- #1

fluidistic

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## Homework Statement

Calculate the Cartesian expressions and the value of the modulus of the angular momentum in cylindrical coordinates of a particle whose coordinates are [itex](r, \phi, z)[/itex].

## Homework Equations

[itex]L=T-V[/itex], [itex]\vec P = \sum _i ^3 \frac{\partial L}{\partial \vec {\dot q_i}}[/itex], [itex]\vec M = \sum _i^3 \vec r_i \times \vec P_i [/itex].

## The Attempt at a Solution

Not sure what they mean with Cartesian expressions. The position of such a particle in Cartesian coordinates is [itex](\sqrt {x^2+y^2},\arcsin \left ( \frac{y}{\sqrt{x^2+y^2}} \right ),z)[/itex] if it's what they ask for.

For the Lagrangian, [itex]V=0[/itex] so [itex]L=\frac{m}{2}v^2[/itex].

I've calculated [itex]v^2[/itex] to be worth [itex]\dot r^2 + r^2 \dot \phi ^2 + \dot z ^2[/itex].

This gave me [itex]\vec P =(m \dot r, m r^2 \dot \phi , m \dot z)[/itex].

As for [itex]\vec M[/itex] I'm not so sure. I took [itex]\vec r[/itex] as [itex](r, \phi, z)[/itex] but this doesn't really make sense to me. Anyway this gave me [itex]\vec M=m(\dot z \phi -z r^2 \dot \phi) \hat i +m(\dot r z-r\dot z) \hat j +m(r^3 \dot \phi - \phi \dot r) \hat k[/itex]. Now I have to take the square of each component, sum them all and take the square root of it. But I'm not confident in what I've done so far.

Could you please enlighten me?