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].

@ __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]; > s.a. types of constrained systems.

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