= Energy balance of coupled RL circuits
{title2=$\dot W=Vi_1-R_1i_1^2-R_2i_2^2$}
The <magnetic energy> of this constant-<inductance> model is
$$
W=\frac12L_1i_1^2+Mi_1i_2+\frac12L_2i_2^2
=\frac12\mathbf i^T\mathbf L\mathbf i.
$$
The <positive-definite matrix> condition makes $W\geq0$. Multiplying the two circuit equations by $i_1$ and $i_2$, then adding, gives the displayed <conservation of energy> statement: input <power> $Vi_1$ supplies stored <magnetic energy> and <Joule heating>. With zero drive, $\dot W\leq0$. After the drive becomes zero, the stored <magnetic energy> supplies the remaining <Joule heating>.
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