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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