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Second class constraints

See first class constraints for the preliminaries.

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 λ = 0

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

The Hamiltonian is given by

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

We can't eliminate $\dot{\lambda}$ at this stage yet. We are here treating $\dot{\lambda}$ as a shorthand for a function of the symplectic space which we have yet to determine and not an independent 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 brackets of these constraints amongst themselves do not vanish on the constrained subspace, we have second class constraints. In fact, since for every constraint (and also every nonzero linear combination of constraints), we have at least one other constraint whose Poisson bracket with it doesn't vanish on the constrained subspace, all the four constraints are second class constraints and we have no first class constraints. Note that these constraints satisfy the regularity condition.

Here, we have a symplectic space where the Poisson bracket bracket doesn't have "nice properties" on the constrained subspace. But Dirac noticed that we can turn the underlying differential manifold of the symplectic space into a Poisson manifold using a different bracket, called the Dirac bracket, such that the Dirac bracket of any (smooth) function with any of the second class constraints always vanishes and a couple of other nice properties.

Categories: Theoretical physics

Science Fair Project Encyclopedia Contents Page

10-26-2009 08:16:03
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Second class constraints