Constrained Systems| 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 (qi,
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, φa
first-class, φm second-class,
secondary ψi and
ψi) from the
Lagrangian formulation; To determine the dynamics, either (a) Modify the
Hamiltonian \(H = p\,\dot q - L\) to
H = p q· + ∑a λa φa + ∑m Um φm − L ;
Here, the λs are arbitrary Lagrange multipliers, and the Us are determined by consistency; or (b) Use \(H = p\,\dot q - L\) and the Dirac bracket
df / dt = {H, f}D = {H, f} + ∑a λa {φa, f} + ∑M, N {H, KM} CMN {KN, f} .
* Remark: If pA
= 0 and pB = 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 Ci, 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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