Lagrangian Dynamics

In General > s.a. [classical mechanics]; Euler-Lagrange; lagrangian systems; quantum mechanics [need for action].
* Idea: A formulation of dynamics based on the principle of least action, in turn based on variational calculus.
* History: The term "action", and the concept that physical laws are a consequence of "economy of means" were introduced by Maupertuis in 1744, with W = p dq, and improved by Euler and Lagrange; In 1832 Hamilton proposed the action S = L(q, q^·) dt.
* Motivation: (1) Elegant way of expressing the dynamical content of a physical theory; (2) Study of symmetries of the theory; (3) Important for the pathy integration quantization program (> but see path integrals); (4) For a field theory, it is a "spacetime covariant" formulation, and for relativistic quantum field theory it is preferable to a Hamiltonian formulation.
* Action: (1) Choose a configuration space, with coordinates q depending on x; (2) If the equations of motion are known, the problem is to find, given a region X of the x's, a Lagrangian (density) or an action functional S[q,q, x], written

S[q, q, x] = _M (q, q, x) dx ,

from which equations of interest are derived using variational principles.
* Formal setting for a field theory: A Lagrangian of order k is a horizontal form L: J^kY → ⁿM, where (Y, , M) is the configuration bundle for the theory, (J^kY, ^k, M) its k-jet prolongation, and M the spacetime manifold; A variation of the fields is a vertical vector field X = yⁱ (/yⁱ), under which the Lagrangian varies by

_XL = e(L, X) + dF(L, X) ,  with  e(L, X) = Euler-Lagrange form ,  and  F(L, X) = Poincaré-Cartan form .

* Normalization constants: They do not affect the equations of motion, but they do appear in the conjugate momenta.
* Interacting systems: To get the Lagrangian, just add the ones for each system and (if they are needed) the extra interaction terms, but we have to be careful with the relative coefficients for each term.

And Canonical Formalism > s.a. constrained systems.
* Legendre transformation: The map F: TQ → T*Q (Q is configuration space) defined, for a Lagrangian L, by F(w)v:= dL(w+tv)/dt|_t=0.
@ General references: Kastrup PRP(87); Geyer et al JPA(03)ht/02 [and gauge symmetries]; Bertin et al MPLA(05)ht [first-order actions, Hamilton-Jacobi analysis]; Compère a0708-PhD [covariant Hamiltonian formulation, symmetries, and black holes].
@ For field theories: Echeverría-Enríquez et al IJMMS(02)mp/01, JMP(04) [unified formalism with Hamiltonian].

Choice of Action and Ambiguities > s.a. variational principles.
* Idea: In practice, in particular for field theories, we get the invariances from nature (e.g., gauge symmetries), and this often is very restrictive; However, classically one can always add a boundary term to the action, or a div to , that depends on the fields; This has the effect of changing the definitions of the canonical momenta and energy but not the equations of motion – the Lagrangians are in the same variational cohomology class; In the quantum theory, it is not so simple.
* Field redefinitions: For example, by conformal transformations.
@ Ambiguity in Lagrangian: Hojman & Shepley in(88), Lunev TMP(91), TMP(92) [for field theories]; in Matzner & Shepley 92; Faraoni CQG(94); Cislo & Lopuszanski JMP(01)mp/00 [and quantization].
@ Lagrangian cohomology classes: in Marathe & Martucci 92; in Kolár et al 93.
@ Related topics: Frieden 98 [L's from Fisher information]; Buniy & Kephart PLA(08) [higher-order topological actions and quantum theory].

Symmetries and Conserved Quantities > s.a. conservation laws; noether theorem; symmetries.
* Symmetric variations: The issue is whether the Principle of Symmetric Criticality holds; It states that for any group-invariant lagrangian the equations obtained by restriction of Euler-Lagrange equations to group-invariant fields are equivalent to the Euler-Lagrange equations of a canonically defined, symmetry-reduced lagrangian.
* Conserved currents and Lie algebras: One may introduce at least three different Lie algebras in any Lagrangian field theory, (i) the Lie algebra of local BRST cohomology classes with the odd Batalin-Vilkovisky antibracket; (ii) the Lie algebra of local conserved currents with the Dickey bracket; and (iii) the Lie algebra of conserved, integrated charges with the Poisson bracket.
* Question: Are there known examples of systems for which symmetric criticality fails?
@ General references: Gràcia & Pons JPA(92); Grigore JPA(95)ht/94; Barnich & Henneaux JMP(96) [Batalin-Vilkovisky antibracket and Poisson bracket]; Banerjee et al PLB(99)ht, PLB(00)ht/99, Banerjee ht/00 [and Hamiltonian].
@ Conservation laws: Lutzky JPA(79), JPA(82), JPA(95), JPA(98) [non-Noether]; Sardanashvily mp/03 [Noether].
@ Symmetric variations: Palais CMP(79); Christodoulakis & Korfiatis NCB(94); Davis GRG(98)gq/96, Fels & Torre CQG(02)gq/01 [gravity].
@ Lagangians from Lie transformation groups: Paal & Virkepu a0706; Nucci & Leach JMP(07)-a0706 [Jacobi last multiplier].

References
@ Textbooks: Lagrange 1788; in Sudarshan & Mukunda 75; Brizard 08 [II].
@ General: Bailey FP(81) ["more precise statement"]; Cariñena et al PRP(95) [need for lagrangian in physics]; Stöltzner SHPMP(03) [principle of least action]; Evans AJP(03)may [interpretation of particle action]; Toffoli IJTP(03) [meaning?]; Butterfield phy/04 [ontology]; Gray & Taylor AJP(07)may [not minimized in general]; Cuell RPMP(07) [as a skew critical problem].
@ And equations of motion: Grigore NCB(96)ht [conditions for first-order Lagrangian]; Gitman & Kupriyanov JPA(07)-a0710; Nucci & Tamizhmani a0809 [finding L by the method of Jacobi Last Multiplier].
@ Geometrical: Cariñena et al JMP(88); del Olmo & Santander JPA(89), JGP(90); Sharipov m.DG/01 [in Riemannian manifolds].
@ And spacetime transformations: Whiston IJTP(72).
@ Inverse variational principle: Santilli 78; Okubo PRD(80) [non-unique]; Cariñena et al PRP(95).
@ Generalizations: Grigore IJMPA(92); Núñez-Yépez et al mp/01-in; Arizmendi et al CSF(03)mp/04; Gravanis & Willison a0901 [distributional fields].
@ Related topics: Gocksch PLB(88), Anagnostopoulos & Nishimura PRD(02) [complex action]; Ferraris et al JMP(00) [dual Lagrangians]; Muñoz Díaz a0801 [and time].

main page – abbreviations – journals – comments – other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 28 jun 2009