  We advance here an algorithm of a synthesis of an electric circuit based on
prescribed quadratic Lagrangian. That is the circuit evolution equations are
equivalent to the relevant Euler-Lagrange equations. The proposed synthesis is
a systematic approach that allows to realize any finite dimensional physical
system described by a Lagrangian in a lossless electric circuit so that their
evolution equations are equivalent. The synthesized circuit is composed of (i)
capacitors and inductors of positive or negative values for the respective
capacitances and inductances, and (ii) gyrators. The circuit topological design
is based on the set of LC fundamental loops (f-loops) that are coupled by
GLC-links each of which is a serially connected gyrator, capacitor and
inductor. The set of independent variables of the underlying Lagrangian is
identified with f-loop charges defined as the time integrals of the
corresponding currents. The EL equations for all f-loops account for the
Kirchhoff voltage law whereas the Kirchhoff current law holds naturally as
consequence of the setup of the coupled f-loops and the corresponding charges
and currents. The proposed synthesis in particular provides for efficient
implementation of the desired spectral properties in an electric circuit. The
synthesis provides also a way to realize arbitrary mutual capacitances and
inductances through elementary capacitors and inductors of positive or negative
respective capacitances and inductances.
