Lagrangian and Hamiltonian Formulations for Classical and Quantum Circuits

The application of Lagrange and Hamilton methods to classical electric circuits and to circuit quantum electrodynamic circuits is presented. The formulation of the Lagrangian and Hamiltonian equations for lossless electric circuits including linear and nonlinear circuit elements is discussed. The considered circuits are subdivided into a linear embedding circuit and the embedded nonlinear circuit elements. The embedding circuits are represented as a Lagrangian or Hamiltonian system with port variables. On the basis of the of the classical Hamiltonian description of circuits a circuit quantum electrodynamic representation is introduced. As examples for classical and quantum circuits, Josephson circuits are discussed.