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(2004) chapter 3. lagrangian dynamics (149-211)(32s)

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This version: 06/01/2004 Chapter 3 Lagrangian dynamics Essentially, the previous two chapters dealt with writing the equations of motion, and had very little to say about the behaviour of the solutions of the resulting equations. Gen- erally, one can say very little about the nature of solutions to the Euler-Lagrange equations. Therefore, in this section we look at special cases, and introduce qualitative methods. The study of the dynamics of mechanical systems fills an enormous number of pages, but despite this, very little is known about the behaviour of such systems. Thus we start our discussion in this section with some rather general ideas. Only when we get to Section 3.4 are we able to apply the general ideas to specific, and very simple, systems. It is only in these very special cases that one is able approach anything like a complete understanding to the dynamics of a class of systems. For these systems, the general ideas of the earlier sections of the chapter become more tangible, so it is hoped that the reader will have some patience in waiting for the application of these ideas. 3.1 The Euler-Lagr ange equation s and differentia l equat ions We begin our investigation of Lagrangian dynamics with a discussion of how one obtains differential equations from the Eule r-Lag range equati ons. For those who think that the Euler-Lagrange equations are differential equations, we bring you the following example. 3.1.1 Example Consider the Lagrangian on Q = R defined by L(t,x,v) = v − x. The Euler- Lagrange equation (there is only one) is then d dt ∂L ∂v − ∂L ∂x = 1 = 0. Clearly this is nonsense, and in this case there can be no solution for the Euler-Lagrange equations. You will say that the preceding example is contrived, as surely it is. However, it nonethe- less points out the need for there to be some consideration given to the matter of when the Euler-Lagrange equations have solutions. The following result is perhaps not surprising given Theorem 2.9.2. 3.1.2 Proposition If L is a hyperregular Lagrangian on Q, then for each q0 ∈ Q and for each v0 ∈ Tq0 Q there exists T > 0 and a solution c: [t0, t0 + T] → Q to the Euler-Lagrang e equations for L with the property that c (t0) = v0. F urth ermo re , if we have ˜ T > 0 and a curve ˜ c : [t 0, t0 + ˜ T] → Q with the properties that ˜ c is a solution to the Euler-Lagrange equations and that ˜ c (t0) = v0, then c(t) = ˜ c(t ) for all t ∈ [t0, t0 + T] ∩ [t0, t0 + ˜ T]. 150 3 Lagrangian dynamics 06/01/2004 Proof We fix a coordinate chart (U, φ) around q . Since L is hyperregular, the map F L, given in coordinates by, (q 1 ,...,q  n , v 1 ,...,v n ) → q 1 ,...,q  n , ∂L ∂v 1 ,..., ∂L ∂v n is invertible. By the inverse function theorem, this means that the Jacobian of the map must also be invertible. The Jacobian will be a matrix of the form I n 0n×n A1 A2 , where A1 is the matrix with components ∂ 2 L ∂v i ∂q j , i, j = 1,...,n, and where A2 is the matrix with components ∂ 2 L ∂v i ∂v j , i, j = 1,...,n. In order that this Jac obian be inv ertib le, it is necessary and sufficient that A2 be invertible when evaluated at any (q, v) ∈ R n × R n . Now let us look at the Euler-Lagrange equations. We expand them to arrive at d dt ∂L ∂v i − ∂L ∂q i = ∂ 2 L ∂v i ∂v  j ˙ v  j + ∂ 2 L ∂v i ∂q  j ˙ q  j + ∂ 2 L ∂v i ∂t − ∂L ∂q i , i = 1,...,n. Therefore, the Euler-Lagrange equations, when L is hyperregular, are equivalent to ¨ q k = A −1 2 ki ∂L ∂q i − ∂ 2 L ∂v i ∂q  j ˙ q  j − ∂ 2 L ∂v i ∂t , k = 1,...,n. Here A −1 2 ki , k, i = 1,...,n, are the components of the inverse of A2. Thus we have shown that the Euler -Lagr ange equations are equi val ent to a secon d-order differ enti al equat ion when the Lagrangian is hyperregular, and thus the result follows by the standard existence and uniqueness theory for ordinary differential equations. The above result has the following corollary which gets us out of the woods as concerns worrying about existence and uniqueness of solutions for simple mechanical systems. 3.1.3 Corollary If (Q, g, V) is a simple mechanical system with Lagrangian L, then for each q0 ∈ Q and for each v0 ∈ Tq0 Q there exists T > 0 and a solution c: [t0, t0 + T] → Q to the Euler-Lagrange equations for L with the property that c (t0) = v0. Furthermore, if we have ˜ T > 0 and a curve ˜ c: [t0, t0 + ˜ T] → Q with the properties that ˜ c is a solution to the Euler- Lagr ange equations and that ˜ c (t0) = v0, then c(t) = ˜ c(t ) for all t ∈ [t0, t0 + T] ∩ [t0, t0 + ˜ T]. Proof This follows directly from Proposition 2.9.1. Also note that the differential equations derived in Proposition 2.4.5 for simple mechanical systems also serve as proof of the previ ous result. In any case, the bottom line is that the equations describing the behaviour of simple mechanical systems, at least in the absence of forces and constraints, satisfy the usual existence and uniqueness properties of ordinary differential equations. Let us explore this just a little bit further in order that we ensure ourselves that we have left some stone s untu rned. Inde ed, the imme diate questi on raised by Proposition 3.1.2 is whether hyperregularity of L is equivalent to there being a unique solution to the Euler- Lagrange equations through each point vq ∈ TQ. The answer is that while hyperregularity is, as we have seen, sufficient for existence and uniqueness, it is not necessary. Some contrived examples illustrate this.

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