Electrostatic potential and charges on conductors that are closed to each other can be put in relation with the capacitance matrix .
Can the energy of the system of two (or more) conductors be rewritten as the sum of a part due to each conductor and another one that is due to a "shared" energy of the two conductors?
Consider two conductors of capacitance ,charges and potentials $q_1$, $C_1$, $V_1$, $q_2$, $C_2$, $V_2$.
The energy of the system is by definition $$U=q_1 V_1+q_2 V_2$$
The matrix of capacitance is the 2x2 symmetric matrix such that
$$\begin{pmatrix} q_1 \\ q_2 \end{pmatrix}=\begin{pmatrix}c_{11} & c_{12}\\ c_{12}& c_{21} \end{pmatrix}\begin{pmatrix}V_1 \\ V_2\end{pmatrix}$$
Can I express $U$ as something like the following?
$$U=\frac{1}{2}\frac{q_1^2}{2C_1}+\frac{1}{2}\frac{q_2^2}{2C_2}+...$$
Where $...$ stays for an expression that includes the charges, the potentials and the coefficients $c_{11},c_{12},c_{23}$. This expression should represent the "shared" energy of the two conductors.
Example (which I wonder how to generalize)
Two conductiong spheres have the parameters indicated above and are at a big distance $x$ (induction influence is neglected). The energy of the system can be written as
$$U=\frac{q_1^2}{2C_1}+\frac{q_2^2}{2C_2}+\frac{q_1 q_2}{4 \pi \epsilon_0 x}$$
In this case the expression I'm looking for is $\frac{q_1 q_2}{4 \pi \epsilon_0 x}$, but how can one in general write this term (if it is possible to do it)?