The Energy of an Arbitrary Electrical Circuit, Classical and Quantum

In this Note, I show an algorithmic method to find the energy and, thus, the Hamiltonian of an arbitrary electrical circuit based on the so-called incidence matrix and the circuit’s total power. This method does not require to find any Lagrangian; instead, it is based on the concept of generalized linear momenta for the kinetic and co-kinetic energy of a circuit. The method can account for superconducting loops by a simple extension of Faraday-Henry-Neumann’s law. Auxiliary (i.e., parasitic) circuit elements are required to deal with circuits with an incomplete set of generalized velocities resulting in an incomplete set of canonical coordinates. This method can be readily automatized to obtain the Hamiltonian of arbitrarily complicated circuits. I also show how to quantize the circuit associated with a resonator capacitively coupled with a qubit.