Hamilton-Jacobi Theory| 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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