LC circuit

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

[itex]C[/itex] and [itex]2C[/itex] represent the capacitance of the respective capacitors. [itex]L[/itex] and [itex]2L[/itex] represent the inductance of the respective inductors. Let, the charge on the capacitors GM and PM be [itex]q_1[/itex] and [itex]q_2[/itex] which are variable. [itex]t[/itex] represents time. Initially (when the switch k was closed and [itex]t = 0[/itex]) , [itex]q_1 = \frac{2q_0}{3} ; q_2 = \frac{q_0}{3}[/itex]. Would it be,
[itex]i_1 = - \frac{dq_1}{dt} ; i_2 = - \frac{dq_2}{dt} [/itex] ?
Explain your answer.


2. Relevant equations
That is conceptual question. No equation is needed.

3. The attempt at a solution
The actual homework was different. But I need to clear my concept before solving the actual problem. My confusion is, the MGHJ segment (which has the current [itex]i_1[/itex]) is connected with both the capacitors. So, won't both the capacitor affect the current [itex]i_1[/itex] ?

 

So, if I change the circuit into the following , I think [itex]i_1[/itex] and [itex]i_2[/itex] won't change. Am I right?