Hamilton equation for a block on an inclined plane

[Cocoleia]

Homework Statement

I am asked to find the Hamilton equations for a block on an inclined plane (no friction)

Homework Equations

The Attempt at a Solution

Please ignore the fact that my steps are written in French (sorry!)

I am no longer sure of what I'm doing when it comes to finding the momentum. I here chose y' and just ignored the x', because that seemed logical to me at the time. How can I correct my solution if it isn't right?

 

[TSny]

You should make clear the orientation of the x and y-axes.

Treat the $\dot{x}$ term similarly to how you treated the $\dot{y}$ term.

 

[Cocoleia]

You should make clear the orientation of the x and y-axes.

Treat the $\dot{x}$ term similarly to how you treated the $\dot{y}$ term.

Will I have two different equations for momentum then? And then add them together for a total momentum?

 

[TSny]

Will I have two different equations for momentum then? And then add them together for a total momentum?

You will have a differential equation for the x-component of momentum and a separate differential equation for the y-component of momentum.
Since you didn't state the orientation of your axes, I am having to guess the orientations based on your equations.

 

[Cocoleia]

You will have a differential equation for the x-component of momentum and a separate differential equation for the y-component of momentum.
Since you didn't state the orientation of your axes, I am having to guess the orientations based on your equations.

x measured horizontally across the slope, and y measured down the slope

 

[TSny]

x measured horizontally across the slope, and y measured down the slope

OK. Thank you.

 

[Cocoleia]

OK. Thank you.

I think I understand, I will have to use the definition:

 

[TSny]

I think I understand, I will have to use the definition:
View attachment 109724

Yes. But $p_i$ should not be "dotted" (time derivative) in $H = \sum p_i \dot{q}_i - L$.

 

[Cocoleia]

Yes. But $p_i$ should not be "dotted" (time derivative) in $H = \sum p_i \dot{q}_i - L$.

This is my final answer:

 

[TSny]

In your second equation for $L$, there is a sign error in the kinetic energy part. But you corrected it later.

In your expressions for $H$ you've dropped $y$ from the potential energy part.

Otherwise, looks OK.

Are you also supposed to write out the equations of motion?

 

[Cocoleia]

In your second equation for $L$, there is a sign error in the kinetic energy part. But you corrected it later.

In your expressions for $H$ you've dropped $y$ from the potential energy part.

Otherwise, looks OK.

Are you also supposed to write out the equations of motion?

Yes. How do we go about doing this?

 

[TSny]

Yes. How do we go about doing this?

You showed at the end of your first post how to get the equations of motion by taking partial derivatives of H. But you have two components of p: p_x and p_y.

 

[Cocoleia]

You showed at the end of your first post how to get the equations of motion by taking partial derivatives of H. But you have two components of p: p_x and p_y.

So will I derive 3 times, once for px, once for py, and then for theta?

 

[TSny]

No, just twice. Theta is not a variable. You could call theta a parameter. Theta is assumed to have a fixed value as the block slides on the incline.