The principle of least action

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This paper elaborates on the principle of least action, particularly through Fermat's principle of least time, which asserts that light travels the path that minimizes travel time. By employing the Calculus of Variations, it derives the equations governing light's straight-line propagation in homogeneous and isotropic media and extends the discussion to refraction phenomena via the refractive index. Additionally, it connects the principles of least action with the motion of charged particles in electromagnetic fields, showcasing the versatility and application of these foundational concepts in classical physics.

Figures (4)

Figure 1: Light sent by a building in H appears as sent by H’, due to the variation of the refractive index with the altitude z.

Figure 2: A bead sliding through a wire hoop. theta is the generalised coordinate which defines completely the system. Qa SSS es Ae? See SS ae ee ee See Se ee See eee ee Me ae oe eee several Euler-Lagrange equations will follow from the action integral. In Classical Mechanics things can become very complicated when constraints are introducec Consider, for instance, a bead sliding without friction in a wire hoop of radius R, lying flat o the plane z = 0 (see Figure 2). The problem can be studied within a newtonian framework, b setting up equations like (9) for both variables x and y. There will be x- and y- components fc external forces. Other forces, arising from the bead being compelled to follow a circular trajector will have to be considered in addition to the external ones. We realise pretty soon, though, the the problem is inherently simpler than what appears from the formalism just described. First « all, one variable, the angle 0, and not two is sufficient to describe the bead’s motion. Seconc we should not, really, need to consider additional forces, as they are only a spurious consequenc of the circular constraint. Thus, a variable 0 can be used to describe our system. This wil automatically, take the fictitious forces into account, as @ is a circular variable. But, what wil at this point, be the equation of motion? It can be shown that these can again be derived bh Hamilton’s principle. A lagrangian L(t, 0,0) will be set up, first, and an Euler-Lagrange equatio will follow. @ cannot be considered a coordinate like x or y, but it is nonetheless a valid variable t describe the bead’s dynamic evolution. To variables like 0, that are not conventional coordinate but fully describe the dynamic evolution of a mechanical system, the name generalised coordinate has been assigned. In general, a mechanical system with n degrees of freedom can be entirel defined using n generalised coordinates. It is customary to use symbols qj, q2,..-,Gn to indicat these variables, while q1, qo,...,gn, are their time derivatives. The lagrangian is the differenc between kinetic and potential energies expressed using generalised coordinates,

Figure 3: Motion of a particle in a uniform gravitational field. Let us consider the bead-in-the-hoop example. First we have to express x, y and z as functions of the generalised coordinate 0:

Figure 4: Motion of a particle in a uniform gravitational field. where m is the particle mass and g gravity acceleration. The lagrangian is therefore defined as: