Equations of motion of a particle over a cylinder+conserved quantities

In summary, the conversation discusses determining the conserved quantities and equations of motion for a particle moving over the surface of a cylinder. The Lagrangian and Euler-Lagrange equations are used to find the conserved quantities, and it is determined that the angular momentum and speed along the z-axis are conserved. There is some confusion regarding deriving the equations of motion, but it is ultimately determined that the particle's trajectory can be described by the equations r(t) = K, phi(t) = c_2t + c_4, and z(t) = c_1t + c_3. It is noted that this solution assumes there are no forces acting on the particle except for the constraints.
  • #1
fluidistic
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Homework Statement


I must determine the conserved quantities+the equations of motion (of the trajectory in fact) of a particle over the surface of a cylinder.

Homework Equations


Lagrangian and Euler-Lagrange's equations.

The Attempt at a Solution


I've found the Lagrangian of the particle to be [itex]L=\frac{m}{2}(r^2\dot \phi ^2 + \dot z^2)[/itex].
Since the Lagrangian doesn't depend explicitly on [itex]\phi[/itex] nor [itex]z[/itex], the generalized momenta conjugate are conserved (I'm currently having under my eyes Goldstein's book, 1st edition, page 49).
So I can already answer this part of the problem, [itex]P_\phi=k_1[/itex] and [itex]P_z=k_2[/itex].
By intuition I know that the angular momentum is conserved and the speed of the particle along the z axis is constant.
I have a problem however with the Lagrange's equations.
For the generalized coordinate [itex]q=r[/itex] I have that [itex]\frac{\partial L}{\partial \dot r}=0[/itex] and [itex]\frac{\partial L}{\partial r}=m r \dot \phi ^2[/itex].
This gives me the first equation of motion, namely [itex]r\dot \phi ^2=0[/itex]. Since [itex]r\neq 0[/itex] (I'm dealing with a cylinder), [itex]\dot \phi =0[/itex].
Similarly, I get for [itex]\phi[/itex]: [itex]\underbrace{2 \dot r \dot \phi}_{=0} + r \ddot \phi =0 \Rightarrow \ddot \phi =0[/itex] which isn't a surprise since I already knew that [itex]\dot \phi=0[/itex].
I also get [itex]\ddot z=0[/itex].
So... the motion equations are [itex]\dot \phi =0[/itex] and [itex]\ddot z=0[/itex]?
I think I made a mistake. If I integrate them I get [itex]\phi = \text{ constant}[/itex] which is obviously wrong.
Hmm I'm confused about what I must do.
Edit: It seems that if I hadn't make any mistake, the motion equations should give me the information I already know: [itex]\dot \phi = constant[/itex] and [itex]\dot z = constant[/itex]. By the way I don't see my mistake for the Lagrange equation regarding [itex]\phi[/itex].
 
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  • #2
The particle moves over the surface of a cylinder, so r is not a coordinate, but a constant, the radius of the cylinder. The only coordinates are phi and z. Do not derive with respect to r.
If r were a variable too, you should include the term 1/2 m r(dot) ^2 into the kinetic energy.


ehild
 
  • #3
is there no gravitational force acting on this particle. And since r is constant you cannot consider it to be one of the D.O.F and you can't derive the equation that gives you 0 angular velocity
 
  • #4
Thank you very much guys, I wasn't aware I shouldn't have derived 3 equations of motion because one of the generalized coordinates was a constant.
So I'm left with [itex]\ddot \phi = \ddot z =0[/itex] which means that both [itex]\dot \phi[/itex] and [itex]\dot z[/itex] are constants.
I realize that [itex]\dot \phi =0[/itex] means that the angular momentum of the particle with respect to the z-axis is conserved. While the z component of the linear momentum of the particle is also conserved.
Integrating the equations of motion, I reach that of course as you said, [itex]r(t)=K[/itex]. Also [itex]\phi(t)=c_2t+c_4[/itex] and [itex]z(t)=c_1t+c_3[/itex].
Thus, [itex]\vec r (t)= \begin{bmatrix} K\\ c_2t+c_4 \\ c_1t+c_3 \end{bmatrix}[/itex].

Is my answer correct considering they asked for the trajectory?
 
  • #5
It should be OK if there is no force except the constraints.

ehild
 
  • #6
Thanks for the confirmation. Problem solved.
 

1. What are the equations of motion for a particle moving over a cylinder?

The equations of motion for a particle moving over a cylinder can be described by the Lagrange equations, which take into account the kinetic and potential energy of the system. These equations involve the position, velocity, and acceleration of the particle, as well as the radius and angular velocity of the cylinder.

2. What are some examples of conserved quantities in this system?

Some examples of conserved quantities in the motion of a particle over a cylinder include the total energy of the system, the angular momentum of the particle, and the angular velocity of the cylinder. These quantities remain constant throughout the motion, regardless of any changes in the system.

3. How are the equations of motion affected by changes in the radius of the cylinder?

The equations of motion for a particle over a cylinder are affected by changes in the radius of the cylinder through the terms involving the radius in the Lagrange equations. As the radius increases, the potential energy of the system increases, resulting in changes to the motion of the particle.

4. Can the equations of motion be solved analytically?

In most cases, the equations of motion for a particle over a cylinder cannot be solved analytically, as they involve complex differential equations. However, numerical methods can be used to approximate the solutions and provide valuable insights into the behavior of the system.

5. How do the conserved quantities affect the motion of the particle over a cylinder?

The conserved quantities in the motion of a particle over a cylinder have a significant impact on the behavior of the system. For example, the conservation of energy ensures that the particle will not gain or lose energy during its motion, while the conservation of angular momentum determines the direction and speed of the particle's rotation around the cylinder.

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