Quick Question- Hamiltonian constant proof

Yes you are right, it's a division by dt, not a differentiation. And for the second part, the terms containing ##\frac{d}{dt}(\frac{\partial L}{\partial q_u})## will cancel out with the corresponding terms containing ##\frac{\partial L}{\partial q_u}##, leaving only the desired terms with ##\ddot{q_u}##.
  • #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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  • #2
You are not taking the derivative of ##dH##. ##dH## in itself is a differential
 
  • #3
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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1. What is the Hamiltonian constant?

The Hamiltonian constant, also known as the Hamiltonian function, is a mathematical concept used in classical mechanics to describe the total energy of a system. It is represented by the letter H and is a function of the system's position and momentum.

2. Why is the Hamiltonian constant important?

The Hamiltonian constant is important because it allows us to determine the behavior of a system over time. By calculating the Hamiltonian constant, we can predict how a system will evolve and understand its stability and energy conservation.

3. How is the Hamiltonian constant calculated?

The Hamiltonian constant is calculated by taking the sum of the kinetic energy (represented by T) and potential energy (represented by V) of a system. This can be expressed as H = T + V.

4. Can the Hamiltonian constant change over time?

In classical mechanics, the Hamiltonian constant is considered to be a constant value and does not change over time. However, in quantum mechanics, the Hamiltonian constant is represented by an operator and can change over time as the system evolves.

5. What is the proof for the Hamiltonian constant?

The proof for the Hamiltonian constant is based on the principles of classical mechanics and the conservation of energy. It involves using the equations of motion and the Lagrangian function to derive the Hamiltonian function and show that it remains constant over time.

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