Pendulum energy question...

[Klaynos]

OK, I know this is simple and I'm just having an issue tonight.

 

A simple pendulum, is damped, Theta is the angle from vertical hanging.

 

There should be a change to either the KE or PE of the system due to the damping...

 

But I can't remember what it is, I *think* the PE should change to:

 

[math]U=-mglCos\theta + A\dot\theta[/math]

 

Where A is some constant of friction...

 

The reason I ask in this way is because I'm trying to do this using Lagrangian mechanics...

[Severian]

That is not how you should be doing it. You cannot write a disapative force in terms of a potential - in fact, iirc that is the definition of a conservative force, that it can be written as a potential. The Euler-Lagrange equations as [math]\frac{d}{dt} \left( \frac{\partial {\cal L}}{\partial \dot{\theta}} \right) - \frac{\partial {\cal L}}{\partial \theta} = 0[/math] are not valid for your system.

 

Instead you need to write [math]\frac{d}{dt} \left( \frac{\partial {\cal L}}{\partial \dot{\theta}} \right) - \frac{\partial {\cal L}}{\partial \theta} = W[/math] where [math]W[/math] is the work done by the disapative force, in your case [math]W=-A \dot{\theta}[/math], where [math]A[/math] is a positive constant.

[Klaynos]

Thanks Severian

 

That'd be why it was giving me so many conceptual problems trying to think about how it should work

 

Every time I sat down and thought about it I just bit a brick wall thinking it can't work any way I thought of...

[Severian]

Sorry, I am getting my terms mixed up. W isn't the work done - it is the disapative force. The integral over it is the work done.

[Resha Caner]

Didn't someone later modify Lagrangian mechanics to add non-conservative forces? I seem to remember someone doing that. Or is that just my heritage with engineering kluges?

 

Caner