Question 8.10: Given the Lagrangian L = 1/2 (x − y)² , (8.178) find the pri......

The Hessian matrix

W =\begin{pmatrix} \frac{\partial^{2}L}{\partial \dot{x}^{2} } & \frac{\partial^{2}L}{\partial \dot{x}\partial\dot{y} } \\ \\ \frac{\partial^{2}L}{\partial \dot{y}\partial\dot{x} } & \frac{\partial^{2}L}{\partial \dot{y}^{2} } \end{pmatrix} =\begin{pmatrix} 1 & -1 \\ -1 & 1 \end{pmatrix}  (8.179)

is clearly singular. The canonical momenta are given by

p_{x}=\frac{\partial L}{\partial \dot{x} }= \dot{x}-\dot{y},  p_{y}=\frac{\partial L}{\partial \dot{y} }= \dot{y}-\dot{x}.   (8.180)

These equations are not independent and from them one derives

\phi = p_{x}+p_{y} =0,   (8.181)

which is the only primary constraint.