Momentum Pk Independence from Gen. Coord. Qk - Physicsforums

[sempiris]

Hi Physicsforums

I am re-learning classical mechanics and having a tough time dealing with a certain line from Thornton/Marion. On page 269 (5th ed), a little after introducing Hamiltonian dynamics and the canonical conjugate equations of motion, the author says: "the qk and the pk are independent, whereas the qk and the [itex]\dot{q}[/itex]k are not". The context is describing the advantages of using Hamiltonian dynamics over Lagrangian

In my mind, I see pk = m[itex]\dot{q}[/itex]k. If this is the case, I don't see how the author's statement stands.

Thank you for your time

 

[eljose79]

and any clarification you can provide.

Hello,

Thank you for reaching out to us for clarification on this statement from Thornton/Marion. I completely understand your confusion and I am happy to help explain this concept to you.

First, let's define the terms used in this statement. In Hamiltonian dynamics, qk represents the generalized coordinates and pk represents the canonical conjugate momentum. The dot above the qk (as in \dot{q}k) represents the time derivative of the generalized coordinate.

Now, let's break down the statement itself. The author is saying that the qk and pk are independent, meaning they are not directly related to each other. However, the qk and \dot{q}k are not independent because they are related through the time derivative.

To better understand this, let's look at the equations of motion in Hamiltonian dynamics:

\dot{q}k = \frac{\partial H}{\partial pk}

and

\dot{p}k = -\frac{\partial H}{\partial qk}

From these equations, we can see that the time derivative of the generalized coordinate is directly related to the partial derivative of the Hamiltonian with respect to the canonical conjugate momentum. This means that the qk and \dot{q}k are not independent, as the value of one is dependent on the value of the other.

On the other hand, the qk and pk are independent because they are not directly related to each other in the equations of motion. The value of the canonical conjugate momentum is not dependent on the value of the generalized coordinate, and vice versa.

I hope this explanation helps to clarify the statement from Thornton/Marion. If you have any further questions or need further clarification, please don't hesitate to ask.