Constrained
Systems| Constrained
Systems |
In General > s.a. [hamiltonian dynamics]; lagrangian
systems; 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, 
a
first-class, m second-class,
secondary 
i and
i)
from the Lagrangian formulation; To determine the dynamics, either
(a) Modify the Hamiltonian H = p q· – L to p q·
+
a 
aa +
m Umm –
L; The 's
are arbitrary Lagrange multipliers, and the U's
are determined by consistency; or
(b) Use H = p q· – L and
the Dirac bracket
df / dt = {H, f}D =
{H, f} + a a
{a, 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 
is
a subset
' of such
that all physical states have to be in ';
It can be specified by the vanishing of some functions Ci, i =
1, ..., r on
.
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; 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-in
[rev].
@ 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].
References > s.a. classical
mechanics; hamiltonian
dynamics; lagrangian
systems; observables.
@ Books: Hanson et al 76; Sundermeyer 82; Papastavridis 02.
@ General: Sniatycki 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 mp/06-in
[holonomic]; Randono a0802 [Lagrangian form, and generally covariant systems].
@ Hamiltonian form: Dirac PRS(58);
Shanmugadhasan JMP(73);
Gogilidze et al IJMPA(89);
Barbashov ht/01-in
[and Lagrangian]; Gitman & Tyutin NPB(02)ht [from
Lagrangian]; Salisbury phy/06-in
[Bergmann's contribution], phy/07-in
[history, Rosenfeld]; Duplij a0804 [Legendre
transform without Lagrange multipliers, Clairaut
equation].
@ Lagrange multipliers: Cariñena & Rañada JPA(93);
Montani & Montemayor PRD(98)
[symplectic structure]; Karabulut EJP(06)
[physical meaning]; Mazars JPA(07) [analytic computation].
@ Equations of motion, dynamics and meaning: Uwadia & Kalaba PRS(92),
s.a. Bucy PRS(94);
Grundling & Hurst JMP(98)ht/97 [constraints
not preserved]; Gràcia et al mp/00 [variational
principles
and geometry]; Udwadia & Phohomsiri PRS(07), PRS(07)
[Poincaré equations
of
motion].
@ Approaches: Sardanashvili mp/00-in
[polysymplectic]; Gitman & Tyutin NPB(02);
de León et al mp/02 [field
theories].
@ 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];
Stoilov ht/06 [re
gauge algebra]; > s.a. hamilton-jacobi, Noether
Theorem, symmetries.
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5 jul 2008