Conservation laws and extremal principles
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It is a consequence of Newton’s laws of motion that the total momentum remains constant in a system completely isolated from external influences. The only forces acting on any part of the system are those exerted by other parts; if these are taken in pairs, according to the third law, A exerts on B a force equal and opposite to that of B on A. Since, according to the second law, the momentum of each changes at a rate equal to the force acting on it, the momentum change of A is exactly equal and opposite to that of B when only mutual forces between these two are considered. Because the effects of separate forces are additive, it follows that for the system as a whole no momentum change occurs. The centre of mass of the whole system obeys the first law in remaining at rest or moving at a constant velocity, so long as no external influences are brought to bear. This is the oldest of the conservation laws and is invoked frequently in solving dynamic problems.
Conservation of angular momentum
The total angular momentum (also called moment of momentum) of an isolated system about a fixed point is conserved as well. The angular momentum of a particle of mass m moving with velocity v at the instant when it is at a distance r from the fixed point is mr ∧ v. The quantity written as r ∧ v is a vector (the vector product of r and v) having components with respect to Cartesian axes
The meaning is more easily appreciated if all the particles lie and move in a plane. The angular momentum of any one particle is the product of its momentum mv and the distance of nearest approach of the particle to the fixed point if it were to continue in a straight line. The vector is drawn normal to the plane. Conservation of total angular momentum does not follow immediately from Newton’s laws but demands the additional assumption that any pair of forces, action and reaction, are not only equal and opposite but act along the same line. This is always true for central forces, but it holds also for the frictional force developed along sliding surfaces. If angular momentum were not conserved, one might find an isolated body developing a spontaneous rotation with respect to the distant stars or, if rotating like the Earth, changing its rotational speed without any external cause. Such small changes as the Earth experiences are explicable in terms of disturbances from without—e.g., tidal forces exerted by the Moon. The law of conservation of angular momentum is not called into question.
Nevertheless, there are noncentral forces in nature, as, for example, when a charged particle moves past a bar magnet. If the line of motion and the axis of the magnet lie in a plane, the magnet exerts a force on the particle perpendicular to the plane while the magnetic field of the moving particle exerts an equal and opposite force on the magnet. At the same time, it exerts a couple tending to twist the magnet out of the plane. Angular momentum is not conserved unless one imagines that the balance of angular momentum is distributed in the space around the magnet and charge and changes as the particle moves past. The required result is neatly expressed by postulating the possible existence of magnetic poles that would generate a magnetic field analogous to the electric field of a charge (a bar magnet behaves roughly like two such poles of opposite sign, one near each end). Then there is associated with each pair, consisting of a charge q and a pole P, angular momentum μ0Pq/4π, as if the electric and magnetic fields together acted like a gyroscope spinning about the line joining P and q. With this contribution included in the sum, angular momentum is always conserved.
Conservation of energy
The device of associating mechanical properties with the fields, which up to this point had appeared merely as convenient mathematical constructions, has even greater implications when conservation of energy is considered. This conservation law, which is regarded as basic to physics, seems at first sight, from an atomic point of view, to be almost trivial. If two particles interact by central forces, for which a potential function ϕ may be defined such that grad ϕ gives the magnitude of the force experienced by each, it follows from Newton’s laws of motion that the sum of ϕ and of their separate kinetic energies, defined as 1/2mv2, remains constant. This sum is defined to be the total energy of the two particles and, by its definition, is automatically conserved. The argument may be extended to any number of particles interacting by means of central forces; a potential energy function may always be found, depending only on the relative positions of the particles, which may be added to the sum of the kinetic energies (depending only on the velocities) to give a total energy that is conserved.
The concept of potential energy, thus introduced as a formal device, acquires a more concrete appearance when it is expressed in terms of electric and magnetic field strengths for particles interacting by virtue of their charges. The quantities 1/2ε0Ε2 and B2/2μ0 may be interpreted as the contributions per unit volume of the electric and magnetic fields to the potential energy, and, when these are integrated over all space and added to the kinetic energy, the total energy thus expressed is a conserved quantity. These expressions were discovered during the heyday of ether theories, according to which all space is permeated by a medium capable of transmitting forces between particles (see above). The electric and magnetic fields were interpreted as descriptions of the state of strain of the ether, so that the location of stored energy throughout space was no more remarkable than it would be in a compressed spring. With the abandonment of the ether theories following the rise of relativity theory, this visualizable model ceased to have validity.
Conservation of mass-energy
The idea of energy as a real constituent of matter has, however, become too deeply rooted to be abandoned lightly, and most physicists find it useful to continue treating electric and magnetic fields as more than mathematical constructions. Far from being empty, free space is viewed as a storehouse for energy, with E and B providing not only an inventory but expressions for its movements as represented by the momentum carried in the fields. Wherever E and B are both present, and not parallel, there is a flux of energy, amounting to E ∧ B/μ0, crossing unit area and moving in a direction normal to the plane defined by E and B. This energy in motion confers momentum on the field, E ∧ B/μ0c, per unit volume as if there were mass associated with the field energy. Indeed, the English physicist J.J. Thomson showed in 1881 that the energy stored in the fields around a moving charged particle varies as the square of the velocity as if there were extra mass carried with the electric field around the particle. Herein lie the seeds of the general mass–energy relationship developed by Einstein in his special theory of relativity; E = mc2 expresses the association of mass with every form of energy. Neither of two separate conservation laws, that of energy and that of mass (the latter particularly the outcome of countless experiments involving chemical change), is in this view perfectly true, but together they constitute a single conservation law, which may be expressed in two equivalent ways—conservation of mass, if to the total energy E is ascribed mass E/c2, or conservation of energy, if to each mass m is ascribed energy mc2. The delicate measurements by Eötvös and later workers (see above) show that the gravitational forces acting on a body do not distinguish different types of mass, whether intrinsic to the fundamental particles or resulting from their kinetic and potential energies. For all its apparently artificial origins, then, this conservation law enshrines a very deep truth about the material universe, one that has not yet been fully explored.
An equally fundamental law, for which no exception is known, is that the total electrical charge in an isolated system is conserved. In the production of a negatively charged electron by an energetic gamma ray, for example, a positively charged positron is produced simultaneously. An isolated electron cannot disappear, though an electron and a positron, whose total charge is zero and whose mass is 2me (twice the mass of an electron), may simultaneously be annihilated. The energy equivalent of the destroyed mass appears as gamma ray energy 2mec2.
For macroscopic systems—i.e., those composed of objects massive enough for their atomic structure to be discounted in the analysis of their behaviour—the conservation law for energy assumes a different aspect. In the collision of two perfectly elastic objects, to which billiard balls are a good approximation, momentum and energy are both conserved. Given the paths and velocities before collision, those after collision can be calculated from the conservation laws alone. In reality, however, although momentum is always conserved, the kinetic energy of the separating balls is less than what they had on approach. Soft objects, indeed, may adhere on collision, losing most of their kinetic energy. The lost energy takes the form of heat, raising the temperature (if only imperceptibly) of the colliding objects. From the atomic viewpoint the total energy of a body may be divided into two portions: on the one hand, the external energy consisting of the potential energy associated with its position and the kinetic energy of motion of its centre of mass and its spin; and, on the other, the internal energy due to the arrangement and motion of its constituent atoms. In an inelastic collision the sum of internal and external energies is conserved, but some of the external energy of bodily motion is irretrievably transformed into internal random motions. The conservation of energy is expressed in the macroscopic language of the first law of thermodynamics—namely, energy is conserved provided that heat is taken into account. The irreversible nature of the transfer from external energy of organized motion to random internal energy is a manifestation of the second law of thermodynamics.
The irreversible degradation of external energy into random internal energy also explains the tendency of all systems to come to rest if left to themselves. If there is a configuration in which the potential energy is less than for any slightly different configuration, the system may find stable equilibrium here because there is no way in which it can lose more external energy, either potential or kinetic. This is an example of an extremal principle—that a state of stable equilibrium is one in which the potential energy is a minimum with respect to any small changes in configuration. It may be regarded as a special case of one of the most fundamental of physical laws, the principle of increase of entropy, which is a statement of the second law of thermodynamics in the form of an extremal principle—the equilibrium state of an isolated physical system is that in which the entropy takes the maximum possible value. This matter is discussed further below and, in particular, in the article thermodynamics.
Manifestations of the extremal principle
The earliest extremal principle to survive in modern physics was formulated by the French mathematician Pierre de Fermat in about 1660. As originally stated, the path taken by a ray of light between two fixed points in an arrangement of mirrors, lenses, and so forth, is that which takes the least time. The laws of reflection and refraction may be deduced from this principle if it is assumed as Fermat did, correctly, that in a medium of refractive index μ light travels more slowly than in free space by a factor μ. Strictly, the time taken along a true ray path is either less or greater than for any neighbouring path. If all paths in the neighbourhood take the same time, the two chosen points are such that light leaving one is focused on the other. The perfect example is exhibited by an elliptical mirror, such as the one in ; all paths from F1 to the ellipse and thence to F2 have the same length. In conventional optical terms, the ellipse has the property that every choice of paths obeys the law of reflection, and every ray from F1 converges after reflection onto F2. Also shown in the figure are two reflecting surfaces tangential to the ellipse that do not have the correct curvature to focus light from F1 onto F2. A ray is reflected from F1 to F2 only at the point of contact. For the flat reflector the path taken is the shortest of all in the vicinity, while for the reflector that is more strongly curved than the ellipse it is the longest. Fermat’s principle and its application to focusing by mirrors and lenses finds a natural explanation in the wave theory of light (see light: Basic concepts of wave theory).
A similar extremal principle in mechanics, the principle of least action, was proposed by the French mathematician and astronomer Pierre-Louis Moreau de Maupertuis but rigorously stated only much later, especially by the Irish mathematician and scientist William Rowan Hamilton in 1835. Though very general, it is well enough illustrated by a simple example, the path taken by a particle between two points A and B in a region where the potential ϕ(r) is everywhere defined. Once the total energy E of the particle has been fixed, its kinetic energy T at any point P is the difference between E and the potential energy ϕ at P. If any path between A and B is assumed to be followed, the velocity at each point may be calculated from T, and hence the time t between the moment of departure from A and passage through P. The action for this path is found by evaluating the integral ∫BA (T - ϕ)dt, and the actual path taken by the particle is that for which the action is minimal. It may be remarked that both Fermat and Maupertuis were guided by Aristotelian notions of economy in nature that have been found, if not actively misleading, too imprecise to retain a place in modern science.
Fermat’s and Hamilton’s principles are but two examples out of many whereby a procedure is established for finding the correct solution to a problem by discovering under what conditions a certain function takes an extremal value. The advantages of such an approach are that it brings into play the powerful mathematical techniques of the calculus of variations and, perhaps even more important, that in dealing with very complex situations it may allow a systematic approach by computational means to a solution that may not be exact but is near enough the right answer to be useful.
Fermat’s principle, stated as a theorem concerning light rays but later restated in terms of the wave theory, found an almost exact parallel in the development of wave mechanics. The association of a wave with a particle by the physicists Louis-Victor de Broglie and Erwin Schrödinger was made in such a way that the principle of least action followed by an analogous argument.
