Why does the principle of least action hold

Since only the first-order variation has to be zero, we can do the calculation by three successive shifts. We get one equation. Or, of course, in any order that you want. Anyway, you get three equations. I think that you can practically see that it is bound to work, but we will leave you to show for yourself that it will work for three dimensions.

If you have, say, two particles with a force between them, so that there is a mutual potential energy, then you just add the kinetic energy of both particles and take the potential energy of the mutual interaction. And what do you vary? You vary the paths of both particles. Then, for two particles moving in three dimensions, there are six equations. 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. The fact that quantum mechanics can be formulated in this way was discovered in by a student of that same teacher, Bader, I spoke of at the beginning of this lecture.

There are many very interesting ones. I will not try to list them all now but will only describe one more. Later on, when we come to a physical phenomenon which has a nice minimum principle, I will tell about it then. I want now to show that we can describe electrostatics, not by giving a differential equation for the field, but by saying that a certain integral is a maximum or a minimum. Now we can use this equation to integrate by parts. Attempts to extend the usual action principles to nonholonomic systems have been controversial and ultimately unsuccessful Papastavridis Hamilton's principle in its standard form 2 is not valid, but a more general and correct Galerkin-d'Alembert form has been derived.

These can be chosen as any two of the particle's three Cartesian coordinates with respect to axes with origin at the center of the sphere, or as latitude and longitude coordinates on the sphere surface, etc.

In essence, the Lagrange multipliers relax the constraints, with one multiplier for each constraint relaxed. In the literature e. The usual action principles are valid for this type of velocity-dependent constraint. The Dirac-type constraints are implemented by the method of Lagrange multipliers. In Section 7 we use Lagrange multipliers to relax the fundamental constraints of the Hamilton and Maupertuis principles.

In general, the action principles do not apply to dissipative systems, i. However, for some dissipative systems, including all one-dimensional ones, Lagrangians have been shown to exist, and Hamilton's principle then applies see Gray et al. More generally, the question of whether a Lagrangian and corresponding action principle exist for a particular dynamical system, given the equations of motion and the nature of the forces acting on the system, is referred to as the "inverse problem of the calculus of variations" Santilli Additional freedom of choice will also often exist.

By putting additional conditions on the Lagrangian we can narrow down the choice. Thus for the free particle in one dimension, using an inertial frame of reference and by requiring the Lagrangian function to be invariant under Galilean transformations, i. In this review we restrict ourselves to the most common case where the Lagrangian depends on the coordinates and their first derivatives, but when higher derivatives occur in the Lagrangian the Euler-Lagrange equation generalizes in a natural way Fox, As an example, in considering the vibrational motion of elastic continuum systems Section 11 such as beams and plates, the standard Lagrangian contains spatial second derivatives, and the corresponding Euler-Lagrange equation of motion contains spatial fourth derivatives Reddy, As defined in Section 1, action is never a local maximum, as we shall discuss.

In relativistic mechanics see Section 9 two sign conventions for the action have been employed, and whether the action is never a maximum or never a minimum depends on which convention is used. In our convention it is never a minimum. Establishing the existence of a kinetic focus using this criterion is discussed by Fox An equivalent and more intuitive definition of a kinetic focus can be given. Based on this definition a simple prescription for finding the kinetic focus can be derived Gray and Taylor , i.

This is the first kinetic focus, usually called simply the kinetic focus. Subsequent kinetic foci may exist but we will not be concerned with them. The other trajectories shown in Figure 1 have their own kinetic foci, i.

For the purpose of determining the true trajectories, the nature of the stationary action minimum or saddle point is usually not of interest. However, there are situations where this is of interest, such as investigating whether a trajectory is stable or unstable Papastavridis , and in semiclassical mechanics where the phase of the propagator Section 10 depends on the true classical trajectory action and its stationary nature; the latter dependence is expressed in terms of the number of kinetic foci occurring between the end-points of the true trajectory Schulman In general relativity kinetic foci play a key role in establishing the Hawking-Penrose singularity theorems for the gravitational field Wald Kinetic foci are also of importance in electron and particle beam optics.

Finally, in seeking stationary action trajectories numerically Basile and Gray , Beck et al. If a minimum is being sought, comparison of the action at successive stages of the calculation gives an indication of the error in the trajectory at a given stage since the action should approach the minimum value monotonically from above as the trajectory is refined. The error sensitivity is, unfortunately, not particularly good, as, due the stationarity of the action, the error in the action is of second order in the error of the trajectory.

Thus a relatively large error in the trajectory can produce a small error in the action. For conservative time-invariant systems the Hamilton and Maupertuis principles are related by a Legendre transformation Gray et al. The two action principles are thus equivalent for conservative systems, and related by a Legendre transformation whereby one changes between energy and time as independent constraint parameters.

The existence in mechanics of two actions and two corresponding variational principles which determine the true trajectories, with a Legendre transformation between them, is analogous to the situation in thermodynamics Gray et al. There, as established by Gibbs, one introduces two free energies related by a Legendre transformation, i. We again restrict the discussion to time-invariant conservative systems. Next one can show Gray et al. In these various generalizations of Maupertuis' principle, conservation of energy is a consequence of the principle for time-invariant systems just as it is for Hamilton's principle , whereas conservation of energy is an assumption of the original Maupertuis principle.

It is possible to derive additional generalized principles Gray et al. As we shall see in the next section and in Section 10, the alternative formulations of the action principles we have considered, particularly the reciprocal Maupertuis principle, have advantages when using action principles to solve practical problems, and also in making the connection to quantum variational principles.

We note that reciprocal variational principles are common in geometry and in thermodynamics see Gray et al. Just as in quantum mechanics, variational principles can be used directly to solve a dynamics problem, without employing the equations of motion. This is termed the direct variational or Rayleigh-Ritz method. The solution may be exact in simple cases or essentially exact using numerical methods , or approximate and analytic using a restricted and simple set of trial trajectories.

We illustrate the approximation method with a simple example and refer the reader elsewhere for other pedagogical examples and more complicated examples dealing with research problems Gray et al. We wish to estimate this dependence. The frequency increases with amplitude, confirming what is seen in Fig.

This problem is simple enough that the exact solution can be found in terms of an elliptic integral Gray et al. Direct variational methods have been used relatively infrequently in classical mechanics Gray et al. These methods are widely used in quantum mechanics Epstein , Adhikari , classical continuum mechanics Reddy , and classical field theory Milton and Schwinger They are also used in mathematics to prove the existence of solutions of differential Euler-Lagrange equations Dacorogna The Hamilton and Maupertuis principles, and the generalizations discussed above in Section 7, can be made relativistic and put in either Lorentz covariant or noncovariant forms Gray et al.

The sign of the Lagrangian and corresponding action can be chosen arbitrarily since the action principle and equations of motion do not depend on this sign; here we choose the sign of Lanczos in 16 , opposite to that of Jackson The Hamilton principle is thus gauge invariant.

Specific examples, such as an electron in a uniform magnetic field, are discussed in the references Gray et al. 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. This explains why action is indeed minimized.

It is minimized because energy has the drive to minimize its presence per state. Thereby waste energy in not usable form is generated and entropy increases.

The second law of thermodynamics follows immediately, which is an important result, because it cannot be deduced from the present day time-invertible physical formalism. We can, of course, imagine, that the number, the scalar named Lagrange function, can vary and minimize.

This is actually done via the variational calculus to find the equations of motion. We are simply pretending that a number has the tendency to minimize. But nature does not care about our imagination. A stone rolls down a hill anyway following the principle of least action. And this can only happen when a law exists in which an infinitesimal action segment 3 has the property to minimize. In other words, the energy available for motion cannot be a scalar, it must have time oriented properties.

If the principle of least action is considered to be fundamentally relevant for physics, the definition for free energy must consequently change:. It is here suggested that the principle of least action is nothing else than the statement that our world is fundamentally time oriented and irreversible.

Rate controlling entropy production is critically and fundamentally shaping the change in our environment and determining the progress of time. How did the energy concept in present-day physics loose its relation to change, from where it was actually born?

The Italian-French mathematician J. Lagrange, around , when studying his famous energy equations for dynamic systems, still considered and investigated conditions, which reflected irreversibility and time orientation.

This means, he paid attention to change. Also the Irish mathematician W. Hamilton, when, during the first half of the 19th century, deriving the now famous Hamilton functions, still argued that external irreversibility should be considered. He also felt that there had to be a relation to change. Other scientists also had the impression that the principle of least action is related to change.

With Jacobi [10] together they recognized the meaning of the principle of least action in the least expenditure of work. Energy in present day physics, in fact, got the capacity to do work, but has no interest to do it and no preference to minimize, to decrease the presence of energy per state.

The principle of least action ended up in that scalar energy is treated with variational calculus to predict motion. This gives the superficial illusion that the principle of least action is compatible with time-reversibility. But it is not. It is a statement on time-orientation and was for that reason linked to teleology, which is always time oriented.

Historically, the question was asked how a system can know in advance via what path it can minimize action. Relation 3 and 4 , which describe minimization of an infinitesimally small interval of action, give a precise answer: it is sufficient to assume that free energy is time oriented, that it minimizes its presence per state. It will do that for any point of its path, subject to the constraints given, and thus find that path which is subject to a minimum of action.

A fundamentally directed and irreversible nature of energetic behavior in dynamic processes is the answer to the strange philosophical mystery around the principle of least action.

Energy must have a relation to change, as also deducible from consideration 5. When the conclusions drawn are reasonable, they can be tested. Quantum physics with its counter-intuitive aspects was selected for an intellectual experiment. Quantum states are presently defined as equilibrium states. What happens when they are considered fundamentally dynamic? When dealing with particle and wave in quantum physics subject to a dynamic energy it is clear that they cannot be equivalent and simultaneous.

The system with inferior energy value inferior capacity to do work will follow from that with higher free energy value. The system with inferior free energy must be the wave with spread out energy distribution in conventional quantum physics energy in particle and wave is considered equivalent. However the overall reversible nature of energy behavior within the particle-wave quantum phenomenon no energy is finally turned over also has to be considered.

Energy in a particle should be able to expand into a wave and it thereby also loses working ability while generating a fundamental type of entropic energy. The formalism for the quantum system, however, must therefore set aside energy in the form of information to guarantee reversibility in the absence of overall energy turnover. A kind of fundamental Maxwell demon is needed to bring back the energy distributed in a wave to the shape of the particle.

Since this demon will need information, and thus energy to do so, this energy has to be set aside by the particle before converting into a wave. It has to be set aside to act from the outside of the expanding energy system. The energy of the particle and of the wave should therefore not be identical and should not ignore space, as seen in Figure 1 a for conventional quantum physics, but the energy of particle E p should be equivalent to the. This would work via a fundamental Maxwell demon.

In other words: the energy in the particle E p converts into the distributed energy of the wave plus the. No energy is exchanged with the outside and the energy of information, which is set aside from the beginning, is tailored in such a way that the total energy as expressed in formula 7 is sustained. We have thus permitted that the energy converted from a particle into a wave loses some ability to perform work it assumes a microscopic form of entropic energy , but simultaneously provides energy in the form of information also called negentropic energy because it behaves and acts somehow in the opposite way to entropic energy , which subsequently reconverts the entropic energy.

The information provided by a hypothetical microscopic Maxwell demon is thus used to re-concentrate the energy into a particle Figure 1 b. This does not contradict the second law of thermodynamics, which states that the entropy increases in a closed space or volume, because information is assumed to come from outside.

The system works as it does with information provided to a three dimensional printer, which. Figure 1.