Solving KE for a Bead Rolling Along an Ellipse Using Lagrange's Method

[elegysix]

I'm trying to find the equation of motion of a bead, which is constrained to roll along the bottom half of this frictionless ellipse, by using langrange's method - $$L(\phi,\dot{\phi})$$.
Here's the setup:
given the bottom half of an ellipse:
$$\mathbf{r}(\phi) = acos(\phi)\mathbf{\hat{i}}-bsin(\phi)\mathbf{\hat{j}}$$
where $$0<\phi<\pi$$, and $$\mathbf{\hat{i}}$$ , $$\mathbf{\hat{j}}$$ are the unit vectors for x and y, and $$\phi$$ is measured from the positive x-axis counterclockwise.

my question is should I use

1) $$KE=\frac{m}{2}(\dot{r}^{2}+r^{2}\dot{\phi}^{2})$$
or
2) $$KE=\frac{m}{2}(\dot{x}^{2}+\dot{y}^{2})$$

where $$\dot{x} \:\&\: \dot{y}$$ come from $$x=acos(\phi)$$ , and $$y=-bsin(\phi)$$

which one?

I'm confused because I have polar type coordinates, $$r(\phi)$$ in terms of Cartesian components.


Assuming that the second equation for KE is not appropriate to use here,

If I use the first the first equation for KE, then is it 'Legal' to find and use $$\mathbf{\dot{r}}$$ like this? or is this wrong?

$$\mathbf{\dot{r}}=-a\dot{\phi}sin(\phi)\mathbf{\hat{i}}-b\dot{\phi}cos(\phi)\mathbf{\hat{j}}$$

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As my only other option I'm aware of, I know that the definitely correct, brute force approach looks like this:
$$\mathbf{r}(\phi)_{polar}=|\mathbf{r}(\phi)_{cartesian}|\mathbf{\hat{r}}=\sqrt{a^{2}cos^{2}(\phi)+b^{2}sin^{2}(\phi)}\mathbf{\hat{r}}$$

which means we're in polar coordinates, so

$$\mathbf{\dot{r}}(\phi)=\dot{r}\mathbf{\hat{r}}+r\dot{\phi}\mathbf{\hat{\phi}}$$

$$\mathbf{\dot{r}}(\phi)=\frac{d(\sqrt{a^{2}cos^{2}(\phi)+b^{2}sin^{2}(\phi)})}{dt}\mathbf{\hat{r}}+\dot{\phi}\sqrt{a^{2}cos^{2}(\phi)+b^{2}sin^{2}(\phi)}\mathbf{\hat{\phi}}$$

and L=KE-PE... and then I have to solve $$\frac{dL}{d\phi}-\frac{d(\frac{dL}{d\dot{\phi}})}{dt}=0$$
which, that route is becoming a nightmare very quickly, and I don't want any trouble with the algebra police... any advice?

 

[Dale]

The idea in these types of problems is to find an expression for the KE purely in terms of your generalized coordinates. Sometimes you may know the KE immediately in those terms, but other times you may have to start from Cartesian coordinates where you know the expression for KE and then you can transform to your generalized coordinates.

That is a long-winded way of saying that I would start with 2) and substitute in the a cos and b sin terms.

 

[charng]

I would suggest using the first equation for KE, as it takes into account the rotational motion of the bead along the ellipse. The second equation only considers the linear motion of the bead in the x and y directions, which may not accurately reflect the dynamics of the system.

In terms of finding and using \mathbf{\dot{r}}, it is correct to use the equation you have provided. The velocity vector \mathbf{\dot{r}} is a combination of the linear and rotational velocities, which can be expressed in terms of the polar coordinates in this case.

However, if you find that the brute force approach is becoming too complicated, you could also try using the Lagrangian method with polar coordinates. This would involve setting up the Lagrangian as L=KE-PE, where KE is given by the first equation and PE is determined by the gravitational potential energy of the bead at any point along the ellipse. Then, solving for \frac{dL}{d\phi}-\frac{d(\frac{dL}{d\dot{\phi}})}{dt}=0 would give you the equation of motion for the bead.

Ultimately, the choice between the two methods may depend on the specific problem and your personal preference. Whichever method you choose, make sure to carefully consider the dynamics of the system and double check your calculations to ensure accuracy.