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Appendix a Classical Mechanics Appendix A Classical Mechanics In this Appendix we review some of the basic concepts from classical mechanics that are used in the text. A.1 Newton's Equations Newton's equations of motion describe the behavior of collections of N point particles in a three dimensional space (3N degrees of freedom) in terms of 3N coupled second order differential equations (the number of equations can be reduced if constraints are present), cf(mQrQ) _ F (A.1) dt2 - 01, where a = 1, ... , N, mo: is the mass and ro: is the displacement of the a th particle, and F 01 is the net force on the ath particle due to the other particles and any external fields that might be present. Eqs. (A.1) only have simple structure in inertial frames of reference and for Cartesian coordinates. For noninertial frames and general orthog­ onal curvilinear coordinate systems they rapidly become extremely complicated and nonintuitive. Because Newton's equations are second order, the state of a col­ lection of N particles at time t is determined once the velocities, VOl = ~ = rQ and displacements rQ are specified at time t. New­ ton's equations allow one to determine the state of the system at time, t, uniquely in terms of the state at time t=O Thus a system composed of N point particles evolves in a phase space composed of 3N velocity and 3N position coordinates. 460 Appendix A. Classical Mechanics A.2 Lagrange's Equations Lagrange showed that it is possible to formulate Newtonian mechan­ ics in terms of a variational principle which vastly simplifies the study of mechanical systems in curvilinear coordinates and nonin­ ertial frames and allows a straightforward extension to continuum mechanics. We assume there exists a function, L( {qi}, {qi}, t), of generalized velocities,qi and positions, qi ({ qil denotes the set of velocities (qI, ... , q3N) and {qil denotes the set of generalized posi­ tions (qI, ... qN)) such that when we integrate L({qi}, {qi}, t) between two points {qi(tl)} and {qi(t2)} in phase space the actual physical path is the one which extremizes the integral (A.2) The function, L( {qi}, {qi}, t) is called the Lagrangian and the inte­ gral, S, has units of action. Extremization of the integral in Eq. (A.2) leads to the requirement that the Lagrangian satisfy the equation -88L --d (8L)8· = 0, (t. = 1, ... , N). (A.3) qi dt qi Eqs. (A.3) are called the Lagrange equations. For a single particle in a potential energy field, V(r), the Lagrangian is simply L = m;2 - V(r). Note that Eqs. (A.3) are expressed directly in terms of curvilinear coordinates. If we write down the Lagrangian in terms of curvilinear coordinates, it is then a simple matter to obtain the equations of motion. Two important quantities obtained from the Lagrangian are the generalized momentum, 8L (A.4) Pi = -8.qi and the total energy, or Hamiltonian 3N H = I)qiPi) - L. (A.5) i=l Generalized coordinates are defined from the differential element oflength, ds, in real space. In cartesian coordinates (dS)2 = (dx)2 + (dy)2 + (dz)2 so that ql = X, q2 = y, and q3 = z. In polar coordinates (ds)2 = (dr)2 + r2(dO)2 + (dz)2 so that ql = r, q2 = 0, and q3 = z. A.4. The Poisson Bracket 461 In spherical coordinates (ds)2 = (dr)2 + r2(d8)2 + r 2sin2 (8) (d4» 2 so that ql = r, q2 = 8, and q3 = 4>. A.3 Hamilton's Equations In the Newtonian and Lagrangian formulations of mechanics dynam­ ical systems are described in terms of a phase space composed of generalized velocities and positions. The Hamiltonian formulation describes such systems in terms of a phase space composed of gen­ eralized momenta, {Pi}, and positions, {qi}. The Hamiltonian phase space has very special properties. If the system has some translational symmetry then some of the momenta may be conserved quantities. In addition, for systems obeying Hamilton's equations motion, volume elements in phase space are preserved. Thus the phase space be­ haves like an incompressible fluid. A Legendre transformation from coordinates {qi}, {qi} to coordinates {Pi}, {qi} yields the following equations of motion for the Hamiltonian phase space coordinates (A.6) (A.7) (0;{) =-(~~). (A.B) Eqs. (A.6) to (A.B) are called Hamilton's equations. A.4 The Poisson Bracket The equation of motion of any phase function (any function of phase space variables) may be written in terms of Poisson brackets. Let us consider a phase function, f( {qi}, {Pi}, t). Its total time derivative is (A.9) Using Hamilton's equations we can write this in the form 462 Appendix A. Classical Mechanics df of dt = at + if, H} Poisson' (A.lO) where 3N (Of oH of OH) if, H} Poisson = L oq. op· - op· oq. (A.ll) i=l " " and if, H} Poisson = -{ H, f} Poisson' The Poisson bracket of any two phase functions, f( {qi}, {Pi}, t) and g( {qi}, {Pi}, t) is written aN (of og of Og) {f, g} Poisson = ~ Oqi OPi - OPi Oqi . (A.12) The Poisson bracket is invariant under canonical transformation. That is, if we make a canonical transformation from coordinates (p, q) to coordinates (P, Q) (that is P = pep, Q), q = q(P, Q», the Poisson bracket is given by Eq. (A.12) but with p-+P and q-+Q and f = f(p(P, Q), q(P, Q». A.S Phase Space Volume Conservation One of the important properties of the Hamiltonian phase space is that volume elements are preserved under the flow of points in phase space. A volume element at some initial time, to, can be written It is related to a volume element, dVt, at time t by the Jacobian, IN(to, t) of the transformation between phase space coordinates at time to, {PiCton, {qi(ton and coordinates at time, t, {Pi(tn, {qi(tn. Thus (A.13) For systems obeying Hamilton's equations (even if they have a time dependent Hamiltonian), the Jacobian is a constant of the motion, dJN(t, to) 0 = (A.14) dt ' and therefore volume elements do not change in time. A.6. Action-Angle Variables 463 A.6 Action-Angle Variables We can write Hamilton's equations in terms of any convenient set of generalized coordinates. We can transform between coordinate systems and leave the form of Hamilton's equations invariant via canonical transformations. There is, however, one set of canonical coordinates which plays a distinctive role both in terms of the anal­ ysis of chaotic behavior in classical nonlinear systems and in terms of the transition between classical and quantum mechanics. These are the action-angle variables. We know that in quantum systems, transitions occur in discrete units of n. If an external field is applied which is sufficiently weak and slow, it is possible that no changes will occur in the quantum system because the field is unable to cause a change in the action of the system by a discrete amount, n. Thus, in the transition from classical to quantum mechanics it is the action variables which are quantized because they are adiabatic invariants and have a similar behavior classically [Landau and Lifschitz 1976], [Born 1960j. If a slowly varying weak external field (with period much longer than and incommensurate with the natural period of the sys­ tem) is applied to a classical periodic system, the action remains unchanged whereas the rate of change of the energy is proportional to the rate of change of the applied field. Thus, of all the possi­ ble mechanical coordinates, the action is the only one unaffected by adiabatic perturbations and is the appropriate variable to quantize. Let us consider a one degree of freedom system described in terms of the usual momentum and position coordinates, (p, q), with Hamil­ tonian, H(p, q). We introduce a generating function, Seq, J), which allows us to transform from coordinates (p, q) to action-angle coor­ dinates, (J, 0) via the equations (A.15) and (A.16) The generating function is path independent so 464 Appendix A. Classical Mechanics p q Fig. A.l. The area enclosed by a periodic orbit is proportional to the action We require that H(p, q) = 'H(J) so that J=constant and (J = w(J)t+ (Jo, where w = ('rf) and (Jo is a constant. Now consider a differential change in S, dS = (M-)Jdq+ (~)qdJ. Find the change in S along a path of fixed J (and therefore fixed energy), (dS) J = (~;) Jdq. Then q Seq, J) - Seq', J) = rS(q,J) dS = l (~S) dq = lqpdq.(A.18) JS(q',J) q' uq J q' We now define the action as J= - pdq. (A.19) 271'If closedpath The integral is over a path of fixed J and therefore fixed energy. The action itself is a measure of the area in phase space enclosed by the path (cf. Figure (A.I». Let us now find an expression for the angle variable . We can write d(J = (~:)Jdq+ (~)qdJ. But (~:)J = (~)q' Thus for a path of fixed J, (dO)J = (~)qdq and we can write 9 81q (J - (Jo = 1dO = oj pdq. (A.20) 80 qo Eqs. (A.19) and (A.20) enable us to construct the canonical transfor­ mation between coordinates (J, (J) and (p, q). The whole discussion can easily be generalized to higher dimensional systems. A.7. Hamilton's Principle F\mction 465 A.7 Hamilton's Principle Function Hamilton's principle function for a system with one degree offreedom is defined R(xo,to;x,t) = ltdr L(x,x,r) = ltdr (Px-H(p,x,r».(A.21) to to We wish to compute partial derivatives of R(xo, to; x, t).
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