First class and second class constraints

Question

Hello I am working on a project that involves the constraints. I checkout the paper of Dirac about the constraints as well as some other resources. But still confuse about the first class and second class constraints.

Suppose, from the Lagrangian, I found two primary constraints $\Phi_a$ and $\Phi_b$. Let $\dot\Phi_a$ and $\dot\Phi_b$ both leads to the secondary constraints $\Sigma_a$ and $\Sigma_b$ respectively. From the consistency condition $\dot\Sigma_a$ leads to a tertiary constraints $\Theta_a$ but $\dot\Sigma_b$ become zero.

Now How can I check which one of them are first-class and which one of them are second-class?

Answer 1

(1) You have a set of irreducible constraints, $\lbrace \phi_j\rbrace$, both primary and secondary This set of constraints defines a submanifold $M$ within the "full" (unconstrained) phase space.

(2) A function on the phase space is set to be weakly zero if it vanishes when restricted to the constrained submanifold $M$. A function is called strongly zero if its derivatives with respect to the unconstrained phase space coordinates are weakly zero. By definition the constraints are weakly zero, $\phi_j \approx 0$, but not necessarily strongly zero.

(3) A function $F$ defined on the full phase-space is called a first-class function if its Poisson brackets with all constraints vanish weakly. So $F$ is first class if

$$\lbrace F,\phi_j \rbrace \approx 0 $$

for all constraints $\phi_j$. A function is called second class if it's not first class, i.e. if it has one or more non-weakly vanishing Poisson brackets with the constraints.

(3') Just as a reminder: the derivatives in the Poisson bracket are calculated in the full phase space, i.e. the momenta and coordinates $(p,q)$ are treated as independent, such that you can calculate the derivatives $\delta F/\delta q$ and $\delta F / \delta p$. Then after this differentiation you apply the constraint equations to see if the Poisson bracket vanishes weakly.

(4) Then finally: a constraint is first or second class if all its Poisson bracket with the remaining constraints vanish weakly.

(5) Second class constrained are not too difficult to deal with (i.e. when quantizing the system). First class constraints form a much larger obstacle. They are the generators of gauge transformations.

I can highly recommend the book by Hennaux and Teitelboim.

Answer 2

The first step is the completion of the whole constraint list (primary, secondary , ternary etc.), and checking that no secondary constraint leads to a contradiction (i.e., empty constraint surface).

Remark: The time evolution of the constraints is performed with the "total" Hamiltonian:

$H_T(p,q) = p\dot{q}-L+\sum_{\alpha}\lambda_{\alpha} \phi^{(1)}_{\alpha} $

where $\phi^{(1)}_{\alpha}$ are the primary constraints and $\lambda_{\alpha}$ are Lagrange multipliers.

The next step is the computation of the Poisson bracket matrix of all constraints:

$P_{\alpha\beta} = \{\phi_{\alpha}, \phi_{\beta}\}|_{\Sigma}$

(for all constraints $\phi_1, ., ., ., \phi_n$). $\Sigma$ is the constraint surface.

The number of first class constraints is equal to the corank of the matrix $P$: $n-\mathrm{rank}(P)$.

Answer 3

In the answer below I will not go into relation between Lagrangian and Hamiltonian formalism for the case of constrained systems but will simply restrict to the meaning of constraints (the way I understand it) in Hamiltonian formalism :-

Suppose you are given with a phase space $P$ with variables $(q_1,..,q_n;p_1,..,p_n)$. Ordinarily one deals with "unconstrained systems". Which means you are only given with a Hamiltonian $H$; and all solutions to equations of motion are allowed. That is there is no restriction on dynamics.

For a constrained system, besides the Hamiltonian $H$ you also have a set of constraint equations :

$\tag{1}\phi_k(q_1,..,q_n;p_1,..,p_n)=0,\quad k=1,...,m.$

along with the conditions

$\{\phi_i,H\} =0\quad\mbox{on the surface of constraints for all $i=1,...,m$}\tag{2}$

What it means it that now you have to restrict yourself to the dynamics on the surface (i.e. submanifold) in phase space defined by (1). Conditions (2) make sure that constraints are consistent with the dynamics, i.e. if you begin from a point $(q_{i0},p_{i0})$ on the surface of constraint then its time evolution too would lie on the surface.

Now (to the extent I understand it) idea is to differentiate between those constraints which do not pose much problem and can be eliminated (called second class constraints), as compared to those which are somewhat more difficult to deal with (first class constraints). The procedure to define them is as follows:

Find maximal subset {$\phi_1,...,\phi_k$} of set {$\phi_1,...,\phi_n$} of constraint functions such that Poisson bracket of any function in {$\phi_1,...,\phi_k$} with a function in {$\phi_1,...,\phi_n$} is linear combination (with coefficients any arbitrary functions of $p_i,q_i$) of functions $\phi_1,...,\phi_n$. Constraints corresponding to {$\phi_1,...,\phi_k$} are called first class constraints while those corresponding to {$\phi_{k+1},...,\phi_n$} are called second class constraints.

Dirac showed that surface of constraint defined by second class constraints {$\phi_{k+1},...,\phi_n$} is a symplectic submanifold $M$ of phase space $P$. What it means is that (except for the dimension) locally $M$ would look just like phase space $P$ (if you choose right coordinates) and so dynamics on it can be studied like in usual phase space without constraints.

So one can simply eliminate second class constraints by restricting to $M$ as the new phase space, and the rest of functions {$\phi_1,...,\phi_k$} (restricted to $M$) as new constraints.

Answer 4

If a constraint vanishes Poisson bracket with all other constraints, it is first-class, otherwise second-classs.