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 φmL ;

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.

* 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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