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 (α,β) 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 Σ-diagonal with a constant sum of the indices α and β. In this case, the Lagrangian is the sum of the state functions of the elements of the L or +R types minus the sum of the state functions of the elements of the C or −R types. The equations of motion generated by this Lagrangian are always of even-order. If all the 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.

Higher-Order Hamiltonian for Circuits with (α,β) Elements

“…The classical form of the theorem, representing the case α = β = 0, means that the sum of instantaneous powers delivered to all elements in the circuit is zero. The generalized theorem (16) replaces the instantaneous power [VA] by a quantity [VA] α + β . This is because the integration and differentiation with respect to time belong to Kirchhoff's operators [19], which do not affect the validity of Kirchhoff's laws and the theorems derived from them.…”

“…Two different representations of potential functions correspond to the concept of functions and co-functions introduced by Millar [13] and Cherry [14] in 1951. [15] consisting of dissipative R, FDNR, FDNC elements [16]; the power preserved in the circuit corresponds to the diagonal Σ = 0.…”

“…which is the essence of Hamilton's variational principle [18]. The higher-order Lagrangian (3) was used for the first time to describe circuits built from HOEs in [16]. This work demonstrated that the Lagrangian, which has the ability to generate equations of motion, can only be drawn for the Σ-circuits that contain only elements from the common Σ-diagonal of Chua's table, where the sum of the indices α and β is preserved; thus, Σ = α + β.…”

“…The summation is done through all ε i elements of the given Σ-diagonal, where i is the position of the element on the diagonal (see Figure 2). The signs in front of the state functions in the sums of (6) are governed by the type (even or odd) of the positions of the elements on the diagonal [16]. The difference between kinetic and potential energy, transferring between the inertial and the accumulating elements, which are the common form of the Lagrangian used in classical mechanics, are merely a special case of (6).…”

“…This is the energy [Volt·Amper·sec] in LC and the power [Volt·Amper] in resistive circuits. An example of the Lagrangian of the Pais-Uhlenbeck oscillator [15], consisting of three resistive elements of R, FDNR, and FDNC types, is given in [16].…”