Quantum Mechanics of Non-Hamiltonian and Dissipative Systems

Journal of the Indian Institute of Science

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This makes the magnetic stored energy function more linear with respect to. We can write , as in Equation These abstract aspects of the meter and their relationships are shown in Figure 8. There is only one generalised coordinate for this system, and the Lagrangian function for the linearised system can be written as. The Lagrangian function for this electromechanical system is written in Equation The current, i , comes from an ideal current source, so it is essentially a constraint, rather than an independent coordinate.

The last term in this Lagrangian function determines the coupling between the electrical and mechanical aspects of this system. We can apply Equation 4 to Equation 36 and obtain the equation of motion for the D'Arsonval meter:. This example shows that it is possible to model mixed mechanical and electrical electromechanical systems using Lagrangian techniques. Further, we show that the presence of vicious damping is no obstacle to Lagrangian analysis.

We have extended the range of applications of Lagrangian analysis, to include non-conservative systems that include dissipative forces. This has been achieved, even though it contradicts many of the accepted ideas in the current literature.

A Variational Approach to the Analysis of Dissipative Electromechanical Systems

We have also provided a systematic method of applying an extended type of Lagrangian analysis to non-conservative electromechanical systems. The successful application of Lagrangians in dissipative, non-conserved systems depends on the appropriate substitution of variables, the choice of Legendre transformations and the use of fractional calculus of variations. It is possible to extend Lagrangian techniques to non-linear dissipative systems, such as memristors or diodes, using Taylor's theorem, or by using repeated integration by parts.

If we could extend fractional calculus of variations to include generalised functions, such as white noise, then we could develop a fractional Malliavin calculus. The greater aim is to analyse electromechanical systems in the presence of noise. We expect that this would lead to the solution of the apparent paradoxes of the Penfield motor [7] , and the Davis electromechanical capacitor [43]. A complete theory should be compatible with Fluctuation Dissipation Theorem, as described by Weber [44] , for example. Is a statistical hypothesis test due to Granger [45] , which can be used to determining whether one time series is useful in forecasting another.

If we had a complete theory, which could model damping forces and fluctuations, then it would be interesting to see whether Granger's sense of causality could be used to allocate a direction to the time variable. Extremal principles can be used to create a number of numerical methods.

A number of numerical methods have recently been proposed in the literature, most notably in Almeida [25] and Pooseh [46] , [47]. The opportunities for numerical solution appear to be very good. The authors have had some success using optimisation packages, such as fincon in Matlab, and sqp in GNU Octave.

Such methods can be iterative, so an approximate solution can always be improved, through further iteration.

Quantization physics

These constants of the motion will be generalised forms of momentum and energy. This is discussed in Frederico [48] , [49]. For noisy electrical systems, with many degrees of freedom, it is of great theoretical interest to write down Liouville's theorem, in its most general form. The greater aim here is to understand the thermodynamics of electrical systems. It should be possible to create a time-average Lagrangian analysis for switched-mode systems. In summary, we argue that the generalised Lagrangian functions described in this paper are expected to have impact on theoretical and practical applications in electrical and mechanical engineering.

The authors thank Chris Illert for his helpful suggestions. Chris suggested that it was possible to construct a consistent and completely variational approach to dynamical problems. He also pointed out that Lagrangian functions can be imaginary, or complex. This is discussed in Illert [40] , for example. The authors also thank Gareth Bridges for helping with the proofing of the diagrams, and the choice of symbols and some aspects of notation.

National Center for Biotechnology Information , U. PLoS One. Published online Feb Pearce , 2 and Derek Abbott 1. Charles E. Teresa Serrano-Gotarredona, Editor. Author information Article notes Copyright and License information Disclaimer.

Competing Interests: The authors have declared that no competing interests exist. Received Nov 9; Accepted Sep 6. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are properly credited.

Abstract We develop a method for systematically constructing Lagrangian functions for dissipative mechanical, electrical, and electromechanical systems. Introduction and Motivation It is a widely believed that the Lagrangian approach to dynamical systems cannot be applied to dissipative systems that include non-conservative forces.

A short summary of the variational approach We can denote a Lagrangian function for a system as , then we can specify the total action of the system as. Fractional Calculus The indices of differentiation in The Euler Lagrange Equation 3 can be fractional, which leads to the formulation:. Discussion and Analysis A mechanical harmonic oscillator We consider a common problem from classical mechanics, of a mass on a spring. Open in a separate window. Figure 1.

A mass on a spring. A homomorphic mapping The example shown in Figure 1 is simple and well known, and lies completely within a mechanical problem domain. Table 1 A homomorphic mapping due to Karnopp et al. A homomorphic mapping: The names and purposes of the most important electromechanical dynamical concepts. An electrical harmonic system If we place a capacitor, in parallel and series with an inductor, , as shown in Figure 2 , then the resulting system will form an electromagnetic harmonic oscillator.

Figure 2. An LC electromagnetic harmonic circuit.

Introduction

Lagrangian terms for some common lumped electromechanical elements We can see from the last example that electrical and mechanical systems can be mapped onto one another but some care has to be taken with regard to what we regard as a coordinate. Table 2 Table of Lagrangian terms, in terms of current. Lagrangian terms, with current: We list the common electrical lumped parameters, and compare the admittance with the corresponding term from the Lagrangian function.

We also list the order of differentiation, k , and the corresponding term from the Euler-Lagrange equation. We can multiply the Lagrangian term by any constant that we like, as long as we do this consistently. In this paper, we rigorously adopt the convention that is used in mechanics, which means that we do not use the sign convention that is common in electrical engineering.

Table 3 Table of Lagrangian terms, in terms of voltage, v. Lagrangian terms, with voltage: We list common lumped electrical parameters, and compare the impedance with the corresponding term from the Lagrangian function. Table 4 Table of mechanical Lagrangian terms. Lagrangian terms for mechanical parameters: We list the common lumped mechanical parameters. From left to right, we list the common symbol for the parameter, the Lagrangian for the 3D vector case in terms of position or momentum , the Lagrangian for the 1D case in terms of the position only , the order of differentiation employed, and the resulting them in the Euler-Lagrange Equation.

Some authors introduce an additional minus sign, in order to make all of these terms positive. A damped mechanical harmonic system We consider the damped mechanical oscillator, as shown in Figure 3 , with mass, , spring constant, , and coefficient of damping,. Figure 3. A damped mechanical harmonic oscillator. The use of constraints The use of the calculus of variations to evaluate extremal functions, subject to constraints is described in a number of references, including Lanczos [2].

Figure 4. Two resistors in series. A problem with two resistors In Figure 4 a , the voltage source, places a constraint on the voltages across the two resistors, and. A damped electrical harmonic system We can use the terms in Table 2 to write the Lagrangian function for the circuit in Figure 5 as:.

Figure 5. A parallel RLC circuit, with source. A ladder filter The use of constraints can be a powerful technique, but it does add some extra complication to the analysis. Figure 6. An electrical ladder-filter circuit. A electromechanical problem, the D'Arsonval galvanometer One of the great advantages of the Lagrangian approach is that it can be easily used to model devices that transduce energy between different forms. Figure 7. Physical layout of the D'Arsonval galvanometer. Figure 8.

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