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Analytical Dynamics: Lagrange's Equation and Its Application

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Analytical Dynamics: Lagrange's Equation and Its Application Analytical Dynamics: Lagrange's Equation and its Application { A Brief Introduction D. S. Stutts, Ph.D. Associate Professor of Mechanical Engineering Missouri University of Science and Technology Rolla, MO 65409-0050 January 24, 2011∗ ∗Corrected typo in Equation (6) { thanks, Shantanu! 1 © Dr. Daniel S. Stutts, 1995 { 2011All Rights Res. 1 THE CALCULUS OF VARIATIONS 1 The Calculus of Variations The calculus of variations is an extensive subject, and there are many fine references which present a detailed development of the subject { see Bibliography. The purpose of this addendum is do provide a brief background in the theory behind Lagrange's Equations. Fortunately, complete understanding of this theory is not absolutely necessary to use Lagrange's equations, but a basic understanding of variational principles can greatly increase your mechanical modeling skills. 1.1 Extremum of an Integral { The Euler-Lagrange Equation Given the Integral of a functional (a function of functions) of the form Z t2 I() = F (U; U;_ t)dt; (1) t1 _ where t1, and t2 are arbitrary, is a small positive, real constant, and U and U are given by U(t) = u(t) + η(t); and U_ (t) =u _(t) + η_(t): (2) The functions U, and u may be thought of as describing the possible positions of a dynamical system between the two instants in time, t1, and t2, where u(t) represents the position when the integral de- scribed by Equation (1) is stationary, i.e. where it is an extremum, and U(t) is u(t) plus a variation η(t). The function U(t) does not by definition render (1) stationary because we shall assume η(t) is indepen- dent of u(t), and we will assume that a unique function renders (1) an extremum. The reasons for these assumptions will become clear below. The important point so far is that have not made any restrictive statements about I() other than it is an integral of a functional of the functions U(t) and U_ (t). We will now specify that the functions u(t), and η(t) are of class C2. That is, they possess continuous second derivatives with respect to t. Further, let us stipulate that η(t) must vanish at t = t1, and t = t2. In other words, u(t) and U(t) coincide at the end points of the interval [ t1, t2], where t1, and t2 are arbitrary. Now that we have the stage more or less set up, lets see what rules the functional F (U; U;_ t) must obey to render (1) extreme. We have, by definition, that the function u(t) renders I stationary, hence, we know this occurs when U(t) = u(t), or = 0. this situation is depicted in Figure1. Figure 1. Relationship between extremizing function u(t), and variation η(t). Thus, assuming that t1, and t2 are not functions of , we set the first derivative of I() equal to zero. Z t2 dI() dF = (U; U;_ t)dt = 0: (3) d =0 t1 d 2 © Dr. Daniel S. Stutts, 1995 { 2011All Rights Res. 1 THE CALCULUS OF VARIATIONS However, dF @F @U @F @U_ (U; U;_ t) = + ; (4) d @U @ @U_ @ so substituting Equation (4) into Equation (3), and setting = 0, we have Z t2 dI() @F @F = η + η_ dt = 0: (5) d =0 t1 @u @u_ Integration of Equation (5) by parts yields: Z t2 t2 dI() @F d @F @F = η(t) − dt + η(t) = 0: (6) d @u dt @u_ @u_ =0 t1 t1 The last term in Equation (6) vanishes because of the stipulation η(t1) = η(t2) = 0, which leaves Z t2 dI() @F d @F = η(t) − dt = 0: (7) d =0 t1 @u dt @u_ By the fundamental theorem of the calculus of variations [1], since η(t) is arbitrary except at the end points t1, and t2, we must have, in general dI() @F d @F = − = 0: (8) d =0 @u dt @u_ Equation (8) is known as the Euler-Lagrange equation. It specifies the conditions on the functional F to extremize the integral I() given by Equation (1). By extremize, we mean that I() may be (1) maxi- mum, (2) minimum, or (3) an inflection point { i.e. neither maximum, nor minimum. In fact, there is no guarantee of the existence of a global extremum; the integral may be only locally extreme for small values of . The determination of the nature of the stationary condition of I() for the general case is beyond the scope of this document. Let it suffice to say that for every case considered in this class, I() will be globally minimum when = 0. Equation (5) is often written Z t2 Z t2 dI() @F @F @F @F δI = = η + η_ dt = δu + δu_ dt = 0; (9) d =0 t1 @u @u_ t1 @u @u_ where δu = η is the variation of u, and [2], d d du δu = (η) = η_ = δ = δu:_ (10) dt dt dt Using Equation (10), and integrating Equation (9) by parts, we obtain Z t2 @F d @F δI = − δudt; (11) t1 @u dt @u_ with the stipulation, as before, that δu(t1) = δu(t2) = 0. 3 © Dr. Daniel S. Stutts, 1995 { 2011All Rights Res. 2 HAMILTON'S PRINCIPLE 2 Hamilton's Principle Hamilton's principal is, perhaps, the most important result in the calculus of variations. We derived the Euler-Lagrange equation for a single variable, u, but we will now shift our attention to a system N particles of mass mi each. Hence, we may obtain N equations of the form mi¨ri = Fi; (12) th where the bold font indicates a vector quantity, and Fi denotes the total force on the i particle. D'Alembert's principle may be stated by rewriting Equation (12) as mi¨ri − Fi = 0 (13) Taking the dot product of each of the Equations (13) with the variation in position δr, and summing the result over all N particles, yields N X (mi¨ri − Fi) · δr = 0: (14) i=1 We note that the sum of the virtual work done by the applied forces over the virtual displacements is given by N X δW = Fi · δr: (15) i=1 Next, we note that N N N X X d 1 X d m ¨r · δr = m (r_ · δr ) − δ r_ · r_ = m (r_ · δr ) − δT; (16) i i i i dt i i 2 i i i dt i i i=1 i=1 i=1 where δT is the variation of the kinetic energy. Hence, Equation (16) may be written, N X d δT + δW = m (r_ · δr ) : (17) i dt i i i=1 In a manner similar to that shown in Figure1, and in view of Equation (10) the possible dynamical paths of each particle may be represented as shown in Figure2, where the varied dynamical path may be thought to occur atemporally. r r r r Figure 2. Possible dynamical paths for a particle between two arbitrary instants in time. 4 © Dr. Daniel S. Stutts, 1995 { 2011All Rights Res. 2 HAMILTON'S PRINCIPLE Since we again have that δr(t1) = δr(t2) = 0, we may multiply Equation (17) by dt, and and integrate between the two arbitrary times t1, and t2 to obtain N t2 Z t2 X (δT + δW ) dt = mi (r_ i · δri) = 0: (18) t1 i=1 t1 If δW can be expressed as the variation of the potential energy, −δV 1, Equation (18) may be written Z t2 (δT − δV ) dt = 0: (19) t1 Introducing the Lagrange function, L = T − V , Equation (19) becomes Z t2 δ Ldt = 0: (20) t1 Equation (20) is the mathematical statement of Hamilton's principal. Hamilton's principal may be defined in words as follows. Definition 1 The actual path a body takes in configuration space renders the value of the definite integral R t2 I = Ldt stationary with respect to all arbitrary variations of the path between two instants t1, and t2 t1 provided that the path variations vanish at t1, and t2 [2]. In all physical systems, the stationary value of I will be a minimum. 2.1 Generalized Coordinates Implicit in the definition of Hamilton's principle is that the system will move along a dynamical path consistent with the system constraints { i.e. along a permissible path. Generalized coordinates render the dynamical path explicitly permissible by describing it using the minimum number of independent coor- dinates. Thus, the ith system position may be described as a function of the N generalized coordinates, and (in general) time, t, as follows: ri = ri(q1; q2; q3; ··· ; qn; t): (21) Hence, the variation of the ith position, which occurs atemporally, may be expressed as N X @ri δr = δq : (22) i @q j j=1 j Consider the following example. Example 1 Given the system shown in Figure3, determine the virtual work δW done by the force F = 4i + 3j, where i, and j are the unit vectors in the x and y directions, over a virtual displacement δr consistent with the constraints: x = r cos θ, and y = r sin θ. Use the generalized coordinates, q1 = r, and q2 = θ. 1The negative sign on δV is chosen to reflect that conservative forces may always be written as the negative gradient of @ @ @ the potential energy: Fci = −r V . Where, for example, in Cartesian coordinates: r = @x i + @y j + @z k. 5 © Dr. Daniel S. Stutts, 1995 { 2011All Rights Res. 3 LAGRANGE'S EQUATIONS OF MOTION m Figure 3. Particle of mass m moving under influence of force F.
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