Lagrangian Mechanics

11 Key excerpts on "Lagrangian Mechanics"

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  Dynamical Systems and Geometric Mechanics

  An Introduction

  • Jared Maruskin(Author)
  • 2018(Publication Date)
  • De Gruyter

    (Publisher)

  Thus, geometric mechanics was of-ficially born. In this chapter, we will present the theory of Lagrangian Mechanics from the modern point of view—as equations of motion for mechanical systems evolving on differentiable manifolds. For a classical treatment of the subject, see texts such as [8], [104], [102], or [293]. 7.1 Hamilton’s Principle Lagrangian Mechanics is variational in nature and is derivable from a variational prin-ciple known as Hamilton’s principle . Variational problems of the calculus of variations are similar to optimization problems of differential calculus, which seek to find paths that are optimal with respect to some sort of performance measure. For example, there are many paths that you may take to go from your home to your local grocery store; one of those paths, however, minimizes the distance that you would need to travel. As it turns out, all laws of nature are variational in nature, which means that nature is always trying to find optimal paths relative to some cost function or performance measure. Before we discuss this further, we will introduce the notion of Lagrangian function. https://doi.org/10.1515/9783110597806-007 184 | 7 Lagrangian Mechanics Let Q be an n -dimensional configuration manifold for a mechanical system. A La-grangian is a function L : TQ → ℝ . For most mechanical systems, the Lagrangian is simply given by the kinetic minus potential energy . Figure 7.1: A simple pendulum. Example 7.1. Let us consider the example of a simple pendulum that was introduced in Example 2.4 and that is depicted in Figure 7.1. The configuration manifold for our system is Q = S 1 , which may be described by the local coordinates ( θ ) , θ ∈ (− π , π ) , where θ is the angle between the pendulum bob and the vertical, as measured from its “down” position. The speed of the bob is l | ̇ θ | , so that its kinetic energy is given by T = 1 2 ml 2 ̇ θ 2 .

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  Analytical Mechanics

  • Nivaldo A. Lemos(Author)
  • 2018(Publication Date)
  • Cambridge University Press

    (Publisher)

  1 Lagrangian Dynamics Lagrange has perhaps done more than any other analyst by showing that the most varied consequences respecting the motion of systems of bodies may be derived from one radical formula; the beauty of the method so suiting the dignity of the results, as to make of his great work a kind of scientific poem. William Rowan Hamilton, On a General Method in Dynamics Mechanical systems subject to restrictions (constraints) of a geometric or kinematic nature occur very often. In such situations the Newtonian formulation of dynamics turns out to be inconvenient and wasteful, since it not only requires the use of redundant variables but also the explicit appearance of the constraint forces in the equations of motion. Lagrange’s powerful and elegant formalism allows one to write down the equations of motion of most physical systems from a single scalar function expressed in terms of arbitrary independent coordinates, with the additional advantage of not involving the constraint forces. 1.1 Principles of Newtonian Mechanics A fair appreciation of the meaning and breadth of the general formulations of classical mechanics demands a brief overview of Newtonian mechanics, with which the reader is assumed to be familiar. Virtually ever since they first appeared in the Principia, Newton’s three laws of motion have been controversial regarding their physical content and logical consistency, giving rise to proposals to cast the traditional version in a new form free from criticism (Eisenbud, 1958; Weinstock, 1961). Although the first and second laws are sometimes interpreted as a definition of force (Marion & Thornton, 1995; Jos´ e & Saletan, 1998; Fasano & Marmi, 2006), we shall adhere to what we believe is the correct viewpoint that regards them as genuine laws and not mere definitions (Feynman, Leighton & Sands, 1963, p. 12–1; Anderson, 1990).

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  Interactions of Photons and Neutrons with Matter

  • Sow-Hsin Chen, Michael Kotlarchyk;;;(Authors)
  • 2007(Publication Date)
  • WSPC

    (Publisher)

  Chapter 2 AN OVERVIEW OF CLASSICAL MECHANICS Theprimarymotivationforthischapteristolaythegroundworknecessaryformaking connections between the classical and quantum-mechanical description of a physical system. In particular, the dynamic description of mechanical systems in terms of La-grangian and Hamiltonian mechanics is presented, with a special emphasis on oscilla-tors. It is only by introducing the so-called canonicalcoordinates and by representing the total energy in Hamilton’s form that one can ultimately make the transition from a classical to a quantum mechanical description of both atomic systems and the interacting electromagnetic radiation field. 2.1 The Lagrangian Formulation In many simple mechanical systems, the most natural way to obtain the equations of motion is to apply Newton’s Second Law to each particle, namely ˙ p i = F i , (2.1) which relates the force F i acting on the i th particle in the system to the rate of change * of the linear momentum p i = m i v i of that particle, where v i is the velocity of particle i . In most cases, the mass m i of each particle is fixed and Newton’s Second Law reduces to the more familiar form F i = m a i , (2.2) where a i = ˙ v i is the acceleration of a given particle in the system. Consider, for example, the simple case of a single particle under the influence of a linear restoring force along the x -axis. The equation of motion is that for a simple harmonic oscillator, i.e., m ¨ x = -Kx, (2.3) where K is a positive (spring) constant. Alternatively, this can be written as d dt parenleftbigg 1 2 m ˙ x 2 + 1 2 Kx 2 parenrightbigg =0 , (2.4) * Each dot above a letter indicates a time derivative. 6 An Overview of Classical Mechanics which means that the expression in the parentheses remains fixed (or conserved) as the system evolves in time. This constant of the motion can be defined as E , the total energy of the system: E = 1 2 m ˙ x 2 + 1 2 Kx 2 .

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  The Semiclassical Way to Dynamics and Spectroscopy

  • Eric J. Heller(Author)
  • 2018(Publication Date)
  • Princeton University Press

    (Publisher)

  PART I Classical Mechanics with an Eye to Quantum Mechanics Chapter 1 The Lagrangian and the Action 1.1 Extremal Action and Equations of Motion Classical mechanics can be introduced in several ways; there is no “proof ” of the equations of motion, only a statement of experience. One road to the equations of motion for classical systems is a variational principle, finding extremes (maxima or minima; usually the latter) of a quantity called the “action” along a path subject to given end points. Looking ahead, the classical action plays a central role in quantum mechanics via the Feynman path integral, which involves sums over complex exponentials of the action for all possible paths (not just classically allowed paths). The action makes its appearance in semiclassical approximations by stationary phase evaluation of Feynman path integrals. Stationary phase evaluation means exploiting extremal action (stationary or unchanging action against small path changes), which causes a buildup of the result with a phase of that action and an amplitude depending on just how stable the action is to changes. Classical trajectories are paths with stationary action, and they appear naturally out of quantum mechanics in this way. This is the best way to understand the correspondence principle, 1 the best way to build intuition for quantum systems having a classical analog. In what follows we focus on the action principles of classical mechanics, making the future transition to quantum and semiclassical mechanics almost seamless. Landau and Lifshitz, in their famously short but spot-on book, Mechanics [2], intro- duce the Lagrangian L = L (q , ˙ q , t ), saying it should be a function of the coordinates q , their velocities ˙ q , and the time only, banking on the observation that specifying the coordinates and velocities completely determines the subsequent motion of any classical system.

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  A Student's Guide to Lagrangians and Hamiltonians

  • Patrick Hamill(Author)
  • 2013(Publication Date)
  • Cambridge University Press

    (Publisher)

  Newton was aware of this problem and he stated that the sec- ond law is a relationship that holds in a coordinate system at rest with respect to the fixed stars. 1.10.2 The equation of motion in Lagrangian Mechanics Another way to obtain the equation of motion is to use the Lagrangian tech- nique. When dealing with particles or with rigid bodies that can be treated as par- ticles, the Lagrangian can be defined to be the difference between the kinetic energy and the potential energy. 6 That is, L = T − V. (1.15) For example, if a mass m is connected to a spring of constant k, the potential energy is V = 1 2 kx 2 and the kinetic energy is T = 1 2 mv 2 = 1 2 m ˙ x 2 . Therefore the Lagrangian is L = T − V = 1 2 m ˙ x 2 − 1 2 kx 2 . It is usually easy to express the potential energy in whatever set of coordi- nates being used, but the expression for the kinetic energy may be somewhat difficult to determine, so whenever possible, one should start with Cartesian 6 Although Equation (1.15) is pefectly correct for a system of particles, we shall obtain a somewhat different expression when considering continuous systems in Chapter 7. In general, the Lagrangian is defined to be a function that generates the equations of motion. 20 1 Fundamental concepts coordinates. In the Cartesian coordinate system the kinetic energy takes on the particularly simple form of a sum of the velocities squared. That is, T = 1 2 m( ˙ x 2 + ˙ y 2 + ˙ z 2 ). To express the kinetic energy in terms of some other coordinate system requires a set of transformation equations. For example, for a pendulum of length l , the potential energy is V = −mgl cos θ and the kinetic energy is T = 1 2 mv 2 = 1 2 m(l ˙ θ) 2 . (Here θ is the angle between the string and the perpendicular.) Therefore the Lagrangian is L = T − V = 1 2 ml 2 ˙ θ 2 + mgl cos θ.

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  Interaction of Photons and Neutrons with Matter

  • Sow-Hsin Chen, Michael Kotlarchyk;;;(Authors)
  • 1997(Publication Date)
  • WSPC

    (Publisher)

  (2.12) dx dt V dx The Lagrangian Formulation X, X Figure 2.1 The variation of x or x in the principle of least action. The first term vanishes because the two end-points are fixed and the variation, 6x, must therefore be zero at these points. Since Eq. 2.12 must hold for arbitrary 6x, the integrand must vanish, and Eq. 2.9 follows. A major advantage of the Lagrangian formalism for doing mechanics is that it can naturally incorporate constraints that exist between various particles. In par-ticular, even though a system may contain M particles, requiring 3M coordinates to specify the particle positions, constraints between the particles may reduce the num-ber of degrees of freedom to a number significantly less than 3M. For example, in a rigid body, all particles of the body are constrained to be at fixed positions relative to each other. Then, even though the number of particle coordinates 3M can be huge, the actual number of independent coordinates is only six. Hence, the Lagrangian is a function of only six generalized coordinates and their associated velocities. The gen-eralized coordinates are the three Cartesian coordinates of the center-of-mass and the three Eulerian angles specifying the orientation of the rigid body. In general, a many-particle system with N degrees of freedom can be characterized by a set of generalized coordinates {gi,#2, •••, Qk, •••,^AT} and generalized velocities {#1,92, •••, #fc> • ••,4w}-For a conservative system (i.e., one with constant total energy), one can define the La-grangian generally by C(q k ,q k )=T(q k )-V(q k ,q k ) (2.13) and Lagrange's equations can be generalized to = 0 (fc = l,2,...,tf). (2.14) dt dq k dq h There is one equation for each degree of freedom. 8 An Overview of Classical Mechanics There are further advantages to working with Lagrange's equations.

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  Relativistic Quantum Mechanics

  An Introduction to Relativistic Quantum Fields

  • Luciano Maiani, Omar Benhar(Authors)
  • 2015(Publication Date)
  • CRC Press

    (Publisher)

  C H A P T E R 3 THE LAGRANGIAN THEORY OF FIELDS CONTENTS 3.1 The action principle ....................................... 19 3.2 Hamiltonian and canonical formalism ..................... 21 3.3 Transformation of fields ................................... 25 3.4 Continuous symmetries .................................... 29 3.5 Noether’s theorem ......................................... 31 3.6 Energy–momentum tensor ................................. 33 3.7 Problems for Chapter 3 .................................... 37 In classical mechanics two types of physical systems are distinguished: material points (particles) which have spatial coordinates x ( t ) as dynamic variables, or fields (waves), which are dynamic systems described by one or more continuous functions of the coordinates and of time. φ = φ ( x , t ) = φ ( x ) . (3.1) The most important example of this second kind of system is the electro-magnetic field described at every point of space by two vectors corresponding to the values of the electric field E ( x , t ) and the magnetic field B ( x , t ). 3.1 THE ACTION PRINCIPLE In analogy with the mechanics of systems with a finite number of degrees of freedom, it is natural to derive the field equations from an action principle. The action is defined as the time integral of the Lagrangian, between two fixed instants, t 1 < t 2 : S = t 2 t 1 L dt. (3.2) The Lagrangian of a system of particles is the sum over the many different 19 20 RELATIVISTIC QUANTUM MECHANICS degrees of freedom. In the case of a field, the degrees of freedom are localised at every point of space, therefore: L = d 3 x L ( φ, φ μ , x ) , (3.3) where we have denoted the derivatives of the field with respect to the coordi-nates as φ μ : φ μ ( x ) = ∂φ ∂x μ . The function L is given the name Lagrangian density or, more simply for brevity, the Lagrangian and depends on the fields (the dynamic variables) and their derivatives.

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  Mechanics

  Classical and Quantum

  • T. T. Taylor, D. Ter Haar(Authors)
  • 2016(Publication Date)
  • Pergamon

    (Publisher)

  This new quantity is called the Lagrangian, jß -%;, 4h 0 = T{q h q h i)- V(q h f) (1.050) It is noteworthy that the Lagrangian is in general a function of the 6N+ 1 quantities q p q p and /. This again recalls the theme expressed (1.046) 20 THE LAGRANGIAN FORMULATION OF MECHANICS [§ 1.05 in Section 1.03. Since Kdoes not contain the q j9 dJl dT ■τττ-= ^-7- = Pj (1.051) dqj dqj and, with appropriate substitutions, the equations of motion of the system expressed by (1.049) may be written very succinctly as follows: (1.052) These 37V equations are called LAGRANGE'S EQUATIONS. As a simple example of the use of the Lagrangian method to obtain the equations of motion of an unconstrained system, consider the problem of the motion of a planet (the Kepler problem) under the simplifying assumption that the mass of the planet is relatively negli-gible compared with that of the Sun. 1 Under this assumption, the Sun can be taken at rest at the origin of an inertial frame of reference and the configuration of the system is described by the position of the planet relative to this frame. At present, the objective is merely to find the equations of motion in spherical polar coordinates by the Lagrangian method. A discussion of the actual motion appears in Chapter 2. The potential energy in this problem is given by : V = -GMmr~ (1.053) since the negative gradient of this function is the well-known inverse square gravitational force, — GMmr~ 2 e r ; G is the constant of univer-sal gravitation, M is the mass of the Sun and m, the mass of the planet. The kinetic energy for this case has already been given in equation (1.037) and the Lagrangian is therefore: J2 = m{r 2 + r 2 Ô 2 + r 2 sin 2 θ φ 2 ) + GMmr~ x . ( 1.054) t An example of a Keplerian problem with a planet of non-negligible mass is provided by the hydrogenic atom. This will be studied in a quantum mechanical context in Chapter 12. § 1.05] LAGRANGE'S EQUATIONS 21 As in equations (1.038): Pi = P2 = Ρθ p r =-w = mr.

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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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  Differential Manifolds and Theoretical Physics

  • W. D. Curtis, F.R Miller(Authors)
  • 1985(Publication Date)
  • Academic Press

    (Publisher)

  2 C I assica I M ec ha n ics As discussed in Chapter 1, we want to develop Lagrangian Mechanics and generalized coordinates (i.e., manifold theory). The purpose of this chapter is to introduce the reader to some of the basic ingredients of Lagrangian and Hamiltonian mechanics for the special case when configuration space is an open set M in Euclidean space R and state space is M x R. We will gen- eralize these constructions to the manifold setting later. For now, we also restrict attention to time-independent systems. We begin with Newton's law of motion and show how to rewrite it so as to get Lagrange's equations. We then introduce phase space, defined to be M x (R)*, and obtain Hamilton's equations. We obtain vector fields on state and phase space whose integral curves are the trajectories of the system. We see that, in the Hamiltonian form, the vector field is obtained in a very simple way from the energy function. The trajectories of the Lagrangian and Hamil- tonian systems are related by the Legendre transformation. These construc- tions easily generalize to the manifold setting of Chapter 7. MECHANICS OF MANY-PARTICLE SYSTEMS Consider a system of n particles moving in space. Conjiguration space is a subset M c R3. A state is a pair (4, u) E M x R3, q = (q', . . . , q3) rep- resenting position of the n particles and u = (d, . . . , u3) representing ve- locities. If the masses of the particles are m,, . . . , m,, define M i , i = 1, . . . , 3n, by M 3 i -2 = M3i-1 = M 3 i = mi. This is just convenient notation. 5 6 2. CLASSICAL MECHANICS EXAMPLE 2.1 (Two-body problem) Consider a system of two parti- cles of masses m, and m, moving freely in space. If q, = (ql, q2, q3), q, = (q4, q5, q6) are positions, then we define configuration space to be M = {(ql, q2) E R~ x R ~ I ~ , z q,). The requirement q1 # q2 means that our model does not describe collisions. Given a position q E M the velocities are arbitrary so our state space is S = M x R6.

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  Classical Mechanics

  • R. Douglas Gregory(Author)
  • 2006(Publication Date)
  • Cambridge University Press

    (Publisher)

  However, any system of equations that can be written in this way has special properties. In particular, it is equivalent to a stationary principle (see Chapter 13), and can also be written in Hamiltonian form (see Chapter 14). This is the form most suitable for advanced developments and for making the transition to quantum mechanics. There is therefore a strong interest in any physical system whose equations can be written in Lagrangian form. Definition 12.7 Lagrangian form If the equations of motion of a holonomic system with generalised coordinates q can be written in the form d dt  ∂ L ∂ ˙ q j  − ∂ L ∂ q j = 0 (1 ≤ j ≤ n), (12.23) for some function L = L (q , ˙ q , t ), then L is called the Lagrangian of the system and the equations are said to have Lagrangian form. For example, the Lagrangian for the driven pendulum is L (θ, ˙ θ, t ) = 1 2 m  a 2 ˙ θ 2 + ˙ Z 2 − 2a ˙ θ ˙ Z sin θ  + mg( Z + a cos θ), where Z = Z (t ) is the displacement of the support point. Velocity dependent potential There are systems whose specified forces are not conservative (so that V does not exist), but their equations of motion can still be written in Lagrangian form. Any standard system with generalised forces { Q j } satisfies the Lagrange equations (12.15). If it happens that the generalised forces can be written in the form Q j = d dt  ∂ U ∂ ˙ q j  − ∂ U ∂ q j (1 ≤ j ≤ n), (12.24) for some function U (q , ˙ q , t ), then clearly the equations (12.15) can be written in Lagrangian form by taking L (q , ˙ q , t ) = T (q , ˙ q , t ) − U (q , ˙ q .t ). The function U (q , ˙ q , t ) is called the velocity dependent potential of the system. This seems to be a mathematical artifice that has no importance in practice. It is true that there is only one important case in which a velocity dependent potential exists, but that case is very important; it is the case of a charged particle moving in electromagnetic fields. The following example proves this to be so for static fields.

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