Question 8.12: (Sundermeyer, 1982) Apply the Dirac-Bergmann algorithm to th......

The Hessian matrix

W = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}   (8.215)

is singular. The canonical momenta are

p_{1}= \frac{\partial L}{\partial\dot{q}_{1} }= \dot{q}_{1}+q_{2},  p_{2}=\frac{\partial L}{\partial\dot{q}_{2} }= \left(1 − \alpha \right) q_{1},   (8.216)

and give rise to the primary constraint

\phi_{1}= p_{2}+\left(\alpha -1\right) q_{1} \approx 0.   (8.217)

The Hamiltonian is given by

H =\frac{1}{2}\left(p_{1} -q_{2}\right)^{2} – \frac{\beta}{2}\left(q_{1} -q_{2}\right)^{2}.  (8.218)

The requirement that the primary constraint be preserved in time takes the form of the consistency condition

0 \approx \left\{\phi _{1},H\right\} + \lambda \left\{\phi _{1},\phi _{1}\right\} = \alpha \left(p_{1} -q_{2}\right)- \beta \left(q_{1} -q_{2}\right).   (8.219)

Situation (a): α = β = 0. The consistency condition is identically satisfied and there are no secondary constraints. The total Hamiltonian is H_{T} = H +{\lambda}{\phi _{1}} and Hamilton’s equations are

\dot{q}_{1}=p_{1} -q_{2} , \dot{q}_{2}=\lambda , \dot{p}_{1}=\lambda , \dot{p}_{2}= p_{1} -q_{2}.  (8.220)

Since λ is arbitrary, it follows that q_{2} is an arbitrary function of time. On the other hand, from (8.220) one derives

\ddot{q}_{1}=\lambda-\lambda=0 \Longrightarrow q_{1} = at + b ,  (8.221)

with a and b arbitrary constants. It is easy to understand the why and wherefore this is the general solution to the equations of motion by noting that

L = \frac{1}{2}\dot{q}^{2}_{1}+ \frac{d}{dt} \left(q_{1}q_{2}\right).  (8.222)

Inasmuch as Lagrangians that differ by a total time derivative yield the same equations of motion, we are led to conclude that the system is equivalent to a free particle with only one degree of freedom. The variable q_{2} does not represent a physical degree of freedom of the system: it is arbitrary and can be discarded. The appearence of an arbitrary function of time is in agreement with the general theory, for when there is only one constraint it is necessarily first class.

Situation (b): α = 0, β≠ 0. In this case, from (8.219) we get the secondary constraint

\phi _{2}= q_{1}-q_{2} \approx0. (8.223)

The consistency condition applied to \phi _{2} gives

\left\{\phi _{2},H\right\}+ \lambda\left\{\phi _{2},\phi _{1}\right\}=0 \Longrightarrow p_{1} -q_{2}-\lambda=0 \Longrightarrow \lambda=p_{1} -q_{2}.   (8.224)

Thus the Lagrange multiplier is determined and the total Hamiltonian becomes

H_{T} =\frac{1}{2}\left(p_{1} -q_{2}\right)^{2} + \left(p_{1} -q_{2}\right)\left(p_{2} -q_{1}\right) – \frac{\beta }{2}\left(q_{1} -q_{2}\right)^{2}.   (8.225)

The constraints \phi _{1} and \phi _{2} are second class because \left\{\phi _{1},\phi _{2}\right\}= 1. An elementary calculation furnishes the constraint matrix and its inverse:

\left\|\left\{\phi _{a},\phi _{b}\right\}\right\|= \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix} \Longrightarrow \left\|C_{ab}\right\|= \left\|\left\{\phi _{a},\phi _{b}\right\}\right\|^{-1}=\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}.   (8.226)

The Dirac bracket is given by

\left\{F,G\right\}^{D}=\left\{F,G\right\}-\sum\limits_{a,b=1}^{2}{\left\{F,\phi _{a}\right\}C_{ab}\left\{\phi _{b},G\right\}}

=\left\{F,G\right\}+\left\{F,\phi _{1}\right\}\left\{\phi _{2},G\right\}- \left\{F,\phi _{2}\right\}\left\{\phi _{1},G\right\}.  (8.227)

With the use of the Dirac bracket the constraints can be set equal to zero in the Hamiltonian (8.225), which reduces to

H = H_{T} =\frac{1}{2}\left(p_{1} -q_{2}\right)^{2}.   (8.228)

The equations of motion (8.196) yield

\dot{F}= \left\{F,H\right\} ^{D}.  (8.196)

\dot{q}_{1}= p_{1} -q_{2}, \dot{q}_{2}= p_{1} -q_{2} , \dot{p}_{1}= p_{1} -q_{2}, \dot{p}_{2}= p_{1} -q_{2}.   (8.229)

With the use of the Dirac bracket the constraints p_{2} -q_{1}=0 and q_{2}-q_{1}=0 become strong equations and can be substituted into (8.229) to give

\dot{q}_{1}= p_{1} -q_{1} , \dot{p}_{1}= p_{1} -q_{1}.  (8.230)

The general solution to these equations is

q_{1}\left(t\right)= q_{2}\left(t\right)= p_{2}\left(t\right) = at + b − a ,  p_{1}\left(t\right)=at + b , (8.231)

with a and b arbitrary constants. The general analysis is confirmed that no arbitrary functions of time are left when all constraints are second class.

If α ≠ 0 the secondary constraint derived from the consistency condition (8.219) takes the form

\phi _{2}= \alpha \left(p_{1} -q_{2}\right)-\beta\left(q_{1} -q_{2}\right) \approx 0.   (8.232)

With the use of the Hamiltonian (8.218) the consistency condition associated with this constraint is

\left\{\phi _{2},H\right\} + \lambda \left\{\phi _{2},\phi _{1}\right\}= -\beta \left[\left(p_{1} -q_{2}\right)-\alpha \left(q_{1} -q_{2}\right)\right] + \left(\beta -\alpha^{2}\right) \lambda \approx 0.   (8.233)

Situation (c): α ≠ 0, β = α². The secondary constraint (8.232) assumes the simpler form

\phi _{2}\equiv p_{1} -q_{2} – \alpha \left(q_{1} -q_{2}\right) \approx 0.   (8.234)

Using the Hamiltonian (8.218) with β = α², the consistency condition applied to \phi _{2} does not generate new constraints because

\left\{\phi _{2},H\right\} + \lambda \left\{\phi _{2},\phi _{1}\right\}=- \alpha \phi _{2} \approx 0  (8.235)

is identically satisfied owing to the constraint \phi _{2} ≈ 0. The equations of motion (8.209) give

\dot{F} \approx  \left\{F, H_{T}\right\}.  (8.209)

\dot{q}_{1} = p_{1} -q_{2}+ \left(\alpha-1\right) \lambda, \dot{q}_{2} =\lambda,    (8.236a)

\dot{p}_{1} =\alpha^{2} \left(q_{1} -q_{2}\right)-\left(\alpha-1\right)\lambda,  \dot{p}_{2} =p_{1} -q_{2}-\alpha^{2}\left(q_{1} -q_{2}\right).   (8.236b)

It follows that λ (therefore q_{2}) is an arbitrary function of time. This is in agreement with the fact that the constraint \phi _{1} is first class.

Situation (d): α ≠ 0, β ≠ α². The Lagrange multiplier λ is uniquely determined by Eq. (8.233):

\lambda= \frac{\beta }{\beta – \alpha^{2}}\left[\left(p_{1} -q_{2}\right)- \alpha\left(q_{1} -q_{2}\right)\right]= \frac{\beta }{\alpha} \left(q_{1} -q_{2}\right),   (8.237)

where we have used (8.232). The constraints \phi _{1}, \phi _{2} are second class and the equations of motion are

\dot{q}_{1} = p_{1} -q_{2} ,  \dot{q}_{2} = \frac{\beta }{\alpha} \left(q_{1} -q_{2}\right),   (8.238a)

\dot{p}_{1} =\frac{\beta }{\alpha}\left(q_{1} -q_{2}\right),  \dot{p}_{2} =\frac{ \left(1-\alpha\right) \beta}{\alpha}\left(q_{1} -q_{2}\right).  (8.238b)

There are no arbitrary functions of time.