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 .

@ General references: Small & Lam AJP(11)jun [simple derivation]; Román-Roy Math(21)-a2101 [rev].
@ Geometric approaches: Marmo, Morandi & Mukunda RNC(90); Butterfield in(04)qp/02; Cariñena et al IJGMP(06)mp [on the tangent bundle, possibly bi-Hamiltonian and singular Lagrangians], in(09)-a0907 [and the time-evolution operator]; Graffi & Zanelli RVMP(11); Barbero-Liñán et al a1209 [using general Lagrangian submanifolds].
@ And quantum mechanics: Kyprianidis PLA(88); Ferraro qp/96, JPA(99)qp/96; Bhalla et al AJP(97)dec [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]; Marmo et al a0907; Guo & Schmidt PRD(12)-a1204 [quantization employing special solutions of the Hamilton-Jacobi equation]; de Souza Dutra et al PTEP(16)-a1510 [and example]; > s.a. origin of quantum mechanics.

Specific Types of Theories > s.a. classical relativistic particles; spinning particles.
@ Constrained systems: Dominici et al JMP(84); Rothe & Scholtz AP(03) [second-class]; Nawafleh et al IJMPA(04); Leok et al JMP(12)-a1109 [holonomic and non-holonomic].
@ Non-holonomic: Pavon JMP(05)mp/04 [linear in velocity]; Iglesias et al PRD(07)-a0705; Cariñena et al a0908 [geometric approach].
@ 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-proc, ap/95-proc [cosmic time and matter], PRD(97)ap/98; Darian CQG(98) [+ electromagnetism + scalar]; Parentani CQG(00)gq/98 [background-field approximation of quantum gravity]; Bertin et al CQG(11)-a1107 [linearized gravity]; > s.a. time in gravity.
@ Field theories: Stoyanovsky in(04)mp/02; Bruno JMP(07) [solutions]; de León et al a0801 [geometrical]; Vitagliano IJGMP(10), IJGMP(12)-a1109-conf [higher-order field theories].
@ 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]; Rajeev AP(08)-a0711 [thermodynamics]; de León et al JMP(13) [singular Lagrangian systems].
> For quantum gravity: see 2D quantum gravity; time in quantum gravity.
> Other quantum systems: see Bloch Theory; quantum systems; first-class and second-class constraints [Hamilton-Jacobi approach].

Techniques and Related Topics
@ Separation of variables: Benenti JMP(97), et al JMP(02), JMP(02), JMP(05); Rastelli a0907-conf [geometrical theory].
@ Other solution methods: Cheng & Shu JCP(07) [discontinuous Galerkin finite element method].
@ Other related topics: Boyer & Kalnins JMP(77) [symmetries]; Ramírez & Ritto RMF(03)mp [fermions]; Barbero-Liñán et al a1110 [kinematic reduction, and non-holonomic systems]; Cortés & Jiménez-Aquino PhyA(14) [equivalence with the Fokker-Planck equation, overdamped Brownian harmonic oscillator]; Lemos AJP(14)sep [incompleteness, motion of a charged particle in an electric dipole field]; > s.a. Diffieties.
@ Generalizations: Chavoya-Aceves qp/04; Rabei et al PS(08) [with fractional derivatives]; Marciniak & Blazsak JGP(08) [non-Hamiltonian systems]; Balseiro et al Nonlin(10)-a1001 [unified framework]; de León et al a1209 [in the setting of almost Poisson manifolds, including non-holonomic mechanical systems and time-dependent systems with external forces]; Esen et al JPA(20)-a1901 [for higher-order implicit systems].

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