Quick Question- Hamiltonian constant proof

  • Thread starter binbagsss
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  • #1
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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:
hcom.png


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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Answers and Replies

  • #2
Orodruin
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You are not taking the derivative of ##dH##. ##dH## in itself is a differential
 
  • #3
mjc123
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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!
 

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