Lagrangian

12 Key excerpts on "Lagrangian"

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  The Semiclassical Way to Dynamics and Spectroscopy

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

    (Publisher)

  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. The action S is defined as the time integral (“action integral”) of the Lagrangian: S = t 0 L (q , ˙ q , τ ) d τ (1.1) 1 Bohr’s correspondence principle asserts that quantum systems will behave classically in the limit that quantum numbers become “large.” Thus, taking the “classical limit” of a quantum system means going to high energy, or if you are a theorist, small . Either way gets you to large quantum numbers, but of course, the dynamics usually changes with energy. A theorist can stay near fixed energy, as is taken smaller. There are subtleties though, and these will be discussed in context 4 Chapter 1 q b q a S = S c S = S c + O(δ q) 2 q( τ ) classical path q( τ ) + δ q( τ ) 0 t Figure 1.1. The classical path q(τ ) going from q(0) = q a to q(t) = q b in time t, with action S = S c , has no first-order difference in action compared to a slightly nonclassical path, q(τ ) + δq(τ )—that is, S = S c + O(δq) 2 for small deviations δq(τ ). The second-order differences in action provide a minimum by which we can find the genuine classical path. Without even knowing the explicit form of the Lagrangian, the form of the equations of motion for the coordinates and velocities can be derived by introducing the principle of least action, also called “Hamilton’s principle”: the action defined earlier must be stationary against small changes in the coordinates and their velocities along the path starting at time 0 with positions q (0), and ending at time t with position q (t ). (see figure 1.1). The initial and final positions are fixed, and for the moment not to be varied.

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  Advanced Modern Physics

  Theoretical Foundations

  • John Dirk Walecka(Author)
  • 2010(Publication Date)
  • WSPC

    (Publisher)

  1 See, for example, [Fetter and Walecka (2003)]. 95 96 Advanced Modern Physics 5.1 Particle Mechanics Classical Lagrangian mechanics for a point particle of mass m was developed in ProbIs. 10.3–10.5. Here we summarize the results from those problems. Suppose one has a system with n degrees of freedom and n generalized coordinates q i with i = 1 , 2 , ··· ,n . This can be any set of n linearly independent coordinates that completely specify the configuration of the system. 2 One introduces the Lagrangian as follows L ( q, ˙ q ; t ) ≡ T ( q, ˙ q ; t ) -V ( q ; t ) ; Lagrangian (5.1) where T is the kinetic energy, and V is the potential energy. We use the following shorthand for the coordinate dependence of the Lagrangian L ( q, ˙ q ; t ) ≡ L ( q 1 , ··· ,q n , ˙ q 1 , ··· , ˙ q n ; t ) ; shorthand (5.2) where ˙ q i ≡ dq i ( t ) /dt , and the last t in the argument denotes a possible explicit dependence on the time. 3 5.1.1 Hamilton’s Principle The action is defined by S ≡ integraldisplay t 2 t 1 dtL ( q, ˙ q ; t ) ; action (5.3) Hamilton’s principle states that the dynamical path the system takes is one that leaves the action stationary δS = 0 ; Hamilton’s principle fixed endpoints (5.4) This expression has the following meaning: (1) Let the coordinates undergo an infinitesimal variation from the actual path q i 0 ( t ) to the path q i ( t ) = q i 0 ( t ) + λη i ( t ) where λ is an infinitesimal and η i ( t ) is arbitrary, except for the fact that η i ( t 1 ) = η i ( t 2 ) = 0 ; (2) Substitute these expressions in L , and make a Taylor expansion in λ ; 2 We could equally well, for example, be discussing a collection of N such particles moving in three dimensions, in which case n = 3 N . 3 The systematic way to arrive at this Lagrangian is to start in a cartesian basis, and then explicitly introduce the transformation to the generalized coordinates.

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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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  Differential Geometry with Applications to Mechanics and Physics

  • Yves Talpaert(Author)
  • 2000(Publication Date)
  • CRC Press

    (Publisher)

  LECTURE 9 Lagrangian AND HAMILTONIAN MECHANICS In this lecture we recall several notions of Lagrangian and Hamiltonian mechanics in order to prepare the reader for the next lecture. So we give the classical principles of Lagrange and Hamilton in analytical mechanics. The equations of motion are deduced. The canonical transformations and integral invariants are also introduced. Moreover, a fluid-dynamical method that I have perfected is shown. To conclude comments on isolating integrals and ergodicity are made. 1. CLASSICAL MECHANICS SPACES AND METRIC 1.1 GENERALIZED COORDINATES AND SPACES Consider a system of particles with a finite number n of degrees of freedom. This system is called rheonomic if the position vectors of particles depend explicitly on time; it is called sc/eronomic in the opposite case. Classical mechanics teaches us that the motion of a system of particles is described at any time by n variables called generalized coordinates denoted by qt. We must specify that there are other systems of generalized coordinates such that 'J _ jJ( 1 n t) . 1 q - q •... ,q , } = , ... ,n with D(j', ... ,F) 0 I =F-. D(q •... ,qn) These conditions should be always present in mind. Recall also the evolution ofa (material) system with n degrees of freedom can be described in spaces of n coordinates and we say: D

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  Phenomenology of Particle Physics

  • André Rubbia(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  The velocity vectors are estimated as total derivatives:  v k ≡ d r k dt = ∂ r k ∂q j dq j dt + ∂ r k ∂t (6.1) where the summation j = 1,...,n is implied. Given this, the kinetic energies and the potential of the system depend on the generalized coordinates and generalized velocities. In case of conservative forces only, the Lagrangian function is defined as the difference between the kinetic energy T and the potential energy V of the system: L  q i , dq i dt  = T − V (i = 1,...,n) (6.2) The time integral of the Lagrangian within a time interval given by t 1 and t 2 is called the action: S([q]) =  t2 t1 Ldt (6.3) S([q]) is a functional because it returns a number from the Lagrangian function integrated over all times between t 1 and t 2 . It depends on the chosen trajectory of the system given by the generalized coordinates. It 1 Joseph-Louis Lagrange (1736–1813), Italian mathematician and astronomer. 184 185 6.2 Relativistic Tensor Fields can be computed for the “real” trajectory as well as for any arbitrary one, or more generally virtual variations of the entire trajectories around the actual one. The principle of least action states that the system’s equation of motion (i.e., the actual path) corresponds to the one for which the action is stationary (at a minimum or maximum): δS = 0. This condition leads to the Euler 2 –Lagrange equations: d dt   ∂L ∂  dqi dt    − ∂L ∂q i = 0 i = 1,...,n (6.4) So, classically, there are n Euler–Lagrange equations for a system with n degrees of freedom. The motion of the system is completely determined with 2n initial conditions (at some given time). In the Hamilton description, the motion is expressed as a set of first-order differential equations. But the number of initial conditions must still remain 2n, so we consider generalized q i and we need n additional coordinates.

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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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  Intermediate Dynamics

  • Patrick Hamill(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  Hamilton’s principle is sometimes referred to as, “the fundamental principle of mechanics.” 11 To make this a bit more explicit, consider a simple physical process, such as the motion of a dropped rock. This physical system can be considered to evolve from some initial situation at time t 1 to a different final situation at time t 2 . Initially the rock is at height h and has zero velocity. Therefore the initial conditions are (z, ˙ z) = (h, 0). The final conditions (just before it hits the ground) are (0, √ 2gh). Hamilton’s principle states that the behavior of the rock minimizes the quantity  t 2 t 1 L(z, ˙ z,t)dt , where the Lagrangian is (1/2)m˙ z 2 − mgz. Applying the methods of the calculus of variations we obtain the Lagrange equation and the Lagrange equation generates the equation of motion, which for the falling rock, is d 2 z/dt 2 = −g. 11 You may have studied Fermat’s principle in optics which states that the path of a ray of light from one point to another is such as to minimize the time of flight. This is a special case of Hamilton’s principle. 100 4 LagrangianS AND HAMILTONIANS 4.8.2 Relation to Newton’s Second Law As mentioned previously, Newton’s second law can be derived from the Lagrange equation. As an illustration, consider the one-dimensional motion of a particle of mass m. Assume the particle is acted upon by a conservative force F . As we have pointed out, for a conservative force in one dimension, the force can be obtained from the potential energy V = V (x) by F = −dV/dx . The Lagrangian is L = T − V = 1 2 m ˙ x 2 − V (x), and Lagrange’s equation is 0 = d dt ∂L ∂ ˙ x − ∂L ∂x = d dt  ∂ ∂ ˙ x  1 2 m ˙ x 2 − V (x)  − ∂ ∂x  1 2 m ˙ x 2 − V (x)  = d dt (m ˙ x) + ∂V (x) ∂x = m ¨ x − F . So F = m ¨ x as expected. Thus, Newton’s second law is a consequence of Lagrange’s equations and Lagrange’s equations are a consequence of Hamilton’s principle.

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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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  A Student's Guide to Lagrangians and Hamiltonians

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

    (Publisher)

  Hamilton’s principle tells us that a mechanical sys- tem will follow a path in configuration space given by q = q(t) such that the integral  t f t i L(t, q, ˙ q)dt is minimized. (Note that here the Lagrangian is to be considered a functional.) Now if q = q(t) is the minimizing path, then L must satisfy the Euler– Lagrange equation d dt  ∂L ∂ ˙ q  − ∂L ∂q = 0. (3.8) It is customary to call this equation simply “Lagrange’s” equation. It can easily be generalized to systems described by an arbitrary number of generalized coordinates. Exercise 3.3 Show that Equation (3.7) leads to Equation (3.8). Use the argument given in Section 2.2 as a guide. 3.4 Generalization to many coordinates Consider a system described by the generalized coordinates q 1 , q 2 , . . . , q n . We assume the coordinates are all independent. The Lagrangian for this system is L(q 1 , q 2 , . . . , q n , ˙ q 1 , ˙ q 2 , . . . , ˙ q n , t). Our plan is to show that Lagrange’s equa- tions in the form d dt  ∂L ∂ ˙ q i  − ∂L ∂q i = 0, i = 1, . . . , n 76 3 Lagrangian dynamics follow from Hamilton’s principle. The derivation is similar to that presented in Section 2.3. Recall the definition of the variation of a function. If f = f (q i , ˙ q i , t), then δf =  i  ∂f ∂q i  δq i +  ∂f ∂ ˙ q i  δ ˙ q i  . Note that in this expression, the time is assumed constant. Consider again Hamilton’s principle in the form of Equation (3.7), which is δ  t 2 t 1 Ldt = δ  t 2 t 1 L(q 1 , q 2 , . . . , q n , ˙ q 1 , ˙ q 2 , . . . , ˙ q n , t)dt = 0, or δI = 0, where the action is represented by I, and the integral is taken over the “path” actually followed by the physical system. Note that for this path the variation in the action is zero. However, as before, there are an infinite number of possible paths between the end points, and we express them in the form Q i (t) = q i (t) + η i (t). In this case, the action can be considered a function of .

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  Foundations of Molecular Quantum Electrodynamics

  • R. Guy Woolley(Author)
  • 2022(Publication Date)
  • Cambridge University Press

    (Publisher)

  The canonical formalism requires a Legendre transformation from position vari- ables and their time derivatives (velocities) to a new description in which the velocities are replaced by the momenta ( p) conjugate to the position variables (x) defined by the differential relation p = ∂ L ∂ ˙ x , (3.113) in terms of which the Euler–Lagrange equations become ˙ p = ∂ L ∂ x . (3.114) Accompanying this change of variables is a completely new viewpoint of dynamics. For a system with N degrees of freedom, the Lagrangian scheme describes the motion as an orbit in the configuration space formed by the N position variables. In the Ham- iltonian scheme, however, both positions and momenta are regarded as independent ‘coordinates’ in a 2N-dimensional phase space. Equation (3.113) must be solved for the velocities in terms of the conjugate position and momentum variables ˙ x = ˙ x(x, p), (3.115) so that the energy function E , (3.30), can be written in terms of the new variables. In this section we will suppose that L is purely a function of position and velocity variables, and that the velocity associated with every coordinate occurs in the Lagrangian in such a way that the inversion of (3.113) can indeed be achieved. 5 The resulting quantity is called the Hamiltonian for the system N ∑ i p i ˙ x i - L = H ≡ H(x, p). (3.116) For a closed system, H has no explicit time dependence and is the (conserved) total energy of the system; this is the only case we discuss. If the Hessian matrix is singular, the Lagrangian is said to be degenerate; in such a case equations of constraint involving 5 The formal requirement is that the Hessian matrix ||W|| ik = ∂ 2 L/∂ ˙ x i ∂ ˙ x k must be non-singular. 85 3.4 Hamiltonian Mechanics the canonical variables, but not the velocities, have to be considered as well. Likewise a more general theory, cast in terms of constraint equations, is required if accelerations or higher derivatives of x are involved in the Lagrangian.

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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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