Lagrange Equations of Motion for a particle in a vessel

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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.
 

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  • #2
MathematicalPhysicist
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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.
 
  • #3
TSny
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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?
 
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  • #4
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OK, my mistake I found a mistake you wrote in the second line ##-2br\dot{r}^2## it should be a plus sign.
 
  • #5
TSny
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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.
 
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