Demonstrating something is a constant of motion

In summary, the conversation discusses how to show that a given Hamiltonian is a constant of motion for orbits defined by the corresponding Hamiltonian equations. The Hamiltonian, specified as H(qk, pk, t), is constant over time if dH/dt = 0. This can be shown by calculating dH/dt in terms of partials and using the Hamiltonian equations to simplify it. It is also mentioned that the Poisson Bracket of H and H is zero and that the Hamiltonian is a constant of motion if it is time-independent.
  • #1
romeo6
54
0
I am given a Hamiltonian and am asked to show that the Hamiltonain is a constant of motion for orbits defined by the corresponding Hamiltonian equations.

can someone decrypt this for me please...

:redface:


i.e How do I define orbits for a given hamiltonian and then how do I show that the Hamiltonian is a constant of motion?

Thanks in advance.
 
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  • #2
You don't have to define the orbits... the Hamiltonian does that along with the corresponding Hamiltonian equations. (These are qk' = partial H / partial pk and -pk' = partial H / partial qk. I apologize for not using latex here...)

Anyway, the Hamiltonian is specified as H(qk, pk, t), where qk are the generalized coordinates and pk are the generalized momenta. If this is constant over time, dH/dt = 0, right? So, calculate dH/dt in terms of partials with respect to the arguments, and simplify it using the Hamiltonian equations to show that it's zero.
 
  • #3
The PB of H and H is zero and iff the Hamiltonian is time-independent, then it is a constant of motion.

Daniel.
 

Related to Demonstrating something is a constant of motion

1. What is a constant of motion?

A constant of motion is a physical quantity that remains unchanged over time in a particular system. It is a property that is conserved and does not vary with time or other variables.

2. How can I demonstrate that something is a constant of motion?

To demonstrate that something is a constant of motion, you can use mathematical equations and principles, as well as experimental data and observations. You would need to show that the quantity remains unchanged under different conditions or at different points in time.

3. What are some common examples of constants of motion?

Some common examples of constants of motion include energy, momentum, angular momentum, and electric charge. These quantities are conserved in various physical systems and play a crucial role in understanding the behavior of objects.

4. Why is it important to identify constants of motion?

Identifying constants of motion is important because they provide valuable insights into the behavior and dynamics of physical systems. They can help us make predictions and understand the underlying principles governing the motion of objects.

5. Can constants of motion change over time?

No, constants of motion cannot change over time. They are defined as quantities that remain constant and do not vary with time or other variables. However, they can change in different systems or under different conditions, but within a specific system, they remain constant.

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