Second class constraints

See first class constraints for the prelimanaries.

Before going on to the general theory, let's look at a specific example step by step to motivate the general analysis.

Let's start with the action describing a Newtonian particle of mass m constrained to a surface of radius R within a uniform gravitational field g.

The action is given by

$S=\int dt{\mathcal {L}}=\int dt\left[{\frac {m}{2}}({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2})-mgz+{\frac {\lambda }{2}}(x^{2}+y^{2}+z^{2}-R^{2})\right]$

where the last term is the Lagrange multiplier term enforcing the constraint.

Of course, we could have just used different coordinates and written it as

$S=\int dt\left[{\frac {mR^{2}}{2}}({\dot {\theta }}^{2}+\sin ^{2}(\theta ){\dot {\phi }}^{2})+mgR\cos(\theta )\right]$

instead, but let's look at the former coordinatization.

The conjugate momenta are given by

$p_{x}=m{\dot {x}}$, $p_{y}=m{\dot {y}}$, $p_{z}=m{\dot {z}}$, $p_{\lambda }=0$.

Note that we can't determine ${\dot {\lambda }}$ from the momenta.

The Hamiltonian is given by

$H={\vec {p}}\cdot {\dot {\vec {r}}}+p_{\lambda }{\dot {\lambda }}-{\mathcal {L}}={\frac {p^{2}}{2m}}+p_{\lambda }{\dot {\lambda }}+mgz-{\frac {\lambda }{2}}(r^{2}-R^{2})$.

Note that we can't eliminate ${\dot {\lambda }}$ at this stage yet. But here, we're treating ${\dot {\lambda }}$ as a shorthand for a function of the symplectic space which we have yet to determine and NOT an independant variable.

We have the off shell primary constraint p_λ=0.

We require that the Poisson bracket of all the constraints with the Hamiltonian vanish at the constrained subspace.

from this, we immediately get the secondary constraint r²-R²=0.

And from the secondary constraint, we get the tertiary constraint ${\vec {p}}\cdot {\vec {r}}=0$.

And from the tertiary constraint, we get the quartanary constraint

${\frac {p^{2}}{m}}+\lambda R^{2}-mgz=0$.

And finally, from the quartanary constraint, we get

$3gp_{z}-{\dot {\lambda }}R^{2}=0$

from which we deduce

${\dot {\lambda }}={\frac {3gp_{z}}{R^{2}}}$.

Putting it all together,

$H={\frac {p^{2}}{2m}}+{\frac {3gp_{\lambda }p_{z}}{R^{2}}}+mgz-{\frac {\lambda }{2}}(r^{2}-R^{2})$

with the constraints

p_λ=0, r²-R²=0, ${\vec {p}}\cdot {\vec {r}}=0$, ${\frac {p^{2}}{m}}+\lambda R^{2}-mgz=0$.

Since the Poisson bracket of these constraints among themselves do not vanish on the constrained subspace, we have second class constraints. Note that these constraints satisfy the regularity condition.