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On the Principle of Least Action International Journal of Physics, 2018, Vol. 6, No. 2, 47-52 Available online at http://pubs.sciepub.com/ijp/6/2/4 ©Science and Education Publishing DOI:10.12691/ijp-6-2-4 On the Principle of Least Action Vu B Ho* Advanced Study, 9 Adela Court, Mulgrave, Victoria 3170, Australia *Corresponding author: [email protected] Abstract Investigations into the nature of the principle of least action have shown that there is an intrinsic relationship between geometrical and topological methods and the variational principle in classical mechanics. In this work, we follow and extend this kind of mathematical analysis into the domain of quantum mechanics. First, we show that the identification of the momentum of a quantum particle with the de Broglie wavelength in 2-dimensional space would lead to an interesting feature; namely the action principle = 0 would be satisfied not only by the stationary path, corresponding to the classical motion, but also by any path. Thereupon the Bohr quantum condition possesses a topological character in the sense that the principal quantum훿푆 number is identified with the winding number, which is used to represent the fundamental group of paths. We extend our discussions into 3-dimensional space and show that the charge of a particle also possesses a topological character and푛 is quantised and classified by the homotopy group of closed surfaces. We then discuss the possibility to extend our discussions into spaces with higher dimensions and show that there exist physical quantities that can be quantised by the higher homotopy groups. Finally we note that if Einstein’s field equations of general relativity are derived from Hilbert’s action through the principle of least action then for the case of = 2 the field equations are satisfied by any metric if the energy-momentum tensor is identified with the metric tensor, similar to the case when the momentum of a particle is identified with the curvature of the particle’s path. 푛 Keywords: principle of least action, Feynman path integral method, classical mechanics, quantum mechanics, general relativity Cite This Article: Vu B Ho, “On the Principle of Least Action.” International Journal of Physics, vol. 6, no. 2 (2018): 47-52. doi: 10.12691/ijp-6-2-4. quantum condition can also be described in a classical way. In classical mechanics, the actual path of a particle is 1. Introduction found by extremising the action integral of the particle [8]. On the other hand, in quantum mechanics, the wave In classical physics, the principle of least action is a equation of a particle can be found by applying the variational principle that can be used to determine Feynman path integral formulation which assumes the uniquely the equations of motion for various physical particle can take any trajectory [9]. The question then systems. However, recently the principle of least action arises as to whether it is possible for the action integral to that is associated with new concepts of Lagrangian be extremised by any path at the quantum level. In this symmetry has been proposed and studied by many authors. work we show that this problem may be investigated in It has also been shown that different formulations of the terms of geometry and topology, and it transpires that principle of least action can be constructed for a physical topology may play an important role in the determination system that is described by a particular system of of the nature of a quantum observable [10,11]. differential equations [1-5]. Furthermore, new approach to the theories of gravitation using the principle of least action has also been considered [6]. To extend these 2. Principle of Least Action in Classical investigations, in this work we will consider the case in Mechanics which the principle of least action can be applied to both classical physics and quantum physics. The investigation of the relationship between physics In the old quantum theory, the Bohr quantum condition and geometry had been carried out and culminated with = , where is the momentum of a particle, is the development of the principle of least action. In 1744, Planck constant and is a positive integer, played a Euler developed and published his work on this crucial∮ 푝푑푠 role푛ℎ in the 푝quantum description of a physicalℎ variational principle for the dynamics of a particle moving system, although it had푛 been introduced into the quantum in a plane curve [12]. On the one hand, the dynamics of a theory in an ad hoc manner [7]. However, except for the particle can be studied by using the principle of least quantum condition imposed on the orbital angular action. On the other hand, the path of a particle can also be momentum, the Bohr model was based entirely on the determined by using geometrical methods, since a path classical dynamics of Newtonian physics. In this case, it can be constructed if the curvature is known at all points seems natural to raise the question as to whether the Bohr on the path. Euler showed the equivalence between these International Journal of Physics 48 two methods by calculating the radius of curvature of the vector ( ), the unit tangent vector ( ), the unit principal path directly and by means of the variational principle. normal vector ( ) and the unit binormal vector ( ), Consider a particle moving in a plane under the influence defined퐫 by푠 the relation ( ) = (퐭)푠× ( ) , satisfy the of a force. Let and be the forces per unit mass in the Frenet equations퐩 [14,15]푠 퐛 푠 x-direction and the y-direction, respectively. The normal 퐛 푠 퐭 푠 퐩 푠 푥 푦 ddtp d b acceleration 퐹 of the 퐹particle is given by =κκp,, =−+ tb =− p (4) ds ds ds 1 푛 − 푎 2 ( ) ( ) dy dy 2 where and are the curvature and the torsion aFnx= −+ F y 1 . (1) respectively, and = . is the linear element. If we dx dx consider휅 the푠 motion휚 푠 of a particle in a plane, as in the case 푑푠 √푑퐫 푑퐫 If is the radius of curvature, then = , of Bohr’s model of a hydrogen-like atom, the Frenet equations reduce to where is the speed of the particle. This result can also2 be 푛 obtained휌 by using the variational principle 푎 = 0−, 푣where⁄휌 ddtp 푣 =κκpt,. = − (5) is defined by ds ds 훿푆 2 푆 dy By differentiation, we obtain the following system of S= pds = p1. + dx (2) differential equations ∫∫dx dd2ttd(lnκ ) The condition = 0 leads to the stationary condition − +=κ 2t 0 (6) ds2 ds ds 훿푆 22 2 d ∂∂dy dy ddpp(lnκ ) d2 vv1+ −+=1 0. (3) − +=κ p 0. (7) dx dy dx∂ y dx 2 ds ds ∂ ds dx If the curvature ( ) is assumed to vary slowly along Employing the relations = 2 and = the curve ( ), so that the condition (ln ) = 0 can 2 , the stationary condition2 given by Equation2 (3) be imposed, then (휅 )푠 and ( ) may be regarded as being 푥 = 휕푣 ⁄휕푥 퐹 휕푣 ⁄휕푦 oscillating퐫 with푠 a spatial period, or wavelength,푑 휅 ⁄푑푠 λ, whose then푦 leads to . This result reveals an intrinsic relationship퐹 between 2 geometrical methods and the relationship to the퐭 curvature푠 퐩 푠 is found as 푎푛 − 푣 ⁄휌 variational principle in classical mechanics. 2π κ = 휅 . (8) λ 3. Principle of Least Action in Quantum In the case of the Bohr’s planar model of a hydrogen- Mechanics like atom with circular orbits, the condition (ln ) = 0 is always satisfied, since the curvature remains constant 푑 휅 ⁄푑푠 We will now extend this geometrical analysis into for each of the orbits. In order to incorporate this the domain of quantum mechanics. We show that the elementary differential geometry into quantum mechanics, identification of the momentum of a quantum particle with we identify the wavelength defined in Equation (8) with the de Broglie wavelength leads to an interesting feature; the de Broglie’s wavelength of a particle. This seems to be namely the action principle = 0 is satisfied not only by a natural identification since the spatial period is the the stationary path corresponding to the classical motion, wavelength of the unit tangent vector ( ) . With this but also by any path. In 훿푆this case the Bohr quantum assumption, the momentum of the particle and휆 the condition possesses a topological character in the curvature are related through the relation퐭 푠 푝 sense that the principal quantum number is identified p = κ. (9) with the winding number, which is used to represent the 휅 fundamental group of paths [13]. Consider 푛a curve in the We now want to show how this result leads to Bohr’s three-dimensional spatial continuum R which is a postulate of the quantisation of angular momentum. It topological image of an open segment of3 a straight line. should be mentioned here that this can only be discussed The curve can be represented by a real vector function in terms of Bohr’s quantum theory, or Feynman’s path ( ) . For the Bohr planar model, we utilise the integral methods, since these formulations do require fundamental homotopy group and it is convenient to concepts employed in classical physics, especially the consider퐫 푠 curves that may have points that correspond to concept of classical paths of a particle [9]. According to more than one value of the parameter . However, the canonical formulation of classical physics, the particle whenever the single-valuedness of the representation ( ) dynamics is governed by the action principle = is required for differential analysis of the푠 motion, the = 0 . Using the relationship = given in 훿푆 uncertainty principle in quantum mechanics may퐫 be푠 Equation (9) and the expression for the curvature of the invoked to shift the self-intersecting points into the third 훿path∫ 푝푑푠 ( ) of a particle in a plane, =푝 ′′ (ℏ1휅+ ′ ) , dimension to make the curve single-valued for the whole where = and = , the 휅 action2 3⁄2 domain of definition.
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