Hamiltonian Equations for Conservative Circuits with Two Degrees of Freedom

Conservative circuits of general form with two degrees of freedom, consisting of two parallel paths coupled through a common reactive element, are considered. They are equivalent to any possible conservative circuits with two degrees of freedom. In Hamilton’s formalism, the canonical variables "coordinate-momentum" are introduced for description of such circuits. In this work, the Hamiltonian (canonical) equations are derived for these variables as applied to the circuits of a specific type. It is natural to pass to Hamilton’s formalism through the Lagrange’s formalism. A charge and flux-linkage are chosen as generalized coordinates and velocities of the Lagrangian approach. The choice is ambiguous and depends on the circuit structure. In this connection, a choice of the generalized coordinates and momenta depends on the type of the circuit. Hamiltonian equations are obtained in vector form. It is shown that a matrix of coefficients is formally similar to the matrix of coefficients of Hamiltonian equations for the chains with a single degree of freedom. For the first time, the matrix elements are explicitly expressed through physical parameters of circuits (for the circuits under consideration), which is the main result of the work.

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