Higher-Order Lagrangian Systems

In General
* Idea: A generalized form of mechanics, which uses higher-order Lagrangians (in the -order case, the theory is non-local in time).
* Motivation: One motivation was to see if non-local theories were free of divergence problems in quantum field theory.
* Remark: One usually wants first order Lagrangians in order to get second order equations of motion, define conserved quantities in the usual way, and perform Legendre transformations; Even the general relativity Lagrangian is of this kind, though it may not be obvious from the Einstein-Hilbert expression, because the second derivative terms are a pure divergence.

Non-Local Theories > s.a. non-commutative physics.
* Issue: Non-local actions generally possess acausal equations of motion and non-real in-out operator amplitudes.
@ Of finite extent: Woodard PRA(00); Llosa PRA(03)ht/02; > s.a. hamiltonian systems.
@ Related topics: Soussa ht/03-in [causality]; Calcagni et al PRD(07)-a0705 [FRW + scalar cosmology].

Other Specific Types of Theories > s.a. higher-order gravity; oscillators; spinning particles.
@ Field theories: in Stelle GRG(78) [gravity]; Grigore FdP(99)ht/96 [trivial second-order Lagrangians]; de Urries & Julve JPA(98) [scalar]; de Urries et al JPA(01) [bosonic, as constrained second-order]; Villaseñor JPA(02)ht [fermionic]; Bazeia et al JPA(03) [2+1, dualities].

References > s.a. hamiltonian and lagrangian formulation.
@ General: Whittaker 37; de León & Rodrigues 85; Negri & da Silva PRD(86); Jaén et al PRD(86); Gràcia et al JMP(91); Hojman et al JMP(92) [Lagrangian from differential equations of any order].
@ Hamiltonian formulation: Ostrogradskii MASP(1850) [momenta]; Coelho de Souza & Rodrigues JPA(69); Jaén et al PRD(87), JMP(89); Llosa & Vives JMP(94); Belvedere et al ZPC(95) [for field theory, canonical transformations]; Schmidt gq/95 [4th-order theories]; Rashid & Khalil IJMPA(96); Woodard PRA(00)ht/00 [finite non-locality]; Bering ht/00; Cheng et al NPB(02) [and non-commutative field theory]; Leclerc gq/06 [modified Ostrograski formulation]; Muslih & El-Zalan IJTP(07); Andrzejewski et al a0710; Morozov a0712 [brief review]; Dunin-Barkowski & Sleptsov a0801 [reparametrization-invariant theories].
@ Noether theorem, symmetries: Miron IJTP(95); de León & Martín de Diego JMP(95); Sardanashvily mp/03.
@ Geometrical: de León & Lacomba JPA(89) [ito sympletic higher-order tangent bundles].
@ Meaning/degrees of freedom: Chervyakov & Nesterenko PRD(93); de Urries & Julve gq/95.
@ Related topics: Nakamura & Hamamoto PTP(96)ht/95 [path integrals]; Benito et al IJGMP(06) [geometric integrators]; Nesterenko PRD(07) [instability].

Quantization [> s.a. formulations of quantum mechanics and quantum systems.]
@ General references: Hayes JMP(69); Tesser JMP(72); Acatrinei a0708 [from phase space path integrals]; Andrzejewski et al PRA(07).
@ In quantum field theory: Pais & Uhlenbeck PR(50).

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