Conservation of flux linkage
I wiil try to go through a simple application of flux conservation using an example.

I will start from something perhaps more familiar to most of us.

When you try to find final status of this circuit using charge conservation principle 







simulate this circuit &ndash; Schematic created using CircuitLab

you will quickly come to the conclusion that charge at time 0 yields
$$
Q(0)=C_1v_1(0)+C_2v_2(0)
$$
and will be conserved and shared between the two capacitor giving a final voltage
$$
v_1(\infty)=v_2(\infty)=\frac{Q(\infty)}{C_1+C_2}=\frac{Q(0)}{C_1+C_2}=\frac{C_1v_1(0)+C_2v_2(0)}{C_1+C_2}
$$

So what have we done?
 We had a two node circuit made of two capacitors:


this two capacitors were initially charged at some initial voltage
initial charge was calculated considering positive the upper
capacitor armatures
this charge is conserved and shared between the two caps in final
status. Upper node will now remain at constant voltage
just note that at final state voltage across resistor is zero,
otherwise dissipation would change system energy and so we were not
at final status yet.

Now le's make it dual and use flux conservation principle.


Now we have a two mesh circuit made of two inductors:


this two inductors were initially charged at some initial current








simulate this circuit


initial flux is calculated considering positive clockwise mesh current


$$
\Phi(0)=L_1i_1(0)+L_2i_2(0)
$$


this flux is conserved and shared between the two coils in final
status Outer mesh will now remain at constant current


$$
i_1(\infty)=i_2(\infty)=\frac{\Phi(\infty)}{L_1+L_2}=\frac{\Phi(0)}{L_1+L_2}=\frac{L_1i_1(0)+L_2i_2(0)}{L_1+L_2}
$$


just note that at final state current through resistor is zero,
otherwise dissipation would change system energy and so we were not
at final status yet.


I understand this is far from a thorough study but I believe could help to make some clear around that.

Notes:


Both electric charge and magnetic flux conservation priciples could
be named toghether flux conservation, electric flux the first one
and magnetic flux the second one.
Resistors in above schematics are there just to avoid singularities
in equations. Those could be rigorously coped with Dirac
distributions but so far I'd rather spare the subject.