Euler-Lagrange Equations

8 Key excerpts on "Euler-Lagrange Equations"

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  Mathematical Methods for Physicists

  A Concise Introduction

  • Tai L. Chow(Author)
  • 2000(Publication Date)
  • Cambridge University Press

    (Publisher)

  The equations of motion of classical mechanics are the Euler–Lagrange di€erential equations of Hamilton’s principle. Similarly, the SchroÈ dinger equation, the basic equation of quantum mechanics, is also a Euler–Lagrange di€erential equation of a variational principle the form of which is, in the case of a system of N particles, the following Z Ld ˆ 0 ; 8 : 43 † 368 THE CALCULUS OF VARIATIONS with L ˆ X N i ˆ 1 p 2 2 m i @ * @ x i @ @ x i ‡ @ * @ y i @ @ y i ‡ @ * @ z i @ @ z i ‡ V * 8 : 44 † and the constraint Z * d ˆ 1 ; 8 : 45 † where m i is the mass of particle I , V is the potential energy of the system, and d is a volume element of the 3 N -dimensional space. Condition (8.45) can be taken into consideration by introducing a Lagrangian multiplier E : Z L E * † d ˆ 0 : 8 : 46 † Performing the variation we obtain the SchroÈ dinger equation for a system of N particles X N i ˆ 1 p 2 2 m i r 2 i ‡ E V † ˆ 0 ; 8 : 47 † where r 2 i is the Laplace operator relating to particle i . Can you see that E is the energy parameter of the system? If we use the Hamiltonian operator ^ H , Eq. (8.47) can be written as ^ H ˆ E : 8 : 48 † From this we obtain for E E ˆ Z * H d Z * d : 8 : 49 † Through partial integration we obtain Z Ld ˆ Z * H d and thus the variational principle can be formulated in another way: R * H E † d ˆ 0. Problems 8.1 As a simple practice of using varied paths and the extremum condition, we consider the simple function y x † ˆ x and the neighboring paths 369 PROBLEMS y ; x † ˆ x ‡ sin x . Draw these paths in the xy plane between the limits x ˆ 0 and x ˆ 2 for ˆ 0 for two di€erent non-vanishing values of . If the integral I † is given by I † ˆ Z 2 0 dy = dx † 2 dx ; show that the value of I † is always greater than I 0 † , no matter what value of (positive or negative) is chosen.

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  Lectures on Quantum Mechanics

  A Primer for Mathematicians

  • Philip L. Bowers(Author)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  This leads to a discussion of generalized coordinates and Hamilton’s action principle, which is used to derive the general Euler–Lagrange equation of motion, the central ingredient 80 6.1 Newtonian Mechanics 81 of Lagrangian mechanics. The action principle is used throughout the development of quantum mechanics, especially in deriving the equations that govern quantum fields. For this reason it is essential that the student of quantum mechanics masters the content of this lecture. After our presentation of Lagrangian mechanics and an examination of the Euler–Lagrange equations, we move on to Hamiltonian mechan- ics. We derive the Hamilton equations of motion and cast them into a particularly elegant form using Poisson brackets. Finally, Noether’s the- orem, which shows how conservation laws arise from symmetries of the Lagrangian, is proved and several examples are presented. The quan- tum version of Noether’s theorem will be used in Lecture 28 to derive several conservation laws in quantum mechanics. Our review of classical mechanics is continued in Lecture 7, where we forge the whole structure of classical mechanics into the mold of the Hamilton–Jacobi equation. This review of classical mechanics will serve us throughout the re- mainder of these lectures as, time and again, quantum mechanics draws from classical mechanics for both inspiration and substantive discourse. 6.1 Newtonian Mechanics and the Euler–Lagrange Equation in Cartesian Coordinates Consider a particle of mass m moving in a one-dimensional potential well determined by the potential V (x), where x is a Cartesian coordinate along the line of motion. The force experienced by the particle is F = −∇V = −dV /dx. Newton’s law for this conservative force acting in the system gives the equation of motion of the particle as m¨ x + dV dx = 0.

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  Advanced Dynamics

  • Donald T. Greenwood(Author)
  • 2006(Publication Date)
  • Cambridge University Press

    (Publisher)

  2 Lagrange’s and Hamilton’s equations In our study of the dynamics of a system of particles, we have been concerned primarily with the Newtonian approach which is vectorial in nature. In general, we need to know the magnitudes and directions of the forces acting on the system, including the forces of constraint. Frequently the constraint forces are not known directly and must be included as additional unknown variables in the equations of motion. Furthermore, the calculation of particle accelerations can present kinematical difficulties. An alternate approach is that of analytical dynamics , as represented by Lagrange’s equa-tions and Hamilton’s equations. These methods enable one to obtain a complete set of equations of motion by differentiations of a single scalar function, namely the Lagrangian function or the Hamiltonian function. These functions include kinetic and potential ener-gies, but ideal constraint forces are not involved. Thus, orderly procedures for obtaining the equations of motion are available and are applicable to a wide range of problems. 2.1 D’Alembert’s principle and Lagrange’s equations D’Alembert’s principle Let us begin with Newton’s law of motion applied to a system of N particles. For the i th particle of mass m i and inertial position r i , we have F i + R i − m i ¨ r i = 0 (2.1) where F i is the applied force and R i is the constraint force. Now take the scalar product with a virtual displacement δ r i and sum over i . We obtain N i = 1 ( F i + R i − m i ¨ r i ) · δ r i = 0 (2.2) This result is valid for arbitrary δ r s; but now assume that the δ r s satisfy the instantaneous or virtual constraint equations , namely, 3 N i = 1 a ji ( x , t ) δ x i = 0 ( j = 1 , . . . , m ) (2.3) 74 Lagrange’s and Hamilton’s equations where the δ x s are the Cartesian components of the δ r s.

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  A Student's Guide to Lagrangians and Hamiltonians

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

    (Publisher)

  (1.16) If there are n coordinates, there are n Lagrange equations, namely, d dt ∂L ∂ ˙ q i − ∂L ∂q i = 0, i = 1, . . . , n. (1.17) It is important to realize that Lagrange’s equations are the equations of motion of a system. 1.10 Obtaining the equation of motion 21 For example, for a mass on a spring the Lagrangian is L = T − V = 1 2 m ˙ x 2 − 1 2 kx 2 . Plugging this into the Lagrange equation we obtain d dt  ∂L ∂ ˙ x  − ∂L ∂x = 0, d dt ∂ ∂ ˙ x  1 2 m ˙ x 2 − 1 2 kx 2  − ∂ ∂x  1 2 m ˙ x 2 − 1 2 kx 2  = 0, d dt (m ˙ x) − kx = 0, so m ¨ x + kx = 0, as expected. (Compare with Equation 1.14.) To illustrate the use of Lagrange’s equations to obtain the equations of motion, we consider several simple mechanical systems. Example 1.2 Atwood’s machine: Figure 1.4 is a sketch of Atwood’s machine. It consists of masses m 1 and m 2 suspended by a massless inextensible string over a frictionless, massless pulley. Evaluate the Lagrangian and obtain the equation of motion. Solution 1.2 The kinetic energy of the masses is T = 1 2 m 1 ˙ x 2 1 + 1 2 m 2 ˙ x 2 2 , and the potential energy is V = −m 1 gx 1 − m 2 gx 2 , where we selected V = 0 at the center of the pulley. The system is subjected to the constraint x 1 + x 2 = l = constant. The Lagrangian is, V = 0 m 2 m 1 x 1 x 2 = l - x 1 Figure 1.4 Atwood’s machine. 22 1 Fundamental concepts L = T − V = 1 2 m 1 ˙ x 2 1 + 1 2 m 2 ˙ x 2 2 + m 1 gx 1 + m 2 gx 2 . But using x 2 = l − x 1 we can rewrite the Lagrangian in terms of a single variable, L = 1 2 m 1 ˙ x 2 1 + 1 2 m 2 ˙ x 2 1 + m 1 gx 1 + m 2 g(l − x 1 ), = 1 2 (m 1 + m 2 ) ˙ x 2 1 + (m 1 − m 2 )gx 1 + m 2 gl. The equation of motion is d dt ∂L ∂ ˙ x 1 − ∂L ∂x 1 = 0, d dt (m 1 + m 2 ) ˙ x 1 − (m 1 − m 2 )g = 0, ¨ x 1 = m 1 − m 2 m 1 + m 2 g. Example 1.3 A cylinder of radius a and mass m rolls without slipping on a fixed cylinder of radius b. Evaluate the Lagrangian and obtain the equation of motion for the short period of time before the cylinders separate.

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  Phenomenology of Particle Physics

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

    (Publisher)

  6 The Lagrangian Formalism If a physical system behaves the same, regardless of how it is oriented in space, the physical laws that govern it are rotationally symmetric; from this symmetry, Noether’s theorem shows the angular momentum of the system must be conserved. The physical system itself need not be symmetric; a jagged asteroid tumbling in space conserves angular momentum despite its asymmetry. Rather, the symmetry of the physical laws governing the system is responsible for the conservation law. As another example, if a physical experiment has the same outcome at any place and at any time, then its laws are symmetric under continuous translations in space and time; by Noether’s theorem, these symmetries account for the conservation laws of linear momentum and energy within this system, respectively. Emmy Noether 6.1 The Euler–Lagrange Equations In quantum mechanics a system is fully described by its Hamiltonian (or Hamilton operator; see Chapter 4). One can also use the Lagrange 1 formalism as is often the case in classical mechanics. The classical motion of a particle in an external potential V is determined by Newton’s law  F = ma, which can be formulated as md 2 x/dt 2 = −∂V/∂x (see Eq. (1.6)). In general, a system can be described by a set of n generalized coordinates q i (i = 1,...,n) which depend on time. The spatial position  r i (t) of each mass as a function of time in the system is given by a function of the generalized coordinates  r i (t) =  r i (q i (t),t) (i = 1,...,n). The time derivatives of the generalized coordinates are called the generalized velocities dq i /dt (also written as ˙ q i ). 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.

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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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  Differential Equations

  A first course on ODE and a brief introduction to PDE

  • Shair Ahmad, Antonio Ambrosetti(Authors)
  • 2019(Publication Date)
  • De Gruyter

    (Publisher)

  16 The Euler–Lagrange equations in the Calculus of Variations: an introduction This chapter is intended to serve as an elementary introduction to the Calculus of Vari-ations, one of the most classical topics in mathematical analysis. For a more complete discussion we refer, e. g., to R. Courant, Calculus of Varia-tions: With Supplementary Notes and Exercises, Courant Inst. of Math. Sci., N. Y. U., 1962 , or B. Dacorogna, Introduction to the Calculus of Variations, 2nd Edition, World Scien-tific, 2008 . Notation: in this chapter we let x denote the independent variable and y = y ( x ) the dependent variable. 16.1 Functionals Given two points in the plane A = ( a , α ) , B = ( b , β ) , the length of a smooth curve y = y ( x ) such that y ( a ) = α , y ( b ) = β is given by ℓ[ y ] = b ∫ a √ 1 + y ? 2 ( x ) dx . The map y 󳨃 → ℓ[ y ] is an example of a functional . In general, given a class of functions Y , a functional is a map defined on Y with values in the set of real numbers ℝ . We will be mainly concerned with functionals of the form I [ y ] = b ∫ a L ( x , y ( x ), y ? ( x ) ) dx , y ∈ Y , (I) where the class Y is given by Y = { y ∈ C 2 ([ a , b ]) : y ( a ) = α , y ( b ) = β } (Y) and the Lagrangian L = L ( x , y , p ) is a function of three variables ( x , y , p ) ∈ [ a , b ]×ℝ×ℝ , such that L ( x , y ( x ), y ? ( x )) is integrable on [ a , b ] , ∀ y ∈ Y . In the preceding arclength example, L is given by L ( p ) = √ 1 + p 2 . To keep the presentation as simple as possible, here and in the sequel we will not deal with the least possible regularity. For example, though I [ y ] would make sense for y ∈ C 1 ([ a , b ]) , we take C 2 functions to avoid technicalities in what follows. The analysis of functionals as (I) is carried out in the Calculus of Variations . This is a branch of mathematical analysis dealing with geometrical or physical problems https://doi.org/10.1515/9783110652864-016

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  The Variational Principles of Mechanics

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

    (Publisher)

  Summary. The method of the Lagrangian multiplier remains valid even in the case of non-holonomic auxiliary conditions. The forces exerted on account of these conditions are once more furnished. These forces are of a polygenic nature. Non-holonomic auxiliary conditions and polygenic forces have the same effect in the Lagrangian equations of motion: they produce a “right-hand side” of these equations.

           10.   Small vibrations about a state of equilibrium. One of the most beautiful examples of the power of the analytical method is the application of the Lagrangian equations to the theory of small vibrations about a state of stable equilibrium. This theory is of basic importance for the elastic behaviour of solids, for the vibrations of structures, for the theory of specific heat and other fundamental problems. The most impressive feature of this theory is its great generality. No matter how simple or how complicated a mechanical system is, its motion near a state of equilibrium is always describable in the same terms. The actual calculation gets complicated if the number of degrees of freedom is large* But the theoretical aspects of the problem remain unchanged.

  The simplifications which occur in this problem are due to the fact that the vibrations are small. We know that the geometry of the configuration space is not Euclidean but Riemannian. But we also know that the curved Riemannian space flattens out more and more if we restrict ourselves to smaller and smaller regions. This behaviour of the Riemannian space finds its analytical expression in the fact that the line element

  where the aik are functions of the qi can be replaced by a separate line element with constant coefficients if we do not leave the immediate neighbourhood of the point P. The aik change so little in that neighbourhood that we can replace them by their values at the point P.

  Let the point P, which is the C-point representing the state of the mechanical system in the configuration space, be a point of equilibrium. For the sake of simplicity, let us agree that the point P shall be at the origin of our reference system, which means that its coordinates are qi = 0. We now consider the line element (510.1 ), with the aik constant, as applicable to the whole of space. The new space is Euclidean, but the error we thus commit tends towards zero as we approach the point P

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