Particle moving inside an inverted cone - Lagrangian

[bigguccisosa]

Homework Statement

Hi there! So I have a problem regarding a particle of mass m moving down an inverted cone under the force of gravity. The cone is linear with equation z(r) = r, in cylindrical coordinates (r, theta, z)

A. Write down the Lagrangian, include the constraint that the particle stays on the surface of the cone via a Lagrange multiplier.
B. Determine the Euler-Lagrange equations for each coordinate and the Lagrange multiplier. Show that the angular momentum remains constant throughout.
C. Simplify these equations to get a 2nd order ordinary differential equation for the radial location of the particle.
D. Show that there are circular orbits and identify their angular velocity as a function of orbital radius.

Homework Equations

Euler-Lagrange equation with one equation of constraint:
$\frac{\partial{L}}{\partial{q}} - \frac{d}{dt}\frac{\partial{L}}{\partial{\dot{q}}} + \lambda\frac{\partial{f}}{\partial{q}} = 0, L = T - V$

Equation of constraint:
$f = tan\alpha - \frac{r}{z} = 0$, where alpha is the half angle of the cone

The Attempt at a Solution

A. I'm not too sure about my Lagrangian here,especially the kinetic energy part and the equation of constraint. I have that $$V = mgz = \frac{mgr}{tan\alpha}$$ and $$T = \frac{1}{2}m(\dot{r}^2 + \frac{\dot{r}^2}{tan^2\alpha} + r^2\dot{\theta}^2)$$ So, $$L = \frac{1}{2}m(\dot{r}^2 + \frac{\dot{r}^2}{tan^2\alpha} + r^2\dot{\theta}^2) - \frac{mgr}{tan\alpha}$$ $$= \frac{1}{2}m(\frac{\dot{r}^2}{sin^2\alpha} + r^2\dot{\theta}^2) - \frac{mgr}{tan\alpha}$$
But I'm not sure if this is correct

B. Applying the Euler-lagrange equation to r and theta I get $$mr\dot{\theta}^2 - \frac{mg}{tan\alpha} - \frac{d}{dt}(\frac{m\dot{r}}{sin^2\alpha} + \lambda\frac{1}{h} = 0$$ and $$0 - \frac{d}{dt}(mr^2\dot{\theta} + \lambda(0) = 0$$
Integrating the second one shows that angular momentum is constant so that means my Lagrangian was correct right?

C. Okay, so I can see that the first Euler-Lagrange equation can be a differential equation for r, but it also has terms of theta in it, so it wouldn't be ordinary then? Here's as far as a I got.

D. I guess that after you solve the correct DE, you can see that the orbit is circular, then invert the equation for angular velocity in terms of r?

Maybe someone can give me something to get me going on c? Thanks

 

[Orodruin]

But I'm not sure if this is correct

It is correct, but not implemented according to what the problem asks you to do. You have implemented the constraint by directly implementing the holonomic constraint into the Lagrangian, not by using a Lagrange multiplier as the problem asks you to do.

Applying the Euler-lagrange equation to r and theta I get

You have now introduced $\lambda$ and $h$ out of nowhere. Your task in A is to construct a Lagrangian which contains the Lagrange multiplier.

 

[bigguccisosa]

It is correct, but not implemented according to what the problem asks you to do. You have implemented the constraint by directly implementing the holonomic constraint into the Lagrangian, not by using a Lagrange multiplier as the problem asks you to do.

So in that case, should I have a Lagrangian with something like L = (unconstrained T-V) + (lagrange multiplier) (constraint)? I'm looking at https://en.wikipedia.org/wiki/Lagrange_multiplier#Example_1

 

[Orodruin]

So in that case, should I have a Lagrangian with something like L = (unconstrained T-V) + (lagrange multiplier) (constraint)? I'm looking at https://en.wikipedia.org/wiki/Lagrange_multiplier#Example_1

Right.

 

[Ark236]

Hi,
This restriction is correct?

May be alpha= constant?

thanks