Variational Principles in Physics  

In General > s.a. constrained systems [including boundary conditions]; jacobi; Maupertuis; schrödinger equation.
* Idea: The equations of motion and/or other equations of interest are given by imposing S = 0; The restrictions chosen on the variations q determine the type of variational principle; Least action: Minimize p · dl, with x1 and x2 fixed.
* Hamilton: (S)t = 0, the usual one, where one fixes t and q at the endpoints of the trajectories; The vanishing of S then gives the Euler-Lagrange equations of motion; Reciprocal: (t)S = 0; Unconstrained: S = –E t.
* Jacobi: Fix the energy E and find the path by extremizing the Jacobi action

S = dx {2m[E–V(x)]}1/2

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

References > s.a. lagrangian dynamics.
@ General: Lanczos 49; Weinstock 52; in Goldstein 80; in Brown & York PRD(89); Kuperschmidt 91; Gray et al AP(96); Lemons 97; Tulczyjew mp/04 [origin = virtual work]; Hanc et al AJP(05) [use of Maupertuis, 1D and 2D]; Basdevant 07.
@ Calculus of variations: De Donder 53; Hermann 68; Goldstine 80 [history]; Struwe 90; Blanchard & Bruning 92; Giusti 02 [direct methods]; Luo et al CTP(04)mp [discrete version, including symplectic].
@ Higher-order calculus of variations: Francaviglia et al DG&A(05); > s.a. higher-order lagrangians.
@ Jacobi vs Hamilton: Bazanski & Jaranowski JPA(94).
@ Invariant derivation of equations of motion: Nester JPA(88).
@ Systems: Ichiyanagi PRP(94) [irreversible processes]; Núñez-Yépez & Salas-Brito PLA(00)mp [Jacobi equations]; Pankrashkin a0710 [Hamiltonians with degenerate lowest-energy states].
@ Inverse problem: Marmo et al CQG(90) [metric from test particle motion].
@ Other: Kaup & Lakoba JMP(96) [caveat re instabilities]; Nishimura IJTP(99) [infinitesimal form]; El-Nabulsi & Torres JMP(08) [fractional actionlike variational problems].

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