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**How does the water in a spinning bucket arrange itself, given that its surface must be perpendicular to the force that holds the water? Solve for the Lagrangian forces of constraint.**

As a convention, I'm writing position, velocity, and acceleration as such:

position: a velocity: a. acceleration: a..

So here's what I've got so far. We know that the lagrangian is:

L = T - U

I chose cylindrical coordinates, so I got (a is the angle):

T = (1/2) * m * r. + (1/2) * m^2 * a^2 + (1/2) * m * z.^2

U = mgz

Constraint1: m = (pi) * r^2 * h * density

dL/dr = 0, dL/dr. = 0, D/DT * dL/dr. = 0

dL/da = 0, dL/a. = m^2 * a. D/DT * dL/a. = m^2 * a..

dL/dz = mg, dL/z. = m * z. D/DT * dL/z. = m * z..

So for my first equation of motion, solving for motion in the a direction:

0 = -m^2 * a..

Which isn't exactly allowing me to solve for much of anything... where did I go wrong?