Higher-Order Lagrangian Systems  

In General > s.a. classical mechanics [higher-order equations of motion].
* Idea: A generalized form of mechanics, in which Lagrangians include higher-order derivatives (in the infinite-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, although higher-derivative terms are often connected with the appearance of instabilities and ghost states.
* 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.
* Ostrogradski theorem: A classical Lagrangian that contains time derivatives higher than the first order and is non-degenerate with respect to the highest-order derivatives leads to an unbounded Hamiltonian linear in the canonical momenta; > s.a. types of higher-order gravity theories.
* Ostrogradski instability / ghost: A classical, linear instability in a higher-derivative theory, that one finds from the Hamiltonian constructed using Ostrogradski's method; However, when multiple fields are present the existence of higher derivatives does not automatically imply the existence of ghosts.

Non-Local Theories > s.a. non-commutative physics and gauge theories; partial differential equations.
* 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.
@ Time-non-local theories: Ferialdi & Bassi EPL(12)-a1112; Heredia & Llosa a2105 [Noether theorem and Hamiltonian]; > s.a. locality.
@ Related topics: Soussa ht/03-proc [causality]; Calcagni et al PRD(07)-a0705 [FLRW + scalar cosmology]; Thieme a2009 [Lagrangian densities depending on pairs of points]; > s.a. FLRW spacetimes.

Other Specific Types of Theories > s.a. gauge transformations; higher-order gravity; oscillators; spinning particles.
* Result: If a Lagrangian exists for an equation of any even order, then it can be derived from the Jacobi last multiplier.
@ General references: Nucci & Arthurs PRS(10) [inverse problem for 4th-order equations]; Motohashi & Suyama PRD(15)-a1411 [3rd-order equations of motion]; Motohashi et al JPSJ(18)-a1711 [ghost-free theory, Lagrangian with third-order time derivatives].
@ Relativistic particle: Beau a1305 [consequences, generalized induction principle and generalization of the concept of inertia].
@ Field theories, second-order: Grigore FdP(99)ht/96 [trivial second-order Lagrangians]; de Urries et al JPA(01) [bosonic, as constrained second-order]; Rosado & Muñoz a1509 [admitting a first-order Hamiltonian formalism].
@ Multiple fields: de Rham & Matas a1604 [gravitational theories like massive gravity and beyond Horndeski].
@ Field theories, other: in Stelle GRG(78) [gravity]; de Urries & Julve JPA(98) [scalar]; Villaseñor JPA(02)ht [fermionic]; Bazeia et al JPA(03) [2+1, dualities]; Nguyen a0807 [principle of least action]; Li et al IJTP(08) [gauge theories]; Campos et al JPA(09)-a0906-conf [unambiguous intrinsic formalism]; Mukherjee & Paul PRD(12)-a1111 [gauge invariances]; Pulgar et al JCAP(15)-a1408 [cosmological scalar field, inspired by the Pais-Uhlenbeck oscillator]; Izadi & Moayedi AP(19)-a1903 [infinite-derivative scalar field]; > s.a. constrained theories.

Hamiltonian Formulation > s.a. hamiltonian systems; Pais-Uhlenbeck Model.
@ General references: Coelho de Souza & Rodrigues JPA(69); Jaén et al PRD(87), JMP(89); Llosa & Vives JMP(94); Rashid & Khalil IJMPA(96); Woodard PRA(00)ht/00 [finite non-locality]; Bering ht/00; Muslih & El-Zalan IJTP(07); Andrzejewski et al a0710; Morozov TMP(08)-a0712 [brief review]; El-Zalan et al IJTP(08); Vitagliano JGP(10)-a0905; Gegelia & Scherer JPA(10)-a1003 [vs Lagrange formalism, and quantum corrections]; Martínez et al IJMPA(11)-a1104 [perturbative Hamiltonian constraints]; Avraham & Brustein PRD(14)-a1401 [generalized Legendre transform].
@ Ostrogradski procedure: Ostrogradskii MASP(1850) [momenta]; Woodard a1506-en [attempts to avoid the instability]; Massa et al IJGMP(18)-a1610 [new geometrical look]; Öttinger JPcomm(18)-a1810 [for fourth-order evolution equations]; Öttinger a1906 [alternative procedure without instabilities]; Donoghue & Menezes a2105 [the instability may be avoided in quantum theory].
@ Field theories: Belvedere et al ZPC(95) [canonical transformations]; Cheng et al NPB(02) [and non-commutative field theory].
@ Infinite-derivative theories: Talaganis & Teimouri a1701 [dynamical degrees of freedom]; Teimouri a1811-PhD.
@ Second-order Lagrangians: Hahne a1306 [from Feynman's path integral]; Cruz et al JMP(16)-a1310 [acceleration-dependent]; Esen & Guha a1607 [Ostrogradsky-Legendre and Schmidt-Legendre transformations].
@ Other types of theories: Schmidt gq/95 [4th-order]; Dunin-Barkowski & Steptsov TMP(09)-a0801 [reparametrization-invariant].
@ Ostrogradski instability: Niedermaier AP(12) [quantum cure]; Chen & Lim JCAP(13)-a1209 [with constraints, strengthening of Ostrogradski's theorem]; Chen et al JCAP(13) [removal by adding constraints].
@ Modified Ostrogradski formulation: Leclerc gq/06; Andrzejewski et al PRD(10)-a1005; Patra et al PRI(14)-a1412 [and Regge-Teitelboim cosmology].

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]; Miron 03-a1003; Prieto-Martínez & Román-Roy JPA(11)-a1106, a1201-conf [Lagrangian-Hamiltonian unified formalism]; Kijowski & Moreno IJGMP(15)-a1408 [symplectic structures].
@ Noether theorem, symmetries: Miron IJTP(95); de León & Martín de Diego JMP(95); Sardanashvily mp/03.
@ Geometrical: de León & Lacomba JPA(89) [in terms of symplectic higher-order tangent bundles]; Prieto-Martínez PhD-a1410.
@ 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]; Kaparulin et al EPJC(14)-a1407 [systems with bounded integral of motion that ensures their stability].

Quantization [> s.a. formulations of quantum mechanics and quantum systems.]
@ General references: Hayes JMP(69); Tesser JMP(72); Acatrinei JPA(07)-a0708 [from phase space path integrals]; Andrzejewski et al PRA(07); Nucci TMP(11); Baaquie IJMPA(13)-a1211, IJMPA(13)-a1211 [action with acceleration term]; Raidal & Veermäe NPB(17)-a1611 [complex classical mechanics and avoiding the Ostrogradsky ghost]; Smilga IJMPA(17)-a1710 [benign ghosts and the Theory of Everything]; Motohashi & Suyama a2001 [quantum Ostrogradsky theorem].
@ In quantum field theory: Pais & Uhlenbeck PR(50); > s.a. Pais-Uhlenbeck Model; path integrals; quantum oscillators.

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