# Hamilton equation for a block on an inclined plane

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1. Nov 30, 2016

### Cocoleia

1. The problem statement, all variables and given/known data
I am asked to find the Hamilton equations for a block on an inclined plane (no friction)

2. Relevant equations

3. 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?

2. Nov 30, 2016

### 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.

3. Nov 30, 2016

### Cocoleia

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

4. Nov 30, 2016

### TSny

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.

5. Nov 30, 2016

### Cocoleia

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

6. Nov 30, 2016

### TSny

OK. Thank you.

7. Nov 30, 2016

### Cocoleia

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

8. Nov 30, 2016

### TSny

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

9. Nov 30, 2016

### Cocoleia

10. Nov 30, 2016

### 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?

11. Dec 1, 2016

### Cocoleia

Yes. How do we go about doing this?

12. Dec 1, 2016

### TSny

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: px and py.

13. Dec 1, 2016

### Cocoleia

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

14. Dec 1, 2016

### 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.