Lagrangian for circuits with higher-order elements

Abstract

The necessary and sufficient conditions of the validity of Hamilton’s variational principle for circuits consisting of (alpha,beta) elements from Chua’s periodical table are derived. It is shown that the principle holds if and only if all the circuit elements lie on the so-called -diagonal with a constant sum of the indices alpha and beta. In this case, the Lagrangian is the sum of the state functions of elements of the L or +R types minus the sum of the state functions of elements of the C or -R types. The equations of motion generated by this Lagrangian are always of even-order. If all elements are linear, the equations of motion contain only even-order derivatives of the independent variable. Conclusions are illustrated on an example of the synthesis of the Pais-Uhlenbeck oscillator via the elements from Chua’s table.The necessary and sufficient conditions of the validity of Hamilton’s variational principle for circuits consisting of (alpha,beta) elements from Chua’s periodical table are derived. It is shown that the principle holds if and only if all the circuit elements lie on the so-called -diagonal with a constant sum of the indices alpha and beta. In this case, the Lagrangian is the sum of the state functions of elements of the L or +R types minus the sum of the state functions of elements of the C or -R types. The equations of motion generated by this Lagrangian are always of even-order. If all elements are linear, the equations of motion contain only even-order derivatives of the independent variable. Conclusions are illustrated on an example of the synthesis of the Pais-Uhlenbeck oscillator via the elements from Chua’s table.