# Quick Question- Hamiltonian constant proof

## Homework Statement

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

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

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Orodruin
Staff Emeritus
Homework Helper
Gold Member
You are not taking the derivative of $dH$. $dH$ in itself is a differential

mjc123
Homework Helper
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!

• binbagsss