Hamilton equation for a block on an inclined plane

In summary: So you would derive the equations of motion once for px, and then for py, and then for theta.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.
  • #1
Cocoleia
295
4

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!)
upload_2016-11-30_16-50-45.png

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?
 
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  • #2
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
TSny said:
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?
 
  • #4
Cocoleia said:
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.
 
  • #5
TSny said:
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
 
  • #6
Cocoleia said:
x measured horizontally across the slope, and y measured down the slope
OK. Thank you.
 
  • #7
TSny said:
OK. Thank you.
I think I understand, I will have to use the definition:
upload_2016-11-30_18-1-41.png
 
  • #8
Cocoleia said:
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 ##.
 
  • #9
TSny said:
Yes. But ##p_i## should not be "dotted" (time derivative) in ## H = \sum p_i \dot{q}_i - L ##.
This is my final answer:
upload_2016-11-30_19-34-15.png
 
  • #10
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
TSny said:
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?
 
  • #12
Cocoleia said:
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: px and py.
 
  • #13
TSny said:
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.
So will I derive 3 times, once for px, once for py, and then for theta?
 
  • #14
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.
 

FAQ: Hamilton equation for a block on an inclined plane

1. What is the Hamilton equation for a block on an inclined plane?

The Hamilton equation for a block on an inclined plane is a mathematical formula that describes the motion of a block sliding down an inclined plane under the influence of gravity. It takes into account the block's position, velocity, and acceleration at any given time.

2. How is the Hamilton equation derived?

The Hamilton equation is derived from the Hamilton's principle, which states that the path a system takes between two points is the one that minimizes the action integral. This principle is based on the conservation of energy and the variational calculus.

3. What are the variables used in the Hamilton equation for a block on an inclined plane?

The variables used in the Hamilton equation for a block on an inclined plane include the block's position (x), velocity (v), acceleration (a), the angle of the inclined plane (θ), and the force of gravity (mg).

4. How does the Hamilton equation differ from the Newton's second law of motion?

The Hamilton equation is a more general form of the Newton's second law of motion. While Newton's law only considers forces acting on a system, the Hamilton equation takes into account the entire path and considers all the forces acting on the system at any given time.

5. Can the Hamilton equation be applied to other systems besides a block on an inclined plane?

Yes, the Hamilton equation can be applied to a wide range of physical systems, including pendulums, springs, and celestial bodies. It is a fundamental equation in classical mechanics and has many applications in physics and engineering.

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