Is the Lagrangian for Parallel LC Circuits Correct?

[Goddar]

Homework Statement

Hi, i have a quick question to see if I'm on the right track (I totally suck at electrical circuits since i never took a formal course so it might seem elementary to you.. anyway):

Three LC systems in parallel with different L and C values, nothing else. closed circuit.
First step in the problem is to formulate the Lagrangian and that's where i need a check, i'll be fine with the rest.

The Attempt at a Solution

Is it correct to write for the system:
L =T+V=$\frac{1}{2}$$\sum L_{i}\dot{q}^{2}_{i}+q^{2}_{i}/C_{i}$
With the constraint:
q$_{1}$+q$_{2}$+q$_{3}$=0
??
It seems too simple to me and i think i should be dealing with cross-terms, so a hint would be greatly appreciated.. thanks!

 

[mark726]

Hi there, it looks like you are on the right track with your formulation of the Lagrangian. The Lagrangian for a system of three LC circuits in parallel would indeed be the sum of the kinetic and potential energies for each individual circuit. However, you also need to take into account the interactions between the circuits, which would involve cross-terms in the Lagrangian. These terms would arise from the coupling between the circuits through their shared voltage or current sources. Without these terms, your Lagrangian would only describe the individual circuits in isolation, and not their behavior as a whole.

To properly formulate the Lagrangian for this system, you may want to consider using a matrix representation, where the coupling terms can be expressed as elements in the matrix. This will allow you to properly account for the interactions between the circuits.

I hope this helps, and good luck with your problem! Don't hesitate to reach out if you have any further questions.