Principle of Least Action

12 Key excerpts on "Principle of Least Action"

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  Feynman's Thesis - A New Approach To Quantum Theory

  • Laurie M Brown(Author)
  • 2005(Publication Date)
  • World Scientific

    (Publisher)

  THE Principle of Least Action IN QUANTUM MECHANICS RICHARD P. FEYNMAN Abstract A generalization of quantum mechanics is given in which the cen-tral mathematical concept is the analogue of the action in classical mechanics. It is therefore applicable to mechanical systems whose equations of motion cannot be put into Hamiltonian form. It is only required that some form of least action principle be available. It is shown that if the action is the time integral of a function of velocity and position (that is, if a Lagrangian exists), the gener-alization reduces to the usual form of quantum mechanics. In the classical limit, the quantum equations go over into the correspond-ing classical ones, with the same action function. As a special problem, because of its application to electrody-namics, and because the results serve as a confirmation of the pro-posed generalization, the interaction of two systems through the agency of an intermediate harmonic oscillator is discussed in de-tail. It is shown that in quantum mechanics, just as in classical mechanics, under certain circumstances the oscillator can be com-pletely eliminated, its place being taken by a direct, but, in general, not instantaneous, interaction between the two systems. The work is non-relativistic throughout. I. Introduction Planck’s discovery in 1900 of the quantum properties of light led to an enormously deeper understanding of the attributes and behaviour of matter, through the advent of the methods of quantum mechanics. When, however, these same methods are turned to the problem of light and the electromagnetic field great difficulties arise which have not been surmounted satisfactorily, so that Planck’s observations still 1 2 Feynman’s Thesis — A New Approach to Quantum Theory remain without a consistent fundamental interpretation.

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  Math Unlimited

  Essays in Mathematics

  • R. Sujatha, H. N. Ramaswamy, C. S. Yogananda(Authors)
  • 2011(Publication Date)
  • CRC Press

    (Publisher)

  A problem that looms large in this search is the very structure of gravity. In this self-contained article we will attempt to explain this prob-lem. 294 Math Unlimited 2 The Principle of Least Action We start with a review of the Principle of Least Action. We are familiar with the concept of “total energy” for a simple mechanical system: it is the sum of kinetic and potential energies. For example, for a simple harmonic oscillator, the potential energy is V = 1 2 kx 2 , (1) where x represents the displacement of the spring from its equilibrium (re-laxed) position and k , the spring constant. The kinetic energy for this sys-tem is T = 1 2 m ˙ x 2 , (2) where ˙ x = dx dt represents the rate of change of position of the mass ‘m’ attached to the spring. The total energy is H = T + V = 1 2 m ˙ x 2 + 1 2 kx 2 . (3) Surprisingly, although the total energy is a useful quantity, it is not essential to our understanding of classical mechanics. Instead, the really important object in mechanics is the Lagrangian (Wikipedia: Lagrange) defined by L = T − V . (4) Explicitly, for the simple harmonic oscillator, this reads L ( x , ˙ x , t ) = 1 2 m ˙ x 2 − 1 2 kx 2 . (5) The reason the Lagrangian is such an important object is because of the “Principle of Least Action” which we now state. Consider a classical system at position x 1 at time t 1 and at position x 2 at time t 2 . This system will always move from x 1 to x 2 in a manner such that the integral S = t 2 t 1 L ( x , ˙ x , t ) dt , (6) The forces of Nature 295 takes the least possible value 1 . This is referred to as the Principle of Least Action ( S being the action) and may be expressed as the following variation δ S = δ t 2 t 1 L ( x , ˙ x , t ) dt = 0 . (7) It is amazing that all of classical mechanics (including laws like the con-servation of energy) follows from this beautiful 2 principle. The entire field of mathematics, called the “Calculus of variations”, is de-voted to extremizing functionals as in (7).

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  The Classical Theory of Fields

  • L D Landau(Author)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  CHAPTER 2

  RELATIVISTIC MECHANICS

  Publisher Summary

  This chapter focuses on the relativistic mechanics of Einstein theory of relativity. The Principle of Least Action asserts that the integral S must be a minimum only for infinitesimal lengths of the path of integration. For paths of arbitrary length, it can be said that S must be an extremum and not necessarily a minimum. For a closed system, in addition to conservation of energy and momentum, there is conservation of angular momentum, that is, of the vector M = ∑ r × p, where r and p are the radius vector and momentum of the particle; the summation runs over all the particles making up the system. The conservation of angular momentum is a consequence of the fact that because of the isotropy of space, the Lagrangian of a closed system does not change under a rotation of the system as a whole. In relativistic mechanics, the definition of the center of inertia of a system of interacting particles requires the explicit inclusion of the momentum and energy of the field produced by the particles. Although in the system K 0 (in which Σ p = 0), the angular momentum is independent of the choice of the point with respect to which it is defined, in the K system (in which Σ p ≠ 0) the angular momentum does depend on this choice.

  § 8 The Principle of Least Action

  In studying the motion of material particles, we shall start from the Principle of Least Action. The Principle of Least Action is defined, as we know, by the statement that for each mechanical system there exists a certain integral S, called the action, which has a minimum value for the actual motion, so that its variation δS is zero.†

  To determine the action integral for a free material particle (a particle not under the influence of any external force), we note that this integral must not depend on our choice of reference system, that is, it must be invariant under Lorentz transformations. Then it follows that it must depend on a scalar. Furthermore, it is clear that the integrand must be a differential of the first order. But the only scalar of this kind that one can construct for a free particle is the interval ds, or α ds,

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  Deterministic and Stochastic Modeling in Computational Electromagnetics

  Integral and Differential Equation Approaches

  • Dragan Poljak, Anna Susnjara, Douglas H. Werner(Authors)
  • 2023(Publication Date)
  • Wiley-IEEE Press

    (Publisher)

  Part I Some Fundamental Principles in Field Theory 1 1 Least Action Principle in Electromagnetics Laws of nature are governed by following fundamental principles – the action principle, locality, Lorentz invariance, and gauge invariance [1]. Hamilton’s prin- ciple, or the least action principle, is originally developed for classical mechanics stating that a particle, among all of the trajectories between fixed time instants t 1 and t 2 , follows the path which minimizes the action. Action is defined as time inte- gral of the difference between the kinetic energy and potential energy, respec- tively. Thus, Hamilton’s principle somehow requires the time averages of the kinetic energy and potential energy to distribute as equally as possible (equiparti- tion) [2]. In classical mechanics, Hamilton’s principle and Newton’s second law represent equivalent formulations. An extension of Hamilton’s principle from classical mechanics to classical elec- tromagnetics can be undertaken starting with the analysis of the motion of single charged particle [3]. Next step is to construct a Lagrangian for the electromagnetic field by extending the Lagrangian pertaining to classical mechanics. From the cor- responding Lagrangians, featuring Noether’s theorem and gauge invariance, it is possible to derive equation of continuity for the charge, Lorentz force, and Maxwell’s equations, which can be found elsewhere, e.g. [2–5]. Generally, when a functional is extremal, Noether’s theorem yields the conser- vation law. Thus, invariance of the system under a time translation results in the energy conservation. It is also worth noting that space translation invariance corresponds to the conservation of linear momentum, rotation invariance corre- sponds to the conservation of angular momentum, while gauge invariance yields the charge conservation [1, 2].

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  Variational and Extremum Principles in Macroscopic Systems

  • Stanislaw Sieniutycz, Henrik Farkas(Authors)
  • 2010(Publication Date)
  • Elsevier Science

    (Publisher)

  Chapter 10

  On the Principle of Least Action and Its Role in the Alternative Theory of Nonequilibrium Processes

  Michael Lauster[email protected]     University of the German Armed Forces Munich, Faculty for Economics, Werner-Heisenberg-Weg 39, 85577 Neubiberg, Germany

  Abstract

  The Principle of Least Action (PLA) is one of the most popular applications of the calculus of variations in physics. Based on the idea that natural processes always run with a minimum amount of action a mathematical formulation for the minimum of an appropriate ‘action integral’ is required. In the context of the alternative theory of nonequilibrium processes (AT) the extremum procedure delivers local conservation laws for energy, linear, and angular momentum as well as the electrical charge and shows—via Noether’s celebrated theorem—how the respective coordinates for space and time have to be constructed. Those are closely connected to the so-called Callen principle that is used to choose suitable extensive quantities for the system under study. This chapter deals with the application of the PLA to a new and promising theory of nonequilibrium processes. It is shown how one of the most common mathematical formulations of the PLA, known as Hamilton’s Principle, can be fitted into the theoretical context of irreversible thermodynamics. This results in an extended version of Noether’s theorem allowing us to express the conservation laws for certain extensive quantities in connection with symmetry transformations of the space and time coordinates. Finally, a formulation for the natural trend to equilibrium is derived from the PLA.

  Keywords nonequilibrium dissipation Principle of Least Action alternative theory

  Dedicated to Howard Brenner, MIT, on the occasion of his 75th birthday Leibniz plus Voltaire lead us to the true optimum—T. S. W. Salomon.

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  Computational Statistical Mechanics

  • W.G. Hoover(Author)
  • 2012(Publication Date)
  • Elsevier Science

    (Publisher)

  Let us illustrate the meaning of the Principle with a simple example, finding the least-action parabolic path for a particle in a gravitational field. See Figure 1.8. We choose the mass and the field strength equal to unity. If we specify both an initial point, (x,y)o = (0,0), and a final point, at t = 1, (x,y) = (1,0), the motion depends on a single variational parameter, a: x = t; y = at(l - t). The corresponding action integral, the integral of the Lagrangian, jLdt = Jdt(l/2)[1 + a 2 (l - 2t)2 - 2at(l -1)] = (3 - a + a 2 ) / 6 , has its minimum value, 11/24, for the Least-Action trajectory with a = 1/2. The general variational statement of the Principle: 1/4 y a = 3/4 a = 1/2 a = 1/4 Figure 1.8 Parabolic trajectories in a unit gravitational field with a values 1/4,1/2, and 3/4. 20 8jL(q,v)dt = 0, can be made more explicit by considering the variations of both arguments of the Lagrangian, 8q and 8v: J[0L(q,v)/3q)8q + 0L(q,v)/3v)5v]dt = 0 . If the second term is integrated by parts, integrating 5v = 6q with respect to time, then, because 6q vanishes at the end points, a simpler form results: J[OL(q,v)/aq) - (d/dt)0L(q,v)/3v)]5qdt = 0. Aside from the two endpoint restrictions, the variation 5q is an arbitrary function of time. It could, for instance, be chosen to vanish everywhere expect in the neighborhood of a particular time. Thus, because the integral must vanish, the coefficient multiplying the arbitrary 5q must vanish also. This requirement gives the Lagrangian equations of motion: (d/dt)OL(q,v)/av) q = OL(q,v)/3q) v . In this form Lagrangian mechanics is specially easy to use in implementing implicit constraints by choosing appropriate generalized coordinates. An alternative form, which implements constraints using Lagrange multipliers is useful too in more complicated cases. This idea can be applied to constraints, or restrictions, involving both the coordinates and the momenta.

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  Advanced Analytical Dynamics

  Theory and Applications

  • Vincent De Sapio(Author)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  (5.254) 5.3 Hamilton’s Principle of Least Action 139 Figure 5.12 (Top) Animation frames from the simulation of the gimballed gyroscope. (Bottom) Time history of the gimbal angles, q 1 and q 2 . The rotor of the gyroscope is spun up to 2π/8 rad/s. Precession of the gyroscope can be observed. 140 Zeroth-Order Variational Principles Figure 5.13 Fluctuations of kinetic and potential energy and conservation of total energy. The potential energy (relative to z = −2.5) of the system is V = g n b  i=1 M i 0 r T G i ˆ e 3 = g[(2.5 − l G 1 )M 1 + 2.5M 2 + 2.5M 3 + (l G 2 M 2 − l 3 M 3 ) sin(q 2 )], (5.255) and the total energy is E = T + V. (5.256) The Lagrangian is L = T − V. (5.257) Figure 5.13 shows the fluctuations of kinetic and potential energy and the conservation of total energy for the gyroscope. 5.3.6 Constrained Least Action Least action can be applied to multibody systems with auxiliary holonomic constraint equations. We introduce a set of m holonomic (and scleronomic) constraint equations, φ(q) = 0. The zeroth-order variation of the constraint equations is δφ = δq = 0, 5.3 Hamilton’s Principle of Least Action 141 where the matrix (q) = ∂ φ/∂ q ∈ R m C ×n is the constraint Jacobian. In this case, the Principle of Least Action can be stated as δI = 0, ∀δ|δq(t o ) = δq(t f ) = 0, and δq = 0. (5.258) Thus, least action seeks the path, q(t ), in configuration space that results in a stationary value of action, I , under all path variations, δq, that vanish at the endpoints and satisfy the constraints. We recall the Euler-Lagrange equations for unconstrained systems: δI = t f  t o  ∂ L ∂ q − d dt ∂ L ∂ ˙ q + τ  · δq dt . (5.259) This makes use of the condition that the path variations vanish at the endpoints. The condition δI = 0 ∀δq|δq = 0 (5.260) applied to (5.259) implies the following orthogonality relation at any instant:  d dt ∂ L ∂ ˙ q − ∂ L ∂ q − τ  · δq = 0 ∀δq ∈ ker(). (5.261) Thus,  d dt ∂ L ∂ ˙ q − ∂ L ∂ q − τ  ∈ ker() ⊥ = im( T ).

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  The Standard Model in a Nutshell

  • Dave Goldberg(Author)
  • 2017(Publication Date)
  • Princeton University Press

    (Publisher)

  He found that the solution was an inverted cycloid (Figure 2.3). The brachistochrone problem demonstrates that dynamics and minimization problems are intimately related. By the 1750s, Leonhard Euler and his student Joseph Louis Lagrange had developed, in essence, a generalized approach to generating Fermat’s theorem. Their methodology was, at least in part, a consequence of the introduction of the concept of action as suggested by Pierre-Louis Moreau de Maupertuis in 1747 (in language reminiscent of the principle introduced by Hero of Alexandria): This is the Principle of Least Action, a principle so wise and so worthy of the supreme Being, and intrinsic to all natural phenomena; one observes it at work not only in every change, but also in every constancy that Nature exhibits. In the collision of bodies, motion is distributed such that the quantity of Action is as small as possible, given that the collision occurs. At equilibrium, the bodies arrange such that, if they were to undergo a small movement, the quantity of Action would be smallest. 2.1 The Principle of Least Action | 27 U q = 0 Figure 2.4. A simple harmonic oscillator, along with its corresponding potential. The action itself wasn’t well defined in the modern sense until 1834, when William Hamilton offered the relation S ≡ dtL ( ˙ q , q , t ) , (2.1) where S is the action, q is the coordinate of a system with one degree of freedom, and L is the Lagrangian (named in Lagrange’s honor by Hamilton). A particle will traverse the path that locally minimizes the action. For particles moving under a conservative force, the Lagrangian is simply the difference of the kinetic and potential energies: L = K − U . (2.2) Historically, particle dynamics began with the development of the equations of motion, and the Lagrangian formalism was created as a more elegant approach.

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  Quantum Field Theory, as Simply as Possible

  • Anthony Zee(Author)
  • 2023(Publication Date)
  • Princeton University Press

    (Publisher)

  So, it is far from a straightforward problem to unify least time and extremal action. In the glare of hindsight, theoretical physicists now know how to do this. 6 Highly satisfying. 7 Two foundational principles are now but one! Notes 1 As I said in chapter II.1, everybody’s utility function differs tremendously. 2 Fermat’s least time principle has a strongly teleological flavor—that light, and particularly daylight, somehow knows how to save time— a flavor totally distasteful to the post rational palate. Things are teleological if they have a pur- pose, or at least act as if they have a purpose. That’s a big no no in Western science. In con- trast, at the time of Fermat, there was a lot of quasi-theological talk about Divine Providence and Harmonious Nature, so there was no ques- tion that light would be guided to follow the most prudent path. Indeed, to some, the least time and action principles provided comforting evidence of Divine guidance. A voice told each particle in the universe to follow the most advantageous path and history. Not surprisingly, the action principle has inspired a considerable amount of quasi-philosophical, quasi-theological writ- ing, a body of writing which, while intriguing, proves to be sterile ultimately. Nowadays, physi- cists generally adopt the conservative, pragmatic position that the action principle is simply a more compact way to formulate physics, and that the quasi-theological interpretation suggested by it is neither admissible nor relevant. 3 Some readers might know that the mag- netic force acting on a charged particle depends on the velocity of the particle. 4 By practical minded physicists, that is. 5 See chapter V.4. Also, Fearful, chapter 14, and QFT Nut, chapters VII.5–7. 6 See GNut, chapter III.5, especially page 212. 7 “Philosophical” satisfaction matters to some, but certainly not all, theoretical physicists.

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  The Variational Principles of Mechanics

  • Cornelius Lanczos(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  Summary.  In the parametric representation of motion the time is an additional position coordinate which can participate in the process of variation. The momentum associated with the time is the negative of the total energy. For scleronomic systems the time becomes a kinosthenic variable and the corresponding momentum a constant. This yields the energy theorem of conservative systems. The elimination of the time as an ignorable coordinate gives a new principle which determines only the path of the mechanical system, not the motion in time. This is Jacobi's principle which is analogous to Fermat’s principle in optics. The same principle can be formulated as the “Principle of Least Action.” In the latter, the time-integral of double the kinetic energy is minimized with the auxiliary condition that both actual and varied motions shall satisfy the energy theorem during the motion. If this principle is treated by the λ-method, the resulting equations are the Lagrangian equations of motion.

           7.   Jacobi”s principle and Riemannian geometry. As pointed out in

  chapter I

  , section 5 , the geometrical structure of the configuration space is not in general Euclidean, but Riemannian. If a mechanical system consists of N free particles, then the configuration space is Euclidean of 3N dimensions. But if there are any constraints between these particles, then the configuration space is a curved subspace of less than 3N dimensions, the geometry of which can be characterized by a Riemannian line element. This line element is defined by the kinetic energy of the mechanical system, expressed in curvilinear coordinates qk :

  Jacobi’s principle brings out vividly the intimate relationship which exists between the motion of conservative holonomic systems and the geometry of curved spaces. We introduce, in addition to the line element of the configuration space, another Riemannian line element defined by

  According to (56.12 ), Jacobi’s principle requires the minimizing of the definite integral

  This is the same as finding the shortest path between two definite end-points in a certain Riemannian space. We can associate with the motion of a mechanical system under the action of the potential energy V the motion of a point along some geodesic of a given Riemannian space. The problem of finding the solution of a given dynamical problem is mathematically equivalent to the problem of finding these geodesics.

  In particular, let us restrict ourselves to the case where the potential energy V vanishes, i.e. where the motion occurs in the absence of any impressed forces. in that case we can dispense with the introduction of the additional line element and can operate directly with the line element of the configuration space. Since V

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  Analytical Dynamics: Course Notes

  Course Notes

  • Samuel D Lindenbaum(Author)
  • 1994(Publication Date)
  • World Scientific

    (Publisher)

  In other words, Rs must be a minimum and therefore WCB (the diminution in work) is a minimum. We have thereby shown that the Principle of Least Constraint implies that any natural motion of a constrained system is such that all the work that can be done in any element of time is done. But we have shown from the Principle of Virtual Work that you cannot have motion unless work is done, i.e. no equilibrium z> L 6r,-ff > 0. Hence, the Principle of Least Constraint here represents the converse of the Principle of 'Virtual Work. [123] Configuration Space and Configuration-Time Space Let the coordinates of a system be specified by qiq2.. . q n , such coordinates comprising a configuration space. Then for every state of the system there corresponds a single point in configuration space and as the system changes, the state points will trace out a trajectory in configuration space. There will be a value of time corresponding to each point on the trajectory. The case of coordinates q and #2 is illustrated in Fig. VII-16. Configuration-time space is illustrated in Fig. VII-17. Fig. VII-16 Fig. vn-17 Additional Principles of General Dynamical Systems 249 Example: Trajectory of a projectile (Fig. VII-18) x = Dot; y = g?/2. = gJ/Zvl , or 2y = gf/Z + gJ/Zvl, yielding the equation of a surface of the form o a an elliptic paraboloid. Fig. VII-18 Gauss's Principle of Least Constraint (Restated): Consider, at any instant of time, all kinematically possible trajectories of a dynamical system for which the coordinates and velocities are the same as those of the actual motion at that instant. Then the actual trajectory is that for which the constraint is a minimum. In what follows we shall be concerned with line integrals of functions of such coordinates, (q r J); hence we initiate here a brief study of the calculus of variations. [124] Review of Notation and Elementary Operations of Calculus of Variations Let a function y =f(x) be defined in the interval XQ

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  The Man Who Flattened the Earth

  Maupertuis and the Sciences in the Enlightenment

  • Mary Terrall(Author)
  • 2006(Publication Date)
  • University of Chicago Press

    (Publisher)

  The condition applies to the whole path, or the whole con fi guration, rather than to each instant or position, so it is global and synthetic rather than analytic. Euler showed that any problem can be viewed from either a reductionist or a global perspective, using di ff erent mathematical methods. This “double method” re fl ected the two kinds of causes operating in physics. The direct method gave the e ff ects of forces (e ffi cient causes), and the indirect method used the calculus of variations to solve extremum problems ( fi nal causes). 28 The mechanics of least action entailed claims about how the world works and about how the philosopher should understand the world. Maupertuis and Euler agreed that mechanics should make use of the double method. In prin-ciple, according to Maupertuis, we should be able to understand God’s designs by investigating mechanics, and vice versa. In practice, given the limitations of human understanding, physics must follow both paths. Mechanics alone doesn’t take us far enough, while teleology alone can be misleading, since “we can be mistaken about the quantity that we should regard as nature’s expenditure in the produc-tion of her e ff ects . . . . Let us calculate the motions of bodies, but let us also con-sult the designs of the Intelligence who causes them to move.” 29 Euler put this metaphysical commitment into practice. Essay on Cosmology Maupertuis regarded the Principle of Least Action, and its application to ratio-nalist theology, as the capstone of his scienti fi c career. In the decade after he pre-sented it to the Berlin Academy, he devoted a great deal of e ff ort to promoting it, defending it from criticism, and reworking the text of his paper for a series of pub-lished versions. In this same period, he was working to build up the reputation of the Academy. These two projects—the Principle of Least Action and the Acad-emy—went hand in hand.

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