Hamilton's Principle

10 Key excerpts on "Hamilton's Principle"

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  eBook - ePub

  Modelling of Mechanical Systems: Discrete Systems

  • Francois Axisa(Author)
  • 2003(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  Chapter 3

  Hamilton’s principle and Lagrange’s equations of unconstrained systems

  One of the most famous fundamental principles of theoretical physics is certainly the law of least action enunciated as a universal principle for the first time by Maupertuis (1746): “when a change occurs in nature, the quantity of action necessary for the change is the least possible”. Here, we shall introduce the exact formulation of this principle as made by Hamilton (1834), which states that the actual motion of a mechanical system between two arbitrarily fixed times t 1 , t 2 makes stationary the time integral over t 1 , t 2 of the extended Lagrangian of the system. Since this integral is suitably identified with the action of the Lagrangian, Hamilton’s principle is thus a principle of stationary action, historically understood as a principle of least action. It can be either postulated as a first principle, or derived from the principle of virtual work. One interesting point for applications in mechanics is that it allows one to introduce kinetic energy in a quite natural way through the Lagrangian. Furthermore, starting from it, Lagrange’s equations can be established by using a few mathematical procedures which are also of major interest in other problems in mechanics, which deal with continuous material systems provided with boundaries.

  3.1 Introduction

  In the preceding chapter, Lagrange’s equations of discrete and unconstrained mechanical systems were formulated starting from the variational principle of virtual work. The latter may be considered as being differential in nature, because it deals with variations that are taken at a fixed time t .

  Here, we shall introduce another variational principle, where we consider the actual motion of the system between two arbitrarily fixed times t 1 , t 2 and small virtual variations about it, see Figure 3.1 . This is the principle of least action in its exact formulation given by Hamilton in 1834 and widely known as Hamilton’s principle,

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  Classical Dynamics of Particles and Systems

  • Jerry B. Marion(Author)
  • 2013(Publication Date)
  • Academic Press

    (Publisher)

  These principlest are closely related to Hamilton's Principle and add nothing to the content of Hamilton's more general formulation ; their mention only serves to emphasize the continual concern with minimal principles in physics. In two papers published in 1834 and 1835, HamiltonÌ announced the dynamical principle upon which it is possible to base all of mechanics and, indeed, most of classical physics. Hamilton's Principle may be stated as follows §: Of all the possible paths along which a dynamical system may move from one point to another within a specified time interval (consistent with any constraints), the actual path followed is that which minimizes the time integral of the difference between the kinetic and potential energies. In terms of the calculus of variations, Hamilton's Principle becomes δ ί ( Τ -U)dt = 0 (9.1) This variational statement of the principle requires only that T — U be an extremum, not necessarily a minimum, but in almost all applications of importance in dynamics the minimum condition obtains. Now, the kinetic energy of a particle expressed in fixed, rectangular coordinates is a function only of the x, and if the particle moves in a con-servative force field, the potential energy is a function only of the x f : T= Tfe); U = U( Xi ) * See, for example, Goldstein (Go50, pp. 228-235) or Sommerfeld (So50, pp. 204-209). f See, for example, Lindsay and Margenau (Li36, pp. 112 120) or Sommerfeld (So50, pp. 210-214). X Sir William Rowan Hamilton (1805-1865), Scottish mathematician and astronomer, and later, Irish Astronomer Royal. § The general meaning of the path of a system will be made clear in Section 9.3. 218 9 · Hamilton's Principle If we define the difference of these quantities to be L=T- U then Eq.

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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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  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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  Advanced Theoretical Mechanics

  A Course of Mathematics for Engineers and Scientists

  • Brian H. Chirgwin, Charles Plumpton(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  Like Hamilton's Principle it concerns the stationary value of an integral, but the variations contemplated here differ from those in Hamilton's Principle. Moreover, the motions to which the present principle applies are those in which the Hamiltonian is conserved. The Hamiltonian function was defined in § 6:6 eqn. (6.26) by o m o T dq t dq i and is taken to represent the total energy, although it only coincides with T + V in the special case in which T = ^ctij-qiCj, a homogeneous function of the velocities. The condition for conservation of H is that L, and therefore H, does not depend explicitly on the time, i.e., 3L 3H The proof of Hamilton's Principle used as the comparison motion a trajectory joining M and N in which the coordinates of the particle at time t were q i + dq^, where dqi = ε η ί (t), and the variations ôq t vanished at the end points. The variations were made at constant time. For the Principle of Least Action we consider a more general variation (but restrict its application to systems with constant H). In the adjacent comparison motion the point B 2 corresponds to the point E of the natural motion (see Fig. 129) while R x would correspond to B in the comparison motion for Hamilton's Principle. The system passes through R 2 in. the (11.10) 464 COUKSE MATHEMATICS varied motion at time At later than it passes through II in the natural motion. .·. Aq t = dq t + q t At. (11.11) Since theZlg^ are arbitrary both eq t and At are arbitrary. Because the system starts at M and finishes at N we impose the conditions (MÎ)M=0, (Aq t ) N =0. (11.12) FIG. 129 But, in general, neither dq t nor A t are zero at M, N. For any function f(q l9 q 2 , ..., q ni t) the change corresponding to Aq t s Δ *~ΙΪ; Δ «< + Ί* Δ *> (1L13) (a summation over i is implied) i.e., Af ^IL 6q%+ L4L + m At ^^i qi+ ^At.

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  Boundary and Eigenvalue Problems in Mathematical Physics

  • Hans Sagan(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  t.)

  Let us now assume that the mass point starts its motion at the point Р1 at a time t1 and arrives at a point P2 at a time t2 . We consider now all possible trial paths joining the space-time points P1, t1 and P2 , t2 and assume that we have evaluated for all those paths the quantity

  which is called the action.

  In accordance with the literature we denote the integrand by L,

  (Euler and Lagrange were the first to formulate the ideas put forth in this subsection, and one calls L, in honor of the latter, the Lagrange function.)

  If we consider all possible continuous paths with a continuous derivative joining the space-time points P1, t1 and P2 , t2 , we can be sure that one of them is the path actually taken by the mass point under consideration. The action will in general have different values for the different paths we consider, and if there is one which yields a “minimum value” for the action, then it is the one actually taken by the mass point. (We put minimum value in quotes because we will have to modify this statement a little later.)

  This constitutes the so-called “principle of least action,” formulated at first by Euler and Lagrange for conservative fields and later generalized by Hamilton for nonconservative fields.1 The way Euler and Lagrange came to formulate this principle can probably be understood on the basis of the general philosophical and religious background of their time, when it was generally believed that God made the world in the most economical way and therefore everything had to obey minimum principles of some kind. The establishment of this principle may appear less artificial in this light. We are now going to show that this principle is equivalent to the leading principle of vectorial mechanics, which guided our considerations at the beginning of this subsection, insofar as it also leads to Newton’s

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  Universe Dynamics

  The Least Action Principle and Lagrange's Equations

  • Jacques Vanier, Cipriana Tomescu (Mandache)(Authors)
  • 2019(Publication Date)
  • CRC Press

    (Publisher)

  It is a question that has intrigued human beings since they first started reflecting on the exact nature of that physical universe we inhabit. No general answers to that profound question covering the entire field of physics have been found. There are many aspects of the behavior of objects in the universe that lead us to believe that the universe may evolve by means of a basic principle that is more general than the specific laws enunciated above. For example, it looks very much that in general nature favors situations of minimum energy. Objects falls on the ground where energy is lower. In searching for minimum energy, nature appears also to look for equilibrium or stationarity. Another observation is that the universe appears to behave in a manner that tends towards simplicity. For example, paths of objects thrown in the air above the earth’s surface seem to be rather regular and uniform. Light travels in a straight line and not in a complicated path; it is reflected from surfaces with a behavior that has been put under the form of rather simple rules. The conservation principles introduced earlier appear to have some origin that deserve deeper reflection than just being the result of assumptions and the elementary analysis that we made. All these considerations lead us to believe that nature is driven by some basic law or principle that just needs to be discovered.

  Such a principle was proposed some time ago following the deep thinking of physicists that followed Newton, such as Bernoulli, Euler, Lagrange, Hamilton and several others. The principle in question is called principle of least action . We will see that a study of that principle leads to differential equations called Lagrange’s equations , which essentially replaces the dynamical equations of motion and conservation laws that we have introduced above. When that principle is applied properly, it also leads to basic laws governing the evolution of the universe as regards its general behavior as well as some of its details. In fact, the approach using that principle is rather universal and appears to regulate the dynamics of material objects and entities such as fields within the whole of physics at both the large scale as well as at the atomic level.

  Before going into the mathematical description of that principle, we need, however, to familiarize ourselves with an interesting mathematical tool called the calculus of variations

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  Variational Methods with Applications in Science and Engineering

  • Kevin W. Cassel(Author)
  • 2013(Publication Date)
  • Cambridge University Press

    (Publisher)

  We begin by stating Hamilton’s principle without proof for discrete and continu- ous systems and showing that the corresponding Euler equation is Newton’s second law. Several examples are included in order to assist in developing some intuition about Hamilton’s principle before providing a formal derivation from the first law of thermodynamics. Along the way, we will encounter the Euler-Lagrange equations of motion, the Hamiltonian formulation, and show that the Euler-Lagrange equations are independent of the coordinate system. The chapter closes with a discussion of the powerful theorem of Noether, which draws a connection between mathematical symmetries in Hamilton’s principle and physical conservation laws, and a brief discussion of the philosophy of science. 117 118 4 Hamilton’s Principle 4.1 Hamilton’s Principle for Discrete Systems Let us begin by considering a single O(1)-sized rigid particle, a point mass m, acted upon by a force f (t ), where r(t ) is the position vector of the system and is the dependent variable. In order to determine the actual trajectory of the system r(t ) over the finite time interval t 1 ≤ t ≤ t 2 , Hamilton’s principle asserts that the trajectory must be such that r(t ) is a stationary function of the functional  t 2 t 1 (δT + f · δr)dt = 0. (4.1) The kinetic energy of the system as a whole is T = 1 2 mv 2 = 1 2 m˙ r 2 , where dots denote differentiation with respect to time, and v = ˙ r is the velocity of the system. Note that v 2 = v · v, which is a scalar. In the second term, f · δr is known as the virtual work owing to application of the actual resultant external force f (t ) acting on the system through the infinitesimal virtual (hypothetical) displacement δr leading to, for example: • Gravitational potential energy owing to the action of gravity on the system. • Potential energy of deformation, or strain energy, owing to deformation of the system by external forces.

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  Symmetries and Laplacians

  Introduction to Harmonic Analysis, Group Representations and Applications

  • D. Gurarie(Author)
  • 1992(Publication Date)
  • North Holland

    (Publisher)

  According to the Hamilton’s principle of Minimal Action: a trajectory (evolution) of the classical system must minimize (or give a stationary path) of the action- functional. So it satisfies the Euler-Lagrange equation: a=P. 6q(t) q -d(P..e=o. dt P (1.1) Equation (1.1) represents a 2-nd order OD system in n variables. In the classical case of P. = “kinetic” - “potential”, (1.1) turns into the Newton’s equation, 9’’ -aV(q) = F- force. The canonical formalism reduces the 2-nd order Euler-Lagrange equations to a 1 - st order system of size 2n. We introduce a new set of variables: ‘11 pi = a. .P.(q;q) - conjugate momenta, (1.2) 370 $8.1. Minimal action principle; Euler- Lagrange equation; and H(q; p ) = p - L, hamiltonian/energy function. Solving a system of equations (1.2) for q-variables’, we get q = Q(q;p), and these are substituted in the hamiltonian H(q;p). Then the Euler-Lagrange equations (1.1) are shown to be equivalent to a hamiltonian system @- -a P H ; Ap= -a p . (1.3) We shall first demonstrate the canonical formalism in the case of N-particle systems. The corresponding Lagrangians are N L = 3’ - V(q), or 3Cmjq: - V, where ( m j } denotes masses of particles. Then the conjugate momenta: p j = rnjqj, and the hamiltonian, H = c l p . 2 +- V(q); “kinetic” + “potential”, a familiar expression from elementary calculus/mechanics. For more general (“kinetic - potential”) Lagrangians on manifolds A, the Euler-Lagrange (1.1) takes the form 2mj J & ~ g i j q j ) -a q i v = 0, while the canonical variables: .. P; = E g i j Q j ; H = + C g ” P i P j + V(q), ( g i j ) denotes the inverse matrix (tensor) to ( g i j ) , and the hamiltonian system becomes pi = - a,v. So geometrically, momentum variables { p } vectors on A, and the (position-momentum) phase can be identified with cotangent space becomes a cotangent bundle I*(A).

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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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