Constrained Hamiltonian systems

A primary constraint follows directly from the momentum definitions before equations of motion are used. For example, if L contains no velocity of a multiplier λ, its momentum πλ vanishes. The canonical Hamiltonian is supplemented with undetermined multipliers times primary constraints to form the total Hamiltonian. Requiring each primary constraint to remain zero under this evolution may determine a multiplier, produce a secondary constraint, or yield an identity. The new constraints must then be checked in turn.

A weak equality means equality only on the constraint surface. Compute Poisson brackets before setting constrained expressions to zero: their derivatives can be nonzero transverse to the surface. Replacing a constraint by zero too early would make its brackets vanish artificially and erase the very equations that determine consistent motion. The procedure also requires regularity and independence checks; a redundant list or changing rank can make a naive count of equations and variables misleading.

Primary versus secondary describes when a constraint is discovered. First class versus second class describes its Poisson brackets with the full constraint set. A first-class constraint has brackets that vanish weakly with all constraints. For a second-class set, the bracket matrix is invertible after an appropriate independent set is chosen. A constraint can be primary and second class, or secondary and first class; these labels answer different questions and should not be identified with one another.

In regular gauge systems, first-class chains are associated with gauge redundancy, but a complete gauge generator can involve linked primary and secondary constraints and derivatives of gauge parameters. It is unsafe to declare that every constraint independently generates a distinct arbitrary gauge function. For a regular independent set with no reducibility complications, physical degrees of freedom can be counted by subtracting one configuration degree per first-class constraint and one half per second-class constraint from the original configuration count.

For an invertible second-class bracket matrix C, subtract a correction from the original Poisson bracket. The resulting Dirac bracket makes the selected constraints bracket to zero with every function. After that modification they may be imposed strongly in the reduced description. This is not the same operation as inserting them into an ordinary Poisson bracket at the start. The correction changes the algebra so that Hamiltonian evolution remains tangent to the constrained surface.

A particle restricted to a circle illustrates the geometry. A radial position constraint is accompanied by a radial momentum constraint so that motion remains tangent. Cartesian position and momentum components then acquire projected brackets rather than the original unrestricted delta pairing. Locally one can instead use an angle and its conjugate angular momentum as a reduced canonical pair. The reduction removes forbidden radial motion while preserving the tangential physical degree of freedom; the circle does not force the particle to remain at a fixed angle.