Why principle of least action

As discussed below, the equations for the field Maxwell equations can also be derived from an action principle. Action principles are important also in general relativity. The proper time is stationary, here a maximum, for the true trajectory which is straight in a Lorentz frame compared to the proper time for all virtual trajectories.

The principle of stationary proper time, or maximal aging, is also valid in general relativity for the motion of a test particle in a gravitational field Taylor and Wheeler ; for "short" true trajectories the proper time is a maximum, and for "long" true trajectories "long" and "short" trajectories are defined in Section 5 the proper time is a saddle point Misner et al.

The corresponding Euler-Lagrange equation of motion is the relativistic geodesic equation. In general relativity the Einstein gravitational field equations can also be derived from an action principle, using the so-called Einstein-Hilbert action Landau and Lifshitz , Misner et al.

General relativity is perhaps the first, and still the best, example of a field where new laws of physics were derived heuristically from action principles, since Einstein and Hilbert were both motivated by action principles, at least partly, in establishing the field equations, and the principle of stationary proper time was used to obtain the equation of motion of a test particle in a gravitational field.

A second example is modern Yang-Mills type gauge field theory. Some of the pioneers e. Some of the early gauge theories were unified field theories of gravitational and electromagnetic fields interacting with matter, and other early unified field theories developed by Einstein, Hilbert and others were also based on action principles Vizgin Modern quantum field theories under development, for gravity alone Rovelli or unified theories Freedman and Van Proeyen , Zwiebach , Weinberg , are usually based on action principles.

The earliest general quantum field theory Heisenberg and Pauli , essentially the theory used in the s for quantum electrodynamics , strong, and weak interactions Wentzel , and the basis of one of the modern methods Weinberg , derives from action principles; commutation relations or anticommutation relations for fermion fields are applied to the field components and their conjugate momenta, with the latter being determined from the Hamilton principle and Lagrangian density for the classical fields Section As for the role of action principles in the creation of quantum mechanics in , in the case of wave mechanics, following hints given in de Broglie's Ph.

Heisenberg did not use action principles in creating matrix mechanics, but his close collaborators Born and Jordan immediately showed that the equations of motion in matrix mechanics can be derived from a matrix mechanics version of Hamilton's principle. Later, following a hint from Dirac in , in his Ph. A very general quantum operator version of Hamilton's principle was devised by Schwinger in Schwinger , Toms As is well known e. The word stationary is used in this section with two different meanings.

Equation 18 is the Euler-Lagrange equation for The reader will notice the striking similarity of 19 to one of the classical variational principles discussed above in Section 7, i. The above heuristic arguments can be tightened up. Conversely, one can "derive" quantum mechanics i. This enabled him to exploit the analogy between ray and wave optics, on the one hand, and particle and wave mechanics, on the other.

A semiclassical variational principle can be based on the reciprocal Maupertuis principle 20 Gray et al. For simplicity, consider first one-dimensional systems. This gives the allowed energies semiclassically as a function of the quantum number. The latter is most easily derived in the Wentzel-Kramers-Brillouin or WKB-like semiclassical approximations in wave mechanics or path integrals, and accounts approximately for some of the quantum effects missing in Bohr-Sommerfeld theory, such as zero-point energy, the uncertainty principle, wave function penetration beyond classical turning points and tunnelling.

The effect of the Morse-Maslov index is more noticeable at smaller quantum numbers. The EBK quantization rule was introduced originally to handle nonseparable, but integrable, multidimensional systems Brack and Bhaduri The classical bound motions are all periodic or quasiperiodic, i. The energy level degeneracies are not given correctly by this first approximation, and simple variational and perturbational improvements are also discussed in the review cited. As discussed in Section 2, in classical mechanics per se there is no particular physical reason for the existence of a principle of stationary action.

However, as first discussed by Dirac and Feynman, Hamilton's principle can be derived in the classical limit of the path integral formulation of quantum mechanics Feynman and Hibbs , Schulman Thus, in the classical limit, the classical Hamilton principle of stationary action is a consequence of the quantum stationary phase condition for constructive interference. There is an extensive literature on a variety of systems particles and fields studied semiclassically via the Feynman path integral expression for the propagator discussed in the preceding paragraph Feynman and Hibbs , Schulman , Brack and Bhaduri A celebrated result of this approach is the Gutzwiller trace formula , which relates the distribution function for the quantized energy levels of the system to the complete set of the system's classical periodic orbits.

The systems can be nonrelativistic or relativistic. These methods are also widely applied in classical continuum mechanics, e. As our first example, we consider the classical nonrelativistic one-dimensional vibrating string with fixed ends, following Brizard Again we have a choice of sign in 27 and have chosen that of Melia , opposite to that of Jackson As mentioned above, the other two Maxwell equations are satisfied identically by the representation of the field in terms of the four-potential, i.

Conservation laws are a consequence of symmetries of the Lagrangian or action. For example, conservation of energy, momentum, and angular momentum follow from invariance under time translations, space translations, and rotations, respectively.

To find an answer and to test consequences for quantum physics is the aim of this publication. During the last three centuries, no other principle has nourished hopes into a universal theory, has constantly been plagued by mathematical challenges, and has ignited metaphysical controversies about causality and teleology more than did the principle of least action [3].

Traditionally, the principle of least action has been thought to have a deeper philosophical significance because it seems to suggest that physical systems are governed by final causes, or that the cause of something has the character of a final cause.

The principle of least action was therefore connected with teleology, which contends that natural phenomena have intrinsic purposes. Why is the principle of least action functioning this way? Leibniz recognized a principle of determination derivable from maxima and minima, such as done in the principle of least action [4].

Planck considered the principle of least action as a significant step towards the aim of attaining knowledge about the real world [6]. He concluded that among the achievements of physical science the principle of least action comes closest to the final goal of theoretical research. Even Einstein came to the conclusion that the principle had to be an essential element in his general theory of relativity [7].

How can the principle of least action be understood in more detail? The path, described by the least action integral, can be split up in many infinitesimally small sections, which equally have to follow the principle of least action. We replace the Lagrange function L, which expresses the dynamically available energy, which generates motion, by a more general energy quantity E, which is dependent on time, position and velocity, since L in physics is treated as a scalar quantity only.

Then one obtains for an infinitesimal section of the action integral variables: position, time :. Its ability to minimize as an infinitesimally small section is a mathematical necessity, if the principle of least action should be generally valid Feynman.

Mathematically this is entirely clear, since a deviation from such a condition for only one infinitesimal section would violate the principle of least action in general. The derivative with respect to time and location has consequently to become a minimum, approaching zero. The behavior of the infinitesimal action remains merely determined by the energy quantity.

This energy has to have the property to decrease and minimize. What does it now mean when the energy, which produces movement, has to approach a minimum. What are the possible interpretations for this minimization process? Feynman comes to the conclusion that the differential statement on the path of least action can only concern the derivatives of the potential, that is the force at a point. The interpretation given here is different and definitively simpler. Relation 4 clearly shows that the dynamic energy quantity has to have itself the ability to minimize.

It is important now to recall first what this energy means. It would anyway stay constant during the dynamic process, which is subject to the minimization of least action, while energy is being converted and entropy generated.

It describes the energy, which becomes available for generating dynamic motion. It is the free energy, which can be converted into other energy forms and into not any more available entropic energy. This free energy can of course and also must decrease during an energy conversion process. Now it becomes clear why the Lagrange function L is, in this publication, replaced by a generalized energy E. The Lagrange function in classical physics is defined as a scalar quantity, a number.

A number has no tendency or ability to minimize. It is defined to act as a mere number. This means that the condition that an infinitesimal segment of action, as described in 3 , is minimized, cannot be fulfilled. It is important to point out that the present physical formalism does not respect this consequence. Free energy, as considered in the Lagrange function L, which is generating motion, is handled as a scalar quantity.

Scalar energy, energy as it is understood in physics now, is defined to have the ability to work, but no interest. Nevertheless an apparent solution was found. It is variational calculus. In fact, variational calculus is imposing and simulating a variation, which a scalar quantity itself cannot perform.

This enables the consequence that the properties of the principle of least action can at least partially and superficially be simulated and exploited. Why is physics doing that? All fundamental physical laws are formulated in such a way as to function in both positive and negative time direction. There is now no fundamental law in physics claiming a preferred time direction, as the here discussed conditions 3 and 4 do. Information, however, has an energy content.

This energy is thrown away, which explains why the system cannot any more return to the beginning, but is then proceeding in one direction only the same mathematics could also allow the function to proceed into opposite time direction, which is not observed.

The derived function can, for this obvious condition, not recover the original situation. The statistical time direction is thus just manipulated mathematically and would anyway not work where self-organization and local reduction of entropy takes place.

Einstein himself compared our understanding of energy with a beggar, who actually is a millionaire, but nobody knows and sees it. This situation is strange, because our concept of energy did evolve from efforts, since antiquity, to understand change.

Therefore the question was asked by philosophers and early scientists as to something, which remains conserved within all changes around us. What remains conserved turned out to be energy. However, during the development and optimization of the energy concept in the course of the 19th century energy lost its relation to changes and irreversibility and became a scalar.

It is now a quantity, which is just a number, without any relation to change. In contrast, it is known that all changes C are originating from conversion of energy E. Changes must consequently be a function of energy.

Mathematically also the inverse relation must therefore hold:. It essentially leads to the same conclusion as to be drawn from relation 4 : Energy has an inherent property related to change provided the constraints of the system allow that. Such a property is today however not recognized. Energy, as handled today, is just a scalar quantity, a number, without relation to change.

But the infinitesimal section of action 3 must necessarily express the ability to minimize. It must be able to decrease, which means that the energy must be able to decrease its presence in this state relation 4. The process is time oriented. When every point on the track of a stone rolling down a hill minimizes the presence of energy per state, a minimum action route will automatically result. It is consequently claimed here, that available energy is fundamentally time oriented and aims at decreasing its presence per state.

This means a paradigm change, since a time orientation is fundamentally imposed. But the principle of least action only works for conservative systems—where all forces can be gotten from a potential function. You know, however, that on a microscopic level—on the deepest level of physics—there are no nonconservative forces. Nonconservative forces, like friction, appear only because we neglect microscopic complications—there are just too many particles to analyze.

But the fundamental laws can be put in the form of a principle of least action. Suppose we ask what happens if the particle moves relativistically. The question is: Is there a corresponding principle of least action for the relativistic case?

There is. Of course, we are then including only electromagnetic forces. This action function gives the complete theory of relativistic motion of a single particle in an electromagnetic field. I will leave to the more ingenious of you the problem to demonstrate that this action formula does, in fact, give the correct equations of motion for relativity. The variations get much more complicated. But I will leave that for you to play with. The question of what the action should be for any particular case must be determined by some kind of trial and error.

It is just the same problem as determining what are the laws of motion in the first place. You just have to fiddle around with the equations that you know and see if you can get them into the form of the principle of least action. So now you too will call the new function the action, and pretty soon everybody will call it by that simple name. There is quite a difference in the characteristic of a law which says a certain integral from one place to another is a minimum—which tells something about the whole path—and of a law which says that as you go along, there is a force that makes it accelerate.

The second way tells how you inch your way along the path, and the other is a grand statement about the whole path. In the case of light, we talked about the connection of these two. Now, I would like to explain why it is true that there are differential laws when there is a least action principle of this kind. The reason is the following: Consider the actual path in space and time. Otherwise you could just fiddle with just that piece of the path and make the whole integral a little lower.

And this is true no matter how short the subsection. Therefore, the principle that the whole path gives a minimum can be stated also by saying that an infinitesimal section of path also has a curve such that it has a minimum action. The only thing that you have to discuss is the first-order change in the potential. The answer can only depend on the derivative of the potential and not on the potential everywhere.

So the statement about the gross property of the whole path becomes a statement of what happens for a short section of the path—a differential statement. And this differential statement only involves the derivatives of the potential, that is, the force at a point. From the differential point of view, it is easy to understand. Every moment it gets an acceleration and knows only what to do at that instant.

But all your instincts on cause and effect go haywire when you say that the particle decides to take the path that is going to give the minimum action. The miracle of it all is, of course, that it does just that.

So our principle of least action is incompletely stated. You remember that the way light chose the shortest time was this: If it went on a path that took a different amount of time, it would arrive at a different phase. And the total amplitude at some point is the sum of contributions of amplitude for all the different ways the light can arrive.

But if you can find a whole sequence of paths which have phases almost all the same, then the little contributions will add up and you get a reasonable total amplitude to arrive. The important path becomes the one for which there are many nearby paths which give the same phase.

The total amplitude can be written as the sum of the amplitudes for each possible path—for each way of arrival. Then we add them all together. What do we take for the amplitude for each path? Our action integral tells us what the amplitude for a single path ought to be. It is the constant that determines when quantum mechanics is important.

One path contributes a certain amplitude. Only those paths will be the important ones. Adam Smith and his followers advocate minimal control of markets. History shows that economies with less control have been far more successful than those under complete government control, such as those in some communist countries. What about politics? Least action can lead to tragedies as we have seen in the events in Srebrenica in Bosnia.

On the other hand, politicians in America, China and Russia suggest that a drastic action, such as the bombing of Iran, can cause major turmoil in the Middle East. Negotiations and economic pressures are preferred. Let us see if this principle of least action works out to the benefit of all.

An issue that may be badly affected by least action is global warming. It has been scientifically established that activities of human beings are causing this warming. If unchecked, it can give rise to severe consequences, such as flooding, severe storms, desertification and spread of disease.

Governments are dragging their feet on this issue. Is this least action beneficial? He is contactable at mhaldar swinburne. Contact us Campuses Staff directory Media enquiries.