Quick Question- Hamiltonian constant proof

[binbagsss]

Homework Statement

Show that if the Lagrangian does not explicitly depend on time that the Hamiltonian is a constant of motion.

Homework Equations

see below

The Attempt at a Solution

method attached here:

Apologies this is probably a bad question, but just on going from the line $dH$ to $dH/dt$ I see the$d/dt$ has hit the $dq_0$ terms only, I don’t understand why a product rule hasn’t been used, so I would get:$\frac{dH}{dt}=\sum_{u} \dot{q_u} (\frac{dp_{u}}{dt}-\frac{\partial L}{\partial q_u} )+ \ddot{q_u}dp_{u}-\frac{d}{dt}(\frac{\partial L}{\partial q_u}) dq_u$Many thanks

 

[Orodruin]

You are not taking the derivative of $dH$. $dH$ in itself is a differential

 

[mjc123]

In going from dH to dH/dt, you are not differentiating, but dividing by dt. Differentiation was in the previous step.
Thus e.g. if f = uv
df = udv + vdu
df/dx = udv/dx + vdu/dx (not uddv/dx + dv du/dx +vddu/dx + du dv/dx)

Edit: Beat me to it!