Variational Principles in Physics| 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[E−V(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];
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].
@ Conceptual:
Wang a0808 [philosophical, dialectical view];
Terekhovich a1909-in [ontology].
@ 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];
Feng & Matzner GRG(18) [Weiss, and gravity].
@ 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];
> s.a. discrete systems below.
@ 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];
Gubbiotti a1808 [inverse problem].
@ 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 CVPDE(17)-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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