Euler Lagrange equation of motion

[skate_nerd]

I have a system with one generalized coordinate, x. In the potential energy part of the lagrangian, I have some constants multiplied by the absolute value of x. That is the only x dependence the lagrangian has, so when I take the partial derivative of the lagrangian with respect to x (to get the euler lagrange differential equation), I get a derivative that is undefined at x=0. Is there anything that I am supposed to do about this? Or do I just leave the derivative (x/|x|) and go on with writing the diff. eq?

Also, let it be known that x is a function of t (x(t)).
I think this may change things but I'm not sure. Why would the partial x derivative of |x(t)| be any different then the direct x derivative of |x(t)|?

 

[Ackbach]

Well, you'll need to treat the corner carefully. The Weierstrass-Erdmann Corner Condition might be useful. Otherwise, I would proceed with the usual calculations.

$x$ is always $x(t)$. You don't need to worry about $t$ showing up in the Lagrangian if it's inside the function $x$.

 

[I like Serena]

when I take the partial derivative of the lagrangian with respect to x (to get the euler lagrange differential equation), I get a derivative that is undefined at x=0. Is there anything that I am supposed to do about this? Or do I just leave the derivative (x/|x|) and go on with writing the diff. eq?

It means that your system explodes if it passes through x=0.

Physically that is not possible, so that suggests there is something special going on with your lagrangian.
Perhaps x=0 is outside of the range in which your lagrangian is valid?
Where is this step function coming from?

Something like that might happen if you model the gravity of an object with mass M with \(\displaystyle V=-\frac {GM}{x}\).
It is only valid if you are outside the object.

 

[skate_nerd]

Thanks for the responses Ackbach + I Like Serena.
@I Like Serena, you're saying I have a step function, so do I need to treat my whole solution like one? With two specific solutions, one for x>0 and x<0, and one for x=0?

 

[I like Serena]

Thanks for the responses Ackbach + I Like Serena.
@I Like Serena, you're saying I have a step function, so do I need to treat my whole solution like one? With two specific solutions, one for x>0 and x<0, and one for x=0?

Looks like it.
Presumably you won't have a solution for x=0 other than that the system behaves like a singularity in space and time.
Can't really say much more without more information.

 

[skate_nerd]

Thanks. I think that's probably all I'll need to know. I've got the info.