^{1}

^{1}

^{2}

^{3}

^{1}

^{2}

^{3}

The classic form of Hamilton’s variational principle does not hold for circuits with dissipative elements. It is shown in the paper that this may not be true in the case of systems consisting of the so-called higher-order elements. Hamilton’s principle is then extended to circuits containing the classical resistors and Frequency Dependent Negative Resistors (FDNRs). The extension is also made to any pair of elements which are the nearest neighbours on any

Hamilton’s variational principle is one of the jewels of classical mechanics [

For a transition between two states, the system seems to be searching for a trajectory for which the action integral acquires a stationary value. For one degree of freedom, the situation is illustrated in Figure

Virtual trajectory (thin line) differs from real trajectory (thick line) by the virtual deviation

It is well-known that condition (

These equations of motion represent the equilibrium of the so-called generalized forces [

However, nonconservative forces cannot be generated from a Lagrangian that is dependent on coordinates and velocities [

Hamilton’s variational principle also holds for systems with external excitations. The potential energy and therefore the Lagrangian

Lagrange’s and Hamilton’s formalisms represent an elegant approach to studying the system dynamics. That is why it is preferred when constructing modern physical theories. As the characteristic attribute of Lagrange’s formalism, the entire information on the time evolution of a system is contained in the scalar function—the Lagrangian. If the Lagrangian exists, and if it can be found, then it can generate the equations of motion of the system. Hamilton’s formalism can also provide information on the quantities that are conserved in the system, on important symmetries, and so forth. No wonder that “Today most physicists would be not only willing to accept as axiomatic the existence of a variational principle but would be also loath to accept any dynamical equations that were not derivable from such a principle” [

In addition to mechanics, Hamilton’s principle is also used in other branches of science. In electric circuits described by methods of loop variables, the coordinate

The following notation is used in (

The coefficients

The dissipative resistive elements are characterized by the constitutive relations

The subscript

The coefficients

The so-called (

All currently known elements from Chua’s table.

The memristor as the (−1, −1) element appeared in 1971 [

Another element termed FDNR (Frequency Dependent Negative Resistor) has been known since 1968 [

In 2002, the discovery of a new mechanical element termed inerter was announced in [

New mechanical and electrical elements bring a new inspiration to these disciplines. This paper demonstrates, on the example of a circuit consisting solely of resistors and FDNRs, that the classical version of Hamilton’s principle can also be applied to some systems that contain dissipative elements.

Consider a circuit consisting of ideal, generally nonlinear resistors, and Frequency Dependent Negative Resistors (FDNRs). The following consideration assumes the circuit to be described by the method of looped charges. However, the choice of the method is not essential.

Let each resistor be described by its constitutive relation

Furthermore, let each FDNR be defined by its constitutive relation

According to the definition in [

The charges

The coefficients

Now consider the vector variation of the trajectory (

The partial derivatives in (

The first factor (multiplicand) on the right side of (

The inner sum in (

A similar consideration of the effect of the trajectory variation on the dissipation function (

The inner sum in (

Since this is a variational problem with fixed endpoints (the coordinates

The

Then, in a circuit built exclusively from resistors and FDNRs, the following variational principle holds:

Hamilton’s variational principle (

It is apparent from Figure

Circuits composed of color-marked pairs of elements located on

The MOVE transformation only changes the generalized coordinates, and it cannot have any effect on the validity of Hamilton’s variational principle. This change will modify only the physical dimension of the Lagrange function, while its form remains unchanged. The color-coded pairs of the elements in Figure

As can be seen in Figure

The MOVE transformation can also yield a circuit made up entirely of ^{2}].

The

Examples of two other dual pairs, _{lin} and _{lin}−

Figure

Analysis of the oscillator formed from a nonlinear resistor and a linear FDNR with the constitutive relations (

The simulation results are shown in Figure

Within the classical mechanics, Hamilton’s variational principle cannot be applied to dissipative processes. The same holds in the related scientific disciplines that have taken up this principle thanks to the fact that their specific generalized forces and coordinates are compatible with Hamilton’s principle. Electrical engineering is no exception. Its link to mechanics through fundamental

In terms of Chua’s table, Hamilton’s principle is only given for systems consisting of (0, −1) and (−1, 0) elements, which lie on the diagonal

Chua’s table provides a new view on the fundamental electrical elements. It brings new challenges not only for electrical engineering, but also for mechanics. A new mechanical element, the inerter, discovered in 2002, cannot be substituted in its nonlinear form by any combination of known inertial, accumulative, or dissipative mechanical elements. While the classical inertia provides the algebraic bond between momentum and velocity, the inerter provides the algebraic bond between the derivatives of those quantities, i.e., between the acting force and acceleration. The classical mechanics did not take into account the existence of this fundamental law. The inerter is therefore a heterogeneous element for Hamilton’s variational principle. As an element that lies on the diagonal

It follows from the analysis in part IV that each circuit consisting of

Within the classical mechanics and electrical engineering, no other element exists for making a pair with the resistor-type dissipative element, which is necessary for the synthesis of systems that do not violate Hamilton’s variational principle. It is shown in the paper that this missing element is the FDNR (1, −1) element. Its linear form, the (0, −2) element, creates a similar pair with the memristor. The applicability of Hamilton’s variational principle to dissipative systems can therefore be considered only after the original set of the three fundamental

Note that the validity of Hamilton’s variational principle for systems consisting of any pair of elements which are the nearest neighbours on arbitrary

No data were used to support this study.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the Czech Science Foundation under grant no. 18-21608S. For research, the infrastructure of K217 Department, UD Brno, was also used. The authors also wish to express their sincere thanks to the Open Access Fund of Brno University of Technology for covering the APC.