# Kinetic energy in parabolic coordinates

1. Jul 16, 2011

### QuArK21343

1. The problem statement, all variables and given/known data

Prove that in parabolic coordinates $\alpha,\beta$ the kinetic energy is $T=m/8(\alpha+\beta)(\dot\alpha^2/\alpha+\dot\beta^2/\beta)$

2. Relevant equations

Parabolic coordinates are defined as follows: $\alpha=r+x, \beta=r-x$ with $r=\sqrt{x^2+y^2}$

3. The attempt at a solution

I don't know how to proceed in this situation: in simpler case (spherical or cylindrical coordinates) I write down the three components of velocity using geometrical intuition (e.g. $v_\rho=\dot \rho, v_\phi=\rho \dot\phi,v_z=\dot z$, because I see they are right...). What if I get only the definition of the new coordinates?

2. Jul 16, 2011

### Redbelly98

Staff Emeritus
Welcome to physics forums.

I think the key here is to get x and y in terms of $\alpha, \beta$. And you hopefully already know how to express the kinetic energy in x, y coordinates, right?

3. Jul 16, 2011

### QuArK21343

That's right! I check explicitly and it works. Now that I have the kinetic energy, it will be doable to write down the equations of motion. Thank you.

4. Jul 16, 2011

### Redbelly98

Staff Emeritus