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
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;
> s.a. locality.

@ __Related topics__: Soussa ht/03-proc
[causality];
Calcagni et al PRD(07)-a0705
[FLRW
+ scalar cosmology]; > 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].

@ __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];
> 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 a1610 [new geometrical look].

@ __Field theories__: Belvedere et al ZPC(95)
[canonical transformations];
Cheng et al NPB(02)
[and non-commutative field theory]; Mazumdar et al a1701 [infinite-derivative theories, dynamical degrees of freedom].

@ __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 a1611 [complex classical mechanics and avoiding the Ostrogradsky ghost].

@ __In quantum field theory__: Pais & Uhlenbeck PR(50);
>
s.a. Pais-Uhlenbeck Model; path
integrals; quantum
oscillators.

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