Constrained Systems |

**In General** > s.a. lagrangian dynamics
/ types of constrained systems.

* __Lagrangian form__:
They arise when the momenta obtained from varying the action are not all
independent functions of (*q*^{i},
*q*^{·}^{i}),
e.g., some vanish; If we don't add the appropriate *N*(constraint) to *H*
in the Legendre transformation, we don't get the most general possible motion.

* __Hamiltonian form__: We first
determine the constraints (primary, *φ*_{a}
first-class, *φ*_{m} second-class,
secondary *ψ*_{i} and
*ψ*_{i}) from the
Lagrangian formulation; To determine the dynamics, either (a) Modify the
Hamiltonian \(H = p\,\dot q - L\) to

*H* = *p* *q*^{·}
+ ∑_{a}
*λ*_{a}
*φ*_{a}
+ ∑_{m}
*U*_{m}
*φ*_{m} − *L* ;

Here, the *λ*s are arbitrary Lagrange multipliers, and the *U*s
are determined by consistency; or (b) Use \(H = p\,\dot q - L\) and the Dirac bracket

d*f* / d*t* = {*H*, *f*}_{D}
= {*H*, *f*} + ∑_{a}
*λ*_{a}
{*φ*_{a}, *f*}
+ ∑_{M, N}
{*H*, *K*_{M}}
*C*_{MN}
{*K*_{N}, *f*} .

* __Remark__: If *p*_{A}
= 0 and *p*_{B} = *A*,
then those constraints just tell us that *A* and *B* are canonically
conjugate.

* __Symplectic form__: A constraint
for a system with a phase space Γ is a subset Γ' of Γ such that
all physical states have to be in Γ'; It can be specified by the vanishing of
some functions *C*_{i}, *i*
= 1, ..., *r* on Γ.

**Reduction**

* __Idea__: The process by
which a theory with constraints and redundant/non-physical variables is
rewritten as an equivalent unconstrained one, without redundant variables,
or the separation of the variables in a theory into true degrees of freedom
and gauge degrees of freedom; It can be achieved by solving the constraints,
or by fixing the gauge if appropriate.

* __Faddeev-Jackiw approach__:
An approach in which constraints are solved; Formally, an approach in which
gauge and reparametrization symmetries are generated by the null eigenvectors
of the sympletic matrix and not by constraints, which provides a way of
dealing systematically with hidden symmetries even when the constraints
do not act as the generators of the corresponding transformations.

* __Different approaches__: The
Dirac and Faddeev-Jackiw approaches are equivalent when the constraints are
effective, but when some are ineffective the Faddeev-Jackiw approach may
lose some constraints or some equations of motion; The inequivalence may be
related to the failure of the Dirac conjecture in this case; An alternative
to both of those approaches is to fix a gauge.

@ __General references__: Cariñena et al IJGMP(07)-a0709-conf [rev];
Falceto et al NCC(13)-a1309,
NCC(13)-a1309 [classical and quantum systems, using Lie-Jordan algebras].

@ __Faddeev-Jackiw approach__: in Sudarshan & Mukunda 75;
Faddeev & Jackiw PRL(88);
refs in Seiler JPA(95);
Wotzasek AP(95)ht [and Polyakov 2D induced gravity];
García & Pons IJMPA(97),
IJMPA(98)ht [compared to Dirac approach, gauge theories];
Natividade et al PRD(99) [examples];
> s.a. symmetries in quantum theories.

@ __Other approaches__:
Banerjee & Barcelos-Neto AP(98) [reducible systems];
Marciniak & Blaszak RPMP(05) [geometric reduction];
Attard et al a1702 [dressing field method].

**References**
> s.a. classical mechanics; hamiltonian dynamics
[symmetries]; lagrangian systems; observables.

@ __Books__: Hanson et al 76;
Sundermeyer 82;
Papastavridis 14.

@ __General__: Śniatycki AIHP(74);
Román-Roy IJTP(88);
Cariñena FdP(90);
Lusanna JMP(90),
PRP(90),
RNC(91);
Charap ed-95;
Pons & Shepley CQG(95)gq;
Tulczyjew in(03)mp/06 [holonomic];
Randono CQG(08)-a0802 [Lagrangian form, and generally covariant systems];
Deriglazov JMP(09)-a0901 [and symmetries];
Date a1010-ln [intro]; Errasti et al a2007 [first-order field theories, including gravity].

@ __Hamiltonian form__: Dirac PRS(58);
Shanmugadhasan JMP(73);
Gogilidze et al IJMPA(89);
Barbashov ht/01-conf [and Lagrangian];
Gitman & Tyutin NPB(02)ht [from Lagrangian];
Duplij in(09)-a0804 [Legendre transform without Lagrange multipliers, Clairaut equation];
Bekaert & Park EPJC(09)-a0902 [rev];
Rothe & Rothe 10;
Cendra et al a1106 [Poisson-algebraic and geometric points of view];
Chandre JPA(13)-a1303 [Casimir invariants and the Jacobi identity];
Allen & Matzner a2007 [Dirac formalism, rev];
> s.a. types of constrained systems.

@ __Hamiltonian form, history__:
Salisbury phy/06-proc [Bergmann],
MGXI(08)phy/07 [Rosenfeld];
Salisbury & Sundermeyer EPJH(17)-a1606 [Rosenfeld].

@ __And gauge symmetries__: Stoilov ht/06 [re gauge algebra];
Banerjee & Roy PRD(11) [Poincaré and Hamiltonian gauge symmetries].

@ __Lagrange multipliers__:
Cariñena & Rañada JPA(93);
Montani & Montemayor PRD(98) [symplectic structure];
Karabulut EJP(06) [physical meaning];
Mazars JPA(07) [analytic computation].

@ __Dynamics, meaning__: Uwadia & Kalaba PRS(92),
s.a. Bucy PRS(94);
Grundling & Hurst JMP(98)ht/97 [constraints not preserved];
Gràcia et al mp/00,
Massa et al IJGMP(15)-a1503 [variational principles and geometry];

Udwadia & Phohomsiri PRS(07),
PRS(07) [Poincaré equations of motion].

@ __Approaches__:
Sardanashvily mp/00-conf [polysymplectic];
Gitman & Tyutin NPB(02);
de León et al mp/02 [field theories];
Díaz & Montesinos JMP(18)-a1710
[field theory, geometric Lagrangian approach to counting the physical degrees of freedom].

@ __Related topics__: Pavlov & Starinetz TMP(95) [phase space geometry];
Marmo et al JPA(97) [as implicit differential equations];
Sheikh-Jabbari & Shirzad EPJC(01)ht/99 [and boundary conditions];
Bojowald & Strobl RVMP(03))ht/01 [Poisson geometry];
Rothe PLB(03)ht [dynamics from limit of unconstrained];
Di Bartolo et al JMP(05)gq/04 [discretizations];
> s.a. hamilton-jacobi theory; noether's
theorem; symmetries.

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