Variational Principles in Physics  

In General > s.a. Variational Principles / jacobi dynamics; Maupertuis Principle.
* Idea: The equations of motion and/or other equations of interest are given by imposing δS = 0; The restrictions chosen on the variations δq determine the type of variational principle; Least action: Minimize ∫ p · dl, with x1 and x2 fixed.
* Hamilton's principle: (δS)t = 0, the usual one, where one fixes t and q at the endpoints of the trajectories; The vanishing of δS then gives the Euler-Lagrange equations of motion; Reciprocal: (δt)S = 0; Unconstrained: δS = –E δt.
* Jacobi principle: Fix the energy E and find the path by extremizing the Jacobi action

S = dx {2m[EV(x)]}1/2

with respect to paths x(s) in configuration space; Time dependence is recovered only after imposing another, metric condition.
* Maupertuis principle: (δW)E = 0; Generalized: (δW)E' = 0; Reciprocal: (δE')W = 0; Unconstrained: δW = t δE'.
* Weiss principle: The endpoints of trajectories are not held fixed; It yields the canonical momenta.

References > s.a. classical and quantum mechanics [unified description]; lagrangian dynamics.
@ Texts: Lanczos 49; Weinstock 52; in Goldstein 80; Kuperschmidt 92; Lemons 97; Basdevant 07.
@ General references: in Brown & York PRD(89); Gray et al AP(96); Tulczyjew mp/04 [origin = virtual work]; Hanc et al AJP(05)jul [use of Maupertuis, 1D and 2D]; Wang a0808 [philosophical, dialectical view]; Gondran & Gondran a1212 ["final causes" vs "efficient causes" and Euler-Lagrange vs Hamilton-Jacobi action]; Bekenstein & Majhi NPB(15)-a1411 [field equations from the action without variation]; Anderson et al AJP(16)sep [direct variational methods and their relation to Galerkin and moment methods, intro].
@ Calculus of variations: de Donder 35; Hermann 68; Goldstine 80 [history]; Struwe 90; Blanchard & Bruning 92; Giusti 03 [direct methods]; Chang 16 [lecture notes].
@ Non-differentiable versions: Luo et al CTP(04)mp [including symplectic]; Almeida & Torres MMAS(11)-a1106 [Cresson approach, on the space of Hölder functions].
@ Higher-order calculus of variations: Francaviglia et al DG&A(05); > s.a. higher-order lagrangians.
@ Hamilton's principle: Bażański & Jaranowski JPA(94) [vs Jacobi]; Wharton a0906/PRL [re quantization]; Kapsa & Skála JPA(09) [from spacetime Fisher information].
@ Other principles: Romano et al RPMP(09) [Maupertuis, new formulation and time-dependent systems].
@ With given initial position and velocity: Galley PRL(13)-a1210 [and application to the Lagrangian and Hamiltonian dynamics of non-conservative systems]; Gondran & Gondran a1210-proc [and quantum theory].
@ Invariant derivation of equations of motion: Nester JPA(88).
@ Inverse problem: Marmo et al CQG(90) [metric from test-particle motion]; Ercolessi et al RNC(10)-a1005 [and quantum commutation relations]; Saunders RPMP(10) [rev].
@ Related topics: Kaup & Lakoba JMP(96) [caveat re instabilities]; Nishimura IJTP(99) [infinitesimal form].

Types of Systems and Generalizations > s.a. constrained systems and types [non-holonomic]; schrödinger equation; Schwinger's Principle.
@ General references: Ichiyanagi PRP(94) [irreversible processes]; Núñez-Yépez & Salas-Brito PLA(00)mp [Jacobi equations]; Pankrashkin a0710 [Hamiltonians with degenerate lowest-energy states]; Esteban et al BAMS(08) [in relativistic quantum mechanics]; > s.a. conservation laws [theories with symmetries].
@ For stochastic processes: Yasue JFA(81); Koide & Kodama JPA(12)-a1105 [and the Navier-Stokes equation]; > s.a. stochastic quantization.
@ For non-conservative systems: Galley et al a1412 [stationary action].
@ For field theories: Vankerschaver et al JMP(12)-a1207 [Hamilton-Pontryagin principle and multi-Dirac structures]; Siringo PRD(14)-a1308, MPLA(14)-a1308 [principle of stationary variance in quantum field theory]; Bäckdahl & Valiente Kroon JMP(16)-a1505 [with spinors].
@ Discrete systems: Dittrich & Höhn JMP(13)-a1303 [constraint analysis].
@ Fractional variational principles: El-Nabulsi & Torres JMP(08); Baleanu RPMP(08); Almeida & Torres AML(09)-a0907; Malinowska & Torres 13 [intro]; Odzijewicz et al C&C-a1304 [generalized].
@ Causal variational principles: Finster JRAM(10)-a0811; Finster & Schiefeneder ARMA(13)-a1012; Finster et al in(12)-a1102; Finster & Grotz JRAM(14)-a1303; Finster & Kleiner a1612 [Hamiltonian formulations]; > s.a. Initial-Value Problem.
> Specific types of systems: see action for general relativity; dissipative systems; types of lagrangian systems.


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