Lagrange Equations of Motion for a particle in a vessel

[Wombat11]

Homework Statement

A particle of mass m moves without friction on the inside wall of an axially symmetric vessel given by Z=(b/2)(x^2+y^2) where b is a constant and z is in the vertical direction. Find the Lagrangian of the system and the Euler-Lagrange equations. (Hint: work in cylindrical polar coordinates with x = r cos θ, y = r sin θ.)

Relevant Equations

I don't know how to put the equations in the computer so i'm just gonna drop a picture.

The final answer should have a negative b^2⋅r(dot)^2⋅r term but I have no idea how that term would become negative. Also I know for a fact that my Lagrangian is correct.

 

[MathematicalPhysicist]

Well, first thing first.
The Lagrangian is a scalar, so I don't understand why did you write $\hat{\phi}, \hat{z}$ etc which are a unit-norm vectors.
As for your solution, from the steps that I checked it seems valid to me.

 

[TSny]

The final answer should have a negative b^2⋅r(dot)^2⋅r term but I have no idea how that term would become negative.

You can usually check the sign of a term in the equation of motion by considering a simple, special case of the motion.

For example, suppose you switch off gravity. In addition, suppose the initial condition is such that the the particle moves only in a fixed vertical plane so that $\dot \theta$ remains zero.

Your equation of motion then simplifies to $\ddot r = - \frac{b^2r}{1+b^2r^2} \dot r^2$. Does the negative sign on the right side make sense?

 

[MathematicalPhysicist]

OK, my mistake I found a mistake you wrote in the second line $-2br\dot{r}^2$ it should be a plus sign.

 

[TSny]

OK, my mistake I found a mistake you wrote in the second line $-2br\dot{r}^2$ it should be a plus sign.

I think that mistake is corrected in the next line.

 

[MathematicalPhysicist]

I think that mistake is corrected in the next line:
https://www.physicsforums.com/attachments/249157

Indeed.