solution methods: Cheng & Shu JCP(07) [discontinuous Galerkin finite element method].
@ Related topics: Boyer & Kalnins JMP(77) [symmetries]; Ramírez & Ritto RMF(03)mp [fermions].
@ Generalized: Chavoya-Aceves qp/04.

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