First-class constraint: Difference between revisions

Revision as of 13:03, 17 November 2009

Revision as of 05:05, 17 December 2009

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==Poisson brackets==

In [[Hamiltonian mechanics]], consider a [[symplectic manifold]] ''M'' with a [[smooth function|smooth]] Hamiltonian over it (for field theories, ''M'' would be infinite-dimensional).

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==Geometric theory==

For a more elegant way, suppose given a [[vector bundle]] over M, with ''n''-dimensional [[fiber]] ''V''. Equip this vector bundle with a [[connection form|connection]]. Suppose too we have a [[smooth section]] ''f'' of this bundle.

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==Intuitive meaning==

What does it all mean intuitively? It means the Hamiltonian and constraint flows all commute with each other '''on''' the constrained subspace; or alternatively, that if we start on a point on the constrained subspace, then the Hamiltonian and constraint flows all bring the point to another point on the constrained subspace.

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One way to get around this is this: For reducible constraints, we relax the condition on the right invertibility of Δ''f'' into this one: Any smooth function which vanishes at the zeros of ''f'' is the fiberwise contraction of ''f'' with (a non-unique) smooth section of a <math>\bar{V}</math>-vector bundle where <math>\bar{V}</math> is the [[dual vector space]] to the constraint vector space ''V''. This is called the ''regularity condition''.

== Constrained Hamiltonian dynamics from a Lagrangian gauge theory ==

First of all, we will assume the [[action (physics)|action]] is the integral of a local [[Lagrangian]] which only depends up to the first derivative of the fields. The analysis of more general cases, while possible is more complicated. When going over to the Hamiltonian formalism, we find there are constraints. Recall that in the action formalism, there are [[on shell]] and [[off shell]] configurations. The constraints which hold off shell are called primary constraints while those which only hold on shell are called secondary constraints.

== Examples ==

Look at the dynamics of a single point particle of mass ''m'' with no internal degrees of freedom moving in a [[pseudo-Riemannian]] spacetime manifold ''S'' with [[metric tensor|metric]] '''g'''. Assume also that the parameter τ describing the trajectory of the particle is arbitrary (i.e. we insist upon [[Parametric_curve#Reparametrization_and_equivalence_relation|reparametrization invariance]]). Then, its [[symplectic space]] is the [[cotangent bundle]] T*S with the canonical symplectic form ω. If we coordinatize ''T'' * ''S'' by its position ''x'' in the base manifold ''S'' and its position within the cotangent space '''p''', then we have a constraint

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See also [[Dirac bracket]], [[second class constraints]], [[BRST]], [[analysis of flows]]

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[[Category:Classical mechanics]]

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[[Category:Theoretical physics]]