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where ϕ̇ (t) is the angular velocity of the flywheel andˆ MI33 x2 y 2 z 2 z 2 dxdydzπa2 dˆ Mx2 y 2 dxdy 2πaˆM ρ3 dρdϕπa22πM a4 πa2 41 M a22while the velocity of the pendulum bob is the combination of the swinging, lθ̇, and the oscillatory motion ofthe suspension point, located at 2a cos ωt. With the postion of m given byx 2a cos ϕ l sin θy l cos θthe velocity has componentsẋ 2aϕ̇ sin ϕ lθ̇ cos θẏ lθ̇ sin θTherefore,T 11mv2 I ϕ̇222 112m ẋ ẏ 2 M a2 ϕ̇224 11 2 2 2m 4a ϕ̇ sin ϕ 4alϕ̇θ̇ sin ϕ cos θ l2 θ̇2 M a2 ϕ̇224and the potential is simplyV mgl cos θup to an arbitrary constant. Therefore, 11 L m 4a2 ϕ̇2 sin2 ϕ 4alϕ̇θ̇ sin ϕ cos θ l2 θ̇2 M a2 ϕ̇2 mgl cos θ24To find the Hamiltonian, we first find the conjugate momenta,pθpϕ L θ̇ ml2 θ̇ 2malϕ̇ sin ϕ cos θ L ϕ̇ 14ma2 ϕ̇ sin2 ϕ 2malθ̇ sin ϕ cos θ M a2 ϕ̇2Next, find the Hamiltonian in terms of both velocities and momenta, 12222H ml θ̇ 2malϕ̇ sin ϕ θ̇ 4ma ϕ̇ sin ϕ 2malθ̇ sin ϕ M a ϕ̇ ϕ̇2 11 2 2 2 m 4a ϕ̇ sin ϕ 4alϕ̇θ̇ sin ϕ l2 θ̇2 M a2 ϕ̇2 mgl cos θ241 2 21 ml θ̇ 2malϕ̇θ̇ sin ϕ 2ma2 ϕ̇2 sin2 ϕ M a2 ϕ̇2 mgl cos θ24 11

Finally, solve for the velocities and eliminate them from H,θ̇ pϕ 2mal pϕ 2malθ̇ sin ϕ 2aϕ̇pθ sin ϕ sin ϕ ml2l2apθpϕ sin ϕ l ϕ̇ pθ2aϕ̇ sin ϕml2l14ma2 ϕ̇ sin2 ϕ M a2 ϕ̇214ma2 ϕ̇ sin2 ϕ M a2 ϕ̇214ma2 ϕ̇ sin2 ϕ M a2 ϕ̇ 4ma2 ϕ̇ sin2 ϕ2 122222ϕ̇ 4ma sin ϕ M a 4ma sin ϕ21M a2 ϕ̇224 sin ϕpϕ pθM a2alMand thenθ̇ 2aϕ̇pθ sin ϕml2lpθ8 sin2 ϕ4psinϕ pθ ϕ2ml2 M al l M418msin2 ϕ pθ pϕ sin ϕ1 ml2MM also finally,1 2 21H ml θ̇ 2malϕ̇θ̇ sin ϕ 2ma2 ϕ̇2 sin2 ϕ M a2 ϕ̇2 mgl cos θ24! 2 1 218sinϕ168m8m ml1 sin2 ϕ p2θ sin2 ϕ pϕ pθ 2 2 2 p2ϕ sin2 ϕ1 22MM mal3MM a l(ml2 ) 24 sin ϕ8m41 2malpϕ pθ1 sin2 ϕ pθ pϕ sin ϕ sin ϕM a2alMml2MM al 424 sin ϕ16 sin2 ϕ 212222p 2pϕpθ 2 2 2 pθ mgl cos θ 2ma sin ϕ M a4M 2 a4 ϕM a2alMa l M 2 4 sin ϕ8m18m8msin2 ϕ p2θ sin2 ϕ pϕ pθ 2 2 p2ϕ sin2 ϕ 1 1 22mlMM alMM a 24 sin ϕ8m8 sin ϕ8m8m32m 1 sin2 ϕ pϕ pθ 1 sin2 ϕ pθ pθ 2 2 pϕ pϕ sin2 ϕ 2 pθ pϕ sin3 ϕM laMM l2MM aM al 8m32m sin4 ϕ 4 sin2 ϕ132m sin3 ϕ 4 sin ϕ22 pp p2θ mgl cos θsinϕ p ϕ θϕM 2 a2M a2alM 2alMl2 M 2l2 M4NoteMy computer has swallowed the next ten pages of notes. Instead of rewriting it all now, I am copying relevantsections of my Mechanics book. I’ll try to fill in any missing details.5Phase space and the symplectic formWe now explore some of the properties of phase space and Hamilton’s equations.12

One advantage of the Hamiltonian formulation is that there is now one equation for each initial condition.This gives the space of all qs and ps a uniqueness property that configuration space (the space spanned by theqs only) doesn’t have. For example, a projectile which is launched from the origin. Knowing only this fact,we still don’t know the path of the object – we need the initial velocity as well. As a result, many possibletrajectories pass through each point of configuration space. By contrast, the initial point of a trajectory inphase space gives us both the initial position and the initial momentum. There can be only one path of thesystem that passes through that point.Systems with any number of degrees of freedom may be handled in this way. If a system has N degrees offreedom then its phase space is the 2N -dimensional space of all possible values of both position and momentum. We define configuration space to be the space of all possible postions of the particles comprising thesystem, or the complete set of possible values of the degrees of freedom of the problem. Thus, configurationspace is the N -dimensional space of all values of qi . By momentum space, we mean the N -dimensional spaceof all possible values of all of the conjugate momenta. Hamilton’s equations then consist of 2N first orderdifferential equations for the motion in phase space.We illustrate these points with the simple example of a one dimensional harmonic oscillator.Let a mass, m, free to move in one direction, experience a Hooke’s law restoring force, F kx. SolveHamilton’s equations and study the motion of system in phase space. The Lagrangian for this system isL T V11mẋ2 kx2 22The conjugate momentum is justp L mẋ ẋso the Hamiltonian isH pẋ Lp211 mẋ2 kx2m 22p21 kx22m 2Hamilton’s equations are Hp pm Hṗ kx x H L 0 t tNote that Hamilton’s equations are two first-order equations. From this point on the coupled linear equationsẋ kxpẋ mmay be solved in any of a variety of ways. Let’s treat it as a matrix system, 1dxxm pp kdt qq kkkThe matrix M has eigenvalues ω and diagonalizes to1m, mṗ m iω00iω 13 AM A 1(1)

where 11i km i km 12i km 1 1 i km i kmrk mAA 1ωTherefore, multiplying eq.(1) on the left by A and inserting 1 A 1 A, 1dxxmA AA 1 App kdt(2)we get decoupled equations in the new variables: aa† ddt qp A 12 x 12 x ipkm ipkm (3)The decoupled equations are aa† iω00iω aa† (4)or simplyȧ iωa†ȧ iωa†a a0 e iωtwith solutionsa† a†0 eiωtThe solutions for x and p may be written asxpp0sin ωtmω mωx0 sin ωt p0 cos ωt x0 cos ωt Notice that once we specify the initial point in phase space, (x0 , p0 ) , the entire solution is determined. Thissolution gives a parameterized curve in phase space. To see what curve it is, note thatm 2 ω 2 x2p2 2mE2mE m 2 ω 2 x2p2 p20 m2 ω 2 x20p20 m2 ω 2 x20 22 2m ωp0xcosωt sinωt0p20 m2 ω 2 x20mω12 2( mωx0 sin ωt p0 cos ωt)p0 m2 ω 2 x20p20m2 ω 2 x20 p20 m2 ω 2 x20p20 m2 ω 2 x20114

orm2 ω 2 x2 p2 2mEThis describes an ellipse in the xp plane. The larger the energy, the larger the ellipse, so the possible motionsof the system give a set of nested, non-intersecting ellipses. Clearly, every point of the xp plane lies on exactlyone ellipse.The phase space description of classical systems are equivalent to the configuration space solutions andare often easier to interpret because more information is displayed at once. The price we pay for this isthe doubled dimension – paths rapidly become difficult to plot. To ofset this problem, we can use Poincarésections – projections of the phase space plot onto subspaces that cut across the trajectories. Sometimes thepatterns that occur on Poincaré sections show that the motion is confined to specific regions of phase space,even when the motion never repeats itself. These techniques allow us to study systems that are chaotic,meaning that the phase space paths through nearby points diverge rapidly.Now consider the general case of N degrees of freedom. Let ξ A q i , pjwhere A 1, . . . , 2N. Then the 2N variables ξ A provide a set of coordinates for phase space. We would liketo write Hamilton’s equations in terms of these, thereby treating all 2N directions on an equal footing.In terms of ξ A , we have i dξ Aq̇ ṗjdt! H pi H qj ΩAB H ξ Bwhere the presence of ΩAB in the last step takes care of the difference in signs on the right. Here ΩAB isjust the inverse of the symplectic form found from the curl of the dilatation, given by 0δjiΩAB δij 0Its occurrence in Hamilton’s equations is an indication of its central importance in Hamiltonian mechanics.We may now write Hamilton’s equations asdξ A H ΩAB Bdt ξ(5)Consider what happens to Hamilton’s equations if we want to change to a new set of phase spacecoordinates, χA χA (ξ) . Let the inverse transformation be ξ A (χ) . The time derivatives becomedξ A ξ A dχB dt χB dtwhile the right side becomesΩABC H HAB χ ΩBB ξ ξ χCEquating these expressions, ξ A dχB χD H ΩAB BB χ dt ξ χD15

we multiply by the Jacobian matrix, χC ξ Ato get χC ξ A dχB ξ A χB dtBC dχδBdt χC AB χDΩ ξ A ξ B χC AB χDΩ ξ A ξ B H χD H χDand finallydχC χC AB χD HΩ dt ξ A ξ B χDDefining the symplectic form in the new coordinate system,Ω̃CD χC AB χDΩ ξ A ξ Bwe see that Hamilton’s equations are entirely the same if the transformation leaves the symplectic forminvariant,Ω̃CD ΩCDAny linear transformation M A B leaving the symplectic form invariant,ΩAB M A C M B D ΩCDis called a symplectic transformation. Coordinate transformations which are symplectic transformations ateach point are called canonical. Therefore those functions χA (ξ) satisfyingΩCD χC AB χDΩ ξ A ξ Bare canonical transformations. Canonical transformations preserve Hamilton’s equations.5.1Poisson bracketsWe may also write Hamilton’s equations in terms of the Poisson brackets. Recall that the Poisson bracketof any two dynamical variables f and g is given by{f, g} ΩAB f g ξ A ξ BThe importance of this product is that it too is preserved by canonical transformations. We see this asfollows.Let ξ A be any set of phase space coordinates in which Hamilton’s equations take the form of eq.(5), andlet f and g be any two dynamical variables, that is, functions of these phase space coordinates, ξ A . ThePoisson bracket of f and g is given above. In a different set of coordinates, χA (ξ) , we have{f, g}0 f g χA χB C D ξ f ξ gΩAB χA ξ C χB ξ D C ξ ξ D f gΩAB BA χ χ ξ C ξ DΩABTherefore, if the coordinate transformation is canonical so that ξ C AB ξ DΩ ΩCD χA χB16

then we have0{f, g} ΩAB f g {f, g} ξ C ξ Dand the Poisson bracket is unchanged. We conclude that canonical transformations preserve all Poissonbrackets.An important special case of the Poisson bracket occurs when one of the functions is the Hamiltonian.In that case, we have{f, H} f H ξ A ξ B f H f H ii x pi p xi dpi f f dxi i xi dt pdtdf f t tΩAB or simply,df f {f, H} t tThis shows that as the system evolves classically, the total time rate of change of any dynamical variable isthe sum of the Poisson bracket with the Hamiltonian and the partial time derivative. If a dynamical variablehas no explicit time dependence, then f t 0 and the total time derivative is just the Poisson bracket withthe Hamiltonian.The coordinates now provide a special case. Since neither xi nor pi has any explicit time dependence,with havedxidtdpidt H, xi {H, pi }and we can check this directly:dqidt H, xi N X xi H xi H xj pj pj xjj 1 NXδijj 1 H pj H pi {H, pi } N X pi H pi H qj pj pj qjj 1 anddpidt H qi17(6)

Notice that since qi , pi and are all independent, and do not depend explicitly on time, qi t qi pj pi qj 0 pi t . Finally, we define the fundamental Poisson brackets. Suppose xi and pj are a set of coordinates onphase space such that Hamilton’s equations hold in the either the form of eqs.(6) or of eqs.(5). Since theythemselves are functions of (xm , pn ) they are dynamical variables and we may compute their Poisson bracketswith one another. With ξ A (xm , pn ) we have i jx ,xξ xi xj ξ A ξ B NX xi xj xi xj xm pm pm xmm 1 ΩAB 0 ΩABfor xi with xj , xi , pjξ xi pj ξ A ξ B N X xi pj xi pj xm pm pm xmm 1NXi mδmδjm 1 δji ΩABfor xi with pj and finally{pi , pj }ξ pi pj ξ A ξ B N X pi pj pi pj xm pm pm xmm 10for pi with pj . The subscript ξ on the bracket indicates that the partial derivatives are taken with respectto the coordinates ξ A xi , pj . We summarize these relations as A Bξ , ξ ξ ΩABWe summarize the results of this subsection with a theorem: Let the coordinates ξ A be canonical. Thena transformation χA (ξ) is canonical if and only if it satisfies the fundamental bracket relation A Bχ , χ ξ ΩABFor proof, note that the bracket on the left is defined by χA , χBξ ΩCD χA χB ξ C ξ Dso in order for χA to satisfy the canonical bracket we must haveΩCD χA χB ΩAB ξ C ξ D18(7)

which is just the condition shown above for a coordinate transformation to be canonical. Conversely, supposethe transformation χA (ξ) is canonical and ξ A , ξ B ξ ΩAB . Then eq.(7) holds and we have A Bχ ,χξ ΩCD χA χB ΩAB ξ C ξ Dso χA satisfies the fundamental bracked relation.In summary, each of the following statements is equivalent:1. χA (ξ) is a canonical transformation.2. χA (ξ) is a coordinate transformation of phase space that preserves Hamilton’s equations.3. χA (ξ) preserves the symplectic form, according toΩAB ξ C ξ D ΩCD χA χB4. χA (ξ) satisfies the fundamental bracket relations A Bχ ,χξ ΩCD χA χB ξ C ξ DThese bracket relations represent a set of integrability conditions that must be satisfied by any new set ofcanonical coordinates. When we formulatethe problemof canonical transformations in these terms, it is not obvious what functions q i xj , pj and πi xj , pj will be allowed. Fortunately there is a simple procedurefor generating canonical transformations, which we develop in the next section.We end this section with three examples of canonical transformations.5.1.1Example 1: Coordinate transformationsLet the new configuration space variable, q i , be and an arbitrary function of the spatial coordinates: q i q i xj and let πj be the momentum variables corresponding to q i . Then q i , πj satisfy the fundamental Poissonbracket relations iff: i jq , q x,p 0 iq , πj x,p δji{πi , πj }x,p 0Check each: i jq ,qx,p N X q i q j q i q j xm pm pm xmm 1 since q j pm0 0. For the second bracket,δji iq , πjNX m 1 x,p q i πj q i πj m x pm pm xmNX q i πj xm pmm 119

Since q i is independent of pm , we can satisfy this only if xm πj pm q jIntegrating givesπj xnpn cj q jwith cj an arbitrary constant. The presence of cj does not affect the value of the Poisson bracket. Choosingcj 0, we compute the final bracket:{πi , πj }x,p N X πi πj πi πj xm pm pm xmm 1 N X xm xm 2 xn 2 xnp p nn xm q i q j q i xm q jm 1 N X xm xn xm xn pn q j xm q i q i xm q jm 1 N X xn xn pn q j q i q i q jm 1 0Therefore, the transformationsqj πj q j (xi ) xnpn cj q jis a canonical transformation for any functions q i (x). This means that the symmetry group of Hamilton’sequations is at least as big as the symmetry group of the Euler-Lagrange equations.5.1.2Example 2: Interchange of x and p.The transformationqi piπi xiis canonical. We easily check the fundamental brackets: i jq , q x,p {pi , pj }x,p 0 i q , πj x,p pi , xj x,p pi , xj x,p xj , pi x,p{πi , πj }x,p δij i x , xjx,p 0Interchange of xi and pj , with a sign, is therefore canonical. The use of generalized coordinates does notinclude such a possibility, so Hamiltonian dynamics has a larger symmetry group than Lagrangian dynamics.20

For our last example, we first show that the composition of two canonical transformations is also canonical.Let ψ (χ) and χ (ξ) both be canonical. Defining the composition transformation, ψ (ξ) ψ (χ (ξ)) , wecompute A B BA ψ χE ψ χFCDCD ψ ψ ΩΩ ξ C ξ D χE ξ C χF ξ D B χE χF CD ψ A ψ ΩCDE ξ ξ χ χF A B ψ ψ ΩEF χE χFΩAB so that ψ (χ (ξ)) is canonical.5.1.3Example 3: Momentum transformationsBy the previous results, the composition of an arbitratry coordinate change with x, p interchanges is canonical. Consider the effect of composing (a) an interchange, (b) a coordinate transformation, and (c) aninterchange.For (a), letq1i piπi1 xiThen for (b) we choose an arbitrary function of q1i :Qi Qi q1j Qi (pj )Pi q1n pnπn i xn Qi QFinally, for (c), another interchange:qiπi pn nx Qi Qi Qi (pj ) Pi This establishes that replacing the momenta by any three independent functions of the momenta, preservesHamilton’s equations.5.2Generating functionsThere is a systematic approach to canonical transformations using generating functions. We will give asimple example of the technique. Given a system described by a Hamiltonian H(xi , pj ), we seek anotherHamiltonian H 0 (q i , πj ) such that the equations of motion have the same form, namelydxidtdpidt H pi H i x 21

in the original system and H 0 πi H 0 i qdq idtdπidt in the transformed variables. The principle of least action must hold for each pair:ˆ S pi dxi Hdtˆ S0 πi dq i H 0 dtwhere S and S 0 differ by at most a constant. Correspondingly, the integrands may differ by the addition ofa total differential, df dfdt dt, since this will integrate to a surface term and therefore will not contribute tothe variation. Notice that this corresponds exactly to a local dilatation, which produces a changeWα0 dxα Wα dxα dfdf Wα dxα dtdtIn general we may therefore writepi dxi Hdt πi dq i H 0 dt dfA convenient way to analyze the condition is to solve it for the differential dfdf pi dxi πi dq i (H 0 H) dtFor the differential of f to take this form, it must be a function of xi , q i and t, that is, f f (xi , q i , t).Therefore, the differential of f is f i f fdf dx i dq i dt xi q tEquating the expressions for df we match up terms to require f xi f i qpi πi H0 H (8)(9) f t(10)The first equation f (xj , q j , t)(11) xigives q i implicitly in terms of the original variables, while the second determines πi . Notice that we may pickany function q i q i (pj , xj , t). This choice fixes the form of πi by the eq.(9), while the eq.(10) gives the newHamiltonian in terms of the old one. The function f is the generating function of the transformation.pi 6General solution in Hamiltonian dynamicsWe conclude with the crowning theorem of Hamiltonian dynamics: a proof that for any Hamiltonian dynamical system there exists a canonical transformation to a set of variables on phase space such that the pathsof motion reduce to single points. Clearly, this theorem shows the power of canonical transformations! Thetheorem relies on describing solutions to the Hamilton-Jacobi equation, which we introduce first.22

6.1The Hamilton-Jacobi EquationWe have the following equations governing Hamilton’s principal function. S pi S xi S tSince the Hamiltonian is a given function of the phasecombine the last two equations:0pi Hspace coordinates and time, H H(xi , pi , t), we S S H(xi , pi , t) H(xi ,, t) t xiThis first order differential equation in s 1 variables (t, xi ; i 1, . . . s) for the principal function S is theHamilton-Jacobi equation. Notice that the Hamilton-Jacobi equation has the same general form as th

Hamiltonian Mechanics December 5, 2012 1 Phase space Phase space is a dynamical arena for classic