Lagrange Equations of Motion for a particle in a vessel

[Wombat11]

I start out by substituting rcos(Θ) and rsin(Θ) for x and y respectively. This gives me z=(b/2)r^2. The Lagrangian of this system is (1/2)m(rdot^2+r^2⋅Θdot^2+zdot^2)-mgz. (rdot and such is the time derivative of said variable). I then find the time derivative of z, giving me zdot=br⋅rdot and plug it into the Lagrangian giving me (1/2)m( rdot^2 + r^2⋅Θdot^2 + b^2r^2⋅rdot^2) - (1/2)mgbr^2. (plugged in regular z too). Finding the Lagrange equation of motion for 'r' gives me (ddot will be the second time derivative of said variable) 0= rddot(1+b^2r^2)-r⋅Θdot^2 + b^2⋅rdot^2⋅r + gbr . This is not correct though, the right answer has a negative term of b^2⋅rdot^2⋅r. Any help would be greatly appreciated.
Sorry about the notation, I have no idea how to put the equations into the computer.

 

[tnich]

Here is a quick easy tutorial on how to type in equations in a readable format: https://www.physicsforums.com/help/latexhelp/.

 

[BvU]

Welcome back in PF !

Lagrange equation of motion

The $\mathcal L$ you work out (*)
$$ (1/2)m( \dot r^2 + r^2\dot \theta^2 + b^2r^2\dot r^2) - (1/2)mgbr^2 $$
looks good to me. What is the Euler-Lagrange equation you then use ?(*) I 'typed':
$$ (1/2)m( \dot r^2 + r^2\dot \theta^2 + b^2r^2\dot r^2) - (1/2)mgbr^2 $$