Lagrangian
Dynamics |

**In General** > s.a. Euler-Lagrange Equations;
formulations of quantum theory; lagrangian systems.

* __Idea__: A formulation of dynamics
based on the principle of stationary action (often, 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* = \(int\)*p *d*q*,
and improved by Euler and Lagrange; In 1832 Hamilton proposed the action
*S* = ∫* L*(*q*,
*q*^{·}) d*t*.

* __Motivation__: (1) It is an elegant
way of expressing the dynamical content of a physical theory; (2) Convenient for the study of
symmetries of the theory; (3) Important for the path 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) Given a region *X* of the *x*s, find a Lagrangian (density)
\(\cal L\) or an action functional
*S*[*q*, ∂*q*,* x*], written as

*S*[*q*, ∂*q*, *x*]
= ∫_{M}
\(\cal L\)(*q*, ∂*q*, *x*) d*x* ;

(3) Derive the equations of motion using variational principles; Physical trajectories are extrema of the action.

* __For Newtonian mechanics__: The true trajectory between two spacetime events is a minimum of the action if the final event occurs before the kinetic focus of the initial event; Otherwise, it is a saddle point of the action.

* __For a field
theory__:
A Lagrangian of order *k* is a horizontal
form *L*: *J*^{ k}*Y* → Λ^{n}*M*,
where (*Y*, *π*, *M*)
is the configuration bundle for the theory, (*J*^{k}*Y*,
*π*^{k}, *M*)
its *k*-jet prolongation, and *M* the spacetime manifold; A
variation of the fields is a vertical vector field *X* = δ*y*^{i} (∂/∂*y*^{i}),
under which the Lagrangian varies by

δ_{X}*L* = *e*(*L*, *X*)
+ d*F*(*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*: T*Q* → T**Q*
(*Q* is configuration space) defined, for a Lagrangian *L*, by *F*(*w*)*v*:=
d*L*(*w*+*tv*)/d*t*|_{t=0}.

@ __General references__: Kastrup PRP(87);
Bertin et al MPLA(05)ht [first-order actions, Hamilton-Jacobi analysis];
Compère PhD(07)-a0708
[covariant Hamiltonian formulation, symmetries, and black holes]; Arcuş a1108 [Legendre duality between Lagrangian and Hamiltonian Mechanics].

@ __For field theories__: Giachetta et al 97; Echeverría-Enríquez et al IJMMS(02)mp/01, JMP(04)
[unified formalism];
Geyer et al JPA(03)ht/02 [and
gauge symmetries].

@ __Related
topics__: Duplij JMPAG-a1002 [degenerate Lagrangian theories, Clairaut-type formalism]; Gallardo IJMPA(12) [timelike boundary terms].

& __Related
topics__: R Skinner and R Rusk, Lagrangian-Hamiltonian unified formalism.

**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 divergence to \(\cal L\),
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 91; Faraoni CQG(94);
Cisło & Łopuszański JMP(01)mp/00 [and
quantization]; Nucci JPCS(12)-a1202 [Jacobi's method for finding Lagrangians, and choices that can be used for quantization].

@ __Lagrangian cohomology classes__: in Marathe & Martucci 92;
in Kolár et al 93.

@ __Related topics__: Frieden 98 [Lagrangians 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]; Torre AIP(11)-a1011 [symmetric criticality in field theory]; Deriglazov & Rizzuti PRD(11)-a1105 [extended Lagrangian formalism as a systematic procedure to look for local symmetries].

@ __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];
Wu PRD(09).

@ __Lagangians from Lie transformation groups__: Paal & Virkepu in(09)-a0706;
Nucci & Leach JMP(07)-a0706 [Jacobi
last multiplier].

**References** > s.a. classical mechanics; history of physics.

@ __Textbooks__: Lagrange 1788;
in Sudarshan & Mukunda 75; Calkin 96 [+
solutions 99];
Curry 13, Hamill 13, Brizard 15 [II]; Nolting 16.

@ __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]; Gondran & Gondran a1203 [Euler-Lagrange and Hamilton-Jacobi actions]; Lachièze-Rey a1411 [history-based]; in Cortés & Haupt book(17)-a1612 [lecture notes, mathematical].

@ __Conceptual__: Terekhovich a1511 [and the Humean view of the laws of nature, Leibniz's concept of the possibles].

@ __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]; Pons JMP(10) [substituting fields in the action];
Gray & Poisson AJP(11)jan [for geodesic worldlines in a curved spacetime];
Bekenstein & Majhi NPB(15)-a1412 [the least action principle is not necessary].

@ __Geometrical__: Cariñena et al JMP(88);
del Olmo & Santander
JPA(89),
JGP(90);
Sharipov
m.DG/01 [in
Riemannian manifolds]; Miron a1203 [geometrization];
Meng JPA(15)-a1405 [Tulczyjew's approach, for charged particles in gauge fields].

@ __And spacetime transformations__: Whiston IJTP(72).

@ __Inverse variational principle__: Santilli 78; Okubo PRD(80) [non-unique];
Cariñena et al PRP(95);
Mestdag et al DG&A(11) [with non-conservative forces];
Saha & Talukdar a1301 [non-standard Lagrangians].

@ __Complex action__: Gocksch PLB(88);
Anagnostopoulos & Nishimura PRD(02);
Nielsen a0911-proc [and retrocausation];
Nagao & Nielsen PTP(11)-a1104, IJMPA(12)-a1105.

@ __Other generalizations__: Grigore IJMPA(92);
Núñez-Yépez et al mp/01-in;
Arizmendi
et al CSF(03)mp/04;
Struckmeier IJMPE(09)-a0811 [extended Hamilton-Lagrange formalism]; Gravanis & Willison JMP(09)-a0901 [distributional fields]; Sardanashvily IJGMP(13) [Grassmann-graded, and reducible degenerate systems]; Polonyi PRD(14)-a1407 [closed-time-path extension of effective theories]; Minguzzi JGM(15)-a1410 [covariant, including friction, non-holonomic constraints and energy radiation];
Surawuttinack et al a1502 [multiplicative form].

@ __Related topics__: Ferraris et al JMP(00)
[dual Lagrangians]; Muñoz Díaz a0801 [and
time]; López et al CiM-a1205 [minima vs critical points, and techniques from non-linear analysis]; Neiman a1212 [with null boundary data];
Díaz et al JMP(14)-a1406 [physical degree of freedom count];
> s.a. force; morse theory [on spaces of braids].

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