Nonlinear Mechanics - Physics at Oregon State University

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10 CHAPTER 1. LAGRANGIAN DYNAMICS 1.4 Conservative Forces and the Lagrangian So far we have made no assumptions about the nature of the forces included in ℑ except that they are not forces of constraint. Equation (16) is therefore quite general, although seldom used in this form. In these notes we are primarily concerned with conservative forces, i.e. forces that can be derived from a potential. Fi = −∇iV (r1 · · · rN) (1.17) Notice that V doesn’t depend on velocity. (Electromagnetic forces are velocity dependent, of course, but they can easily be accommodated into the Lagrangian framework. I will return to this issue later on.) Now calculate the work done by changing some of the q’s. ∫ ∑ W = Fi · dri = − ∑ ∫ ∇iV · dri i = − ∑ ∫ ∇iV · i ∑ ∂ri dqk ∂qk k = − ∑ ∫ k ( ∑ ∇iV · i ∂ri ) dqk ∂qk = − ∑ ∫ ∂V dqk ∂qk i k (1.18) The integral is a multidimensional definite integral over the various q’s that have changed. Summing over (1.12) then gives Comparison with (1.18) yields Finally define the Lagrangian δW = ∑ δWk = ∑ ℑkδqk k W = ∑ ∫ k ℑk = − ∂V ∂qk k ℑkdqk (1.19) (1.20) (1.21) L = T − V (1.22)
1.4. CONSERVATIVE FORCES AND THE LAGRANGIAN 11 Equation (??) becomes d dt ( ∂L ∂ ˙qk ) − ∂L ∂qk = 0. (1.23) Equation (1.23) represents a set of 3N −l second order partial differential equations called Lagrange’s equations of motion. I can summarize this long development by giving you a “cookbook” procedure for using (1.23) to solve mechanics problems: First select a convenient set of generalized coordinates. Then calculate T and V in the usual way using the ri’s. Use equation (1.3) to eliminate the ri’s in favor of the qk’s. Finally substitute L into (1.23) and solve the resulting equations. Classical mechanics texts are full of examples in which this program is carried to a successful conclusion. In fact, most of these problems are contrived and of little interest except to illustrate the method. The vast majority of systems lead to differential equations that cannot be solved in closed form. The modern emphasis is to understand the solutions qualitatively and then obtain numerical solutions using the computer. The Hamiltonian formalism described in the next section is better suited to both these ends. 1.4.1 The Central Force Problem in a Plane Consider the central force problem as an example of this technique. V = V (r) F = −∇V (1.24) L = T − V = 1 2 m ( ˙r 2 + r 2 ϕ˙ 2 ) − V (r) (1.25) Let’s choose our generalized coordinates to be q1 = r and q2 = ϕ. Equation (1.23) becomes m¨r − mr ˙ ϕ 2 + dV = 0 dr d ( mr dt (1.26) 2 ) ϕ˙ = 0 (1.27) This last equation tells us that there is a quantity mr 2 ˙ ϕ that does not change with time. Such a quantity is said to be conserved. In this case we have rediscovered the conservation of angular momentum. This reduces the problem to one dimension. mr 2 ˙ ϕ ≡ lz = constant (1.28) m¨r = l2 z dV − mr3 dr (1.29)
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