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Least Action Principle in Electromagnetics

Laws of nature are governed by following fundamental principles - the action principle, locality, Lorentz invariance, and gauge invariance [1]. Hamilton's principle, or the least action principle, is originally developed for classical mechanics stating that a particle, among all of the trajectories between fixed time instants t1 and t2, follows the path which minimizes the action. Action is defined as time integral of the difference between the kinetic energy and potential energy, respectively. Thus, Hamilton's principle somehow requires the time averages of the kinetic energy and potential energy to distribute as equally as possible (equipartition) [2]. In classical mechanics, Hamilton's principle and Newton's second law represent equivalent formulations.

An extension of Hamilton's principle from classical mechanics to classical electromagnetics can be undertaken starting with the analysis of the motion of single charged particle [3]. Next step is to construct a Lagrangian for the electromagnetic field by extending the Lagrangian pertaining to classical mechanics. From the corresponding Lagrangians, featuring Noether's theorem and gauge invariance, it is possible to derive equation of continuity for the charge, Lorentz force, and Maxwell's equations, which can be found elsewhere, e.g. [2-5].

Generally, when a functional is extremal, Noether's theorem yields the conservation law. Thus, invariance of the system under a time translation results in the energy conservation. It is also worth noting that space translation invariance corresponds to the conservation of linear momentum, rotation invariance corresponds to the conservation of angular momentum, while gauge invariance yields the charge conservation [1, 2].

These derivations are recently reviewed in [6-8].

This chapter first deals with derivation of continuity equation and Lorentz force, and a derivation of Maxwell's equations from the electromagnetic field functional is carried out. Finally, a variational basis of numerical solution methods in electromagnetics is discussed.

1.1 Hamilton Principle

Hamilton variational principle represents not only the basis of modern analytical dynamics but also of universal physical laws, i.e. fundamental laws of classical physics can be understood in terms of action.

This section first deals with Hamilton's variational principle in mechanics, Newton's equation of motion, and Noether's theorem. Then the variational principle in electromagnetics is discussed.

For simplicity, a system with one degree of freedom represented by a generalized coordinate q is considered together with related function of position, velocity, and time where denotes the time derivative.

The task is to determine how a point particle should move in this one-dimensional space so that the time integral of L is minimized compared with the integral over the conceivable paths between the same starting and end points, as depicted in Fig. 1.1. The solution is given by stating q as a function of time q = q(t).

To compare all paths having the same starting and end points, the variation of function q is zero at both ends [1-4]

i.e. all alternatives start at instant t1 and arrive together at instant t2.

The minimum condition is then given by a functional F expressed in terms of the integral [1-4]

Figure 1.1 The varied function q(t).

In classical mechanics, function L is referred to as Lagrangian and is expressed as

where Wkin and Wpot are the kinetic energy and potential energy, respectively.

According to the calculus of variation, the functional approaches minimum value

when its variation vanishes, i.e.:

which can also be written as

Therefore, function q(t) minimizes functional (1.2) or (1.4), respectively, and it follows

For the simplest case given by the variation of function L is given by

And by performing some further mathematical manipulation, one readily obtains

As dq = 0 at the ends of the path, the second term at the right-hand side automatically vanishes.

Furthermore, according to the fundamental lemma of variational calculus [4] the first integral term at the right-hand side of (1.9) vanishes if the following condition is satisfied

It is worth noting that the second order differential Eq. (1.10) relates the position q for the time t, and also determines the true path of the system when two end positions and times are given. This equation is known as Lagrange-Euler equation of motion [1, 2].

In the case of a single function of multiple variables, (1.10) becomes

Finally, if few functions of multiple independent variables are considered, it follows

which, for example, pertains to three-dimensional problems in electromagnetics.

Hamilton variational principle can be considered a general law not only for particle dynamics but also for the dynamics of continuous materials.

An extension to three-dimensional problems in continuous materials, i.e. for physical fields, the variational principle corresponding to Eq. (1.6) is given by

where is the so-called Lagrange density defined as

and has a unit of energy per volume.

It is worth noting that the variational principle is an invariant for coordinate transformations [1].

1.2 Newton's Equation of Motion from Lagrangian

Lagrangian in classical mechanics for a particle with mass m with displacement at time t is of the form

where is the particle velocity and stands for potential energy (not due to electromagnetic field).

The corresponding action F0 related to Lagrangian (1.15) is given by integral

Now varying the action (1.16)

It can be written as:

and it follows:

which simply gives:

If mass m is regarded as constant quantity, right-hand side of (1.21) represents the components of mechanical force:

As the mechanical force is defined as a negative gradient of the potential energy

one simply obtains a set of Newton's equations of motion

where the right-hand side of (1.24) represents a force acting on the particle.

1.3 Noether's Theorem and Conservation Laws

There are physical quantities that do not change throughout the time development of physical systems. These quantities are stated to be conserved under certain conditions which are governed by conservation laws. It can be shown that conservation laws are a consequence of the symmetry properties of a physical system (invariance properties of a system under a group of transformations [2, 9]). The symmetry properties of the system and conservation laws are connected with Noether's theorem, e.g. [2, 9].

Let uk(x) (k = 1,2,.n) be a set of differentiable functions of the independent variable x and let vk(x) be the first derivatives, i.e. it can be written as

Now Lagrangian L is defined as a function of x, and n functions of uk and n functions of vk.

If one considers an infinitesimal transformation T

Furthermore, under T, one consequently has:

Now assuming the functional

to be invariant under T so that the transformation (1.30) which maps the interval O into O´ does not change, it follows

Performing some mathematical manipulations, one obtains the following expression

Finally, Noether's theorem states that if functional S is invariant under the infinitesimal one-parameter group of transformations T, then the set of n equations

simply gives

i.e. it can be written as

namely, the expression is...