# Nomenclature for Lagrangian of system with rotational and translational motion

i am trying to wrap my head around how you would go about writing the euler-lagrange equation for a system with both translational and rotational motion. It is for calculating the response of the spring mass in a half car mass spring damper system.

The only way i can get it to work is to take the total potential energy as the sum of the translational and potential energies;

$$V = V_t + V_r$$

then the potential energy wrt to the coordinate used $$q_i$$ would be the sum of the two partial derivatives e.g.

$$\frac{\partial{V}}{\partial{q_i}} = \frac{\partial{V_t}}{\partial{x}} + \frac{\partial{V_r}}{\partial{\theta}}$$

In my head this makes sense but I'm wondering if it's right as those two things aren't always the same! Is there an accepted notation for these things?

• What do you mean 'translational and rotational' potential energies? There are translational and rotational kinetic energies, but one wouldn't normally say that for potential (perhaps potential energy associated with centre of mass position and with orientation? This need not split like you've shown it.) – jacob1729 Oct 13 at 10:55
• If you have translational and rotational kinetic energies, couldn't you consider the potential that way? I don't know if it was something my lecturer used when I was at uni for conceptual understanding. I think I have found how you would write it for my case, I'll update my question to include this – Ross Hanna Oct 13 at 14:10