 Magnetic coenergy

For reversible quasistatic magnetic circuits with flux linkages ${\lambda }_i$, currents $I_i$ and mechanical coordinate $q$, write $d{W}_B={\sum }_i{I}_{i}d{\lambda }_i-F_{q}dq$. The Legendre transform $W_B^{\prime }={\sum }_i{I}_i{\lambda }_i-W_B$ obeys $d{W}_B^{\prime }={\sum }_i{\lambda }_{i}d{I}_i+F_{q}dq$. Thus $F_q=(\partial W_B^{\prime }/\partial q{)}_{I_i}$, while $F_q=-(\partial W_B/\partial q{)}_{{\lambda }_i}$. In a linear reciprocal system $W_B^{\prime }=W_B=\frac{1}{2}{\mathbf{I}}^{\mathsf{T}}L(q)\mathbf{I}$ numerically, but the variables held fixed in their force derivatives differ. The coenergy formulation includes the effect of electrical sources maintaining the currents; it avoids reversing the sign of the magnetic force by differentiating stored energy under the wrong constraint.