PX285 - D2 - a constant of the motion

taking the time derivative of the hamiltonian:

\frac{d H}{d t} = \frac{d}{d t} (\dot{x} \frac{\partial L}{\partial \dot{x}} - L)

for a function, $f (a, b, t)$ , its time derivative is:

\frac{d f}{d t} = \frac{\partial f}{\partial a} \frac{\partial a}{\partial t} + \frac{\partial f}{\partial b} \frac{\partial b}{\partial t} + \frac{\partial f}{\partial t}

for $L (x, \dot{x}, t) :$

\begin{aligned} \frac{d H}{d t} & = \frac{d}{d t} (\dot{x} \frac{\partial L}{\partial \dot{x}}) - (\frac{\partial L}{\partial x} \frac{\partial x}{\partial t} + \frac{\partial L}{\partial \dot{x}} \frac{\partial \dot{x}}{\partial t} + \frac{\partial L}{\partial t}) \\ = \dot{x} (\frac{d}{d t} \frac{\partial L}{\partial \dot{x}} - \frac{\partial L}{\partial \dot{x}}) - \frac{\partial L}{\partial t} \\ ∴ \frac{d H}{d t} & = - \frac{\partial L}{\partial t} \end{aligned}

using the euler-lagrange equation, $\frac{\partial L}{\partial x} - \frac{d}{d t} \frac{\partial L}{\partial \dot{x}} = 0$

if the lagrangian has no explicit time dependence, ie: $L (x, \dot{x}) :$

\frac{d H}{d t} = 0

the hamiltonian is conserved for the case