Question 8.11: Discuss the consistency conditions and obtain the general so......
Discuss the consistency conditions and obtain the general solution to Hamilton’s equations for the system described by the Lagrangian of Example 8.10.
The Hamiltonian associated with the Lagrangian (8.178) is
L = \frac{1}{2} \left(\dot{x} -\dot{y} \right)^{2} , (8.178)
H =\dot{x}p_{x}+\dot{y}p_{y}-L = \left(\dot{x}-\dot{y}\right)p_{x} – \frac{1}{2} p^{2}_{x}=\frac{1}{2} p^{2}_{x}, (8.188)
where we have used (8.180) and (8.181). The consistency condition
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)
\phi = p_{x}+p_{y} =0, (8.181)
{\phi ,H} + λ{\phi, \phi} ≈ 0 (8.189)
for the constraint (8.181) is identically satisfied because {\phi,H} = 0. There are no other constraints and Hamilton’s equations (8.186) take the form
\dot{F} =\left\{F,H\right\} +\sum\limits_{m}{\lambda _{m}\left\{F,\phi _{m}\right\} } \approx \left\{F,H _{T}\right\}, (8.186)
\dot{x}\approx \left\{x,H\right\} + \lambda \left\{x,\phi\right\} = p_{x}+\lambda, \dot{p}_{x} \approx \left\{p_{x},H\right\} + \lambda \left\{p_{x},\phi\right\} =0, (8.190a)
\dot{y}\approx \left\{y,H\right\} + \lambda \left\{y,\phi\right\} =\lambda, \dot{p}_{y} \approx \left\{p_{y},H\right\} + \lambda \left\{p_{y},\phi\right\} =0. (8.190b)
The general solution to these equations is p_{x}=-p_{y}=a , x(t) = y(t) + at + b with a, b arbitrary constants and y(t) an arbitrary function.