Hamiltonian systems. Marc R. Roussel. October 25, 2005

Transcription

1 Hamiltoia systems Marc R. Roussel October 25, 25 1 Itroductio Today s otes will deviate somewhat from the mai lie of lectures to itroduce a importat class of dyamical systems which were first studied i mechaics, amely Hamiltoia systems. There is a large literature o Hamiltoia systems. The itetio here is ot to comprehesively survey this literature, which would be quite impossible eve if we devoted a whole course to the subject, but to discuss some of the key ideas uderlyig the treatmet of this class of systems. We begi with some defiitios: Defiitio 1 Suppose that H(x,p) is a smooth fuctio of its argumets for x ad p R. The the dyamical system ẋ i = H p i, (1a) ṗ i = H x i (i = 1,2,...,) is called a Hamiltoia system ad H is the Hamiltoia fuctio (or just the Hamiltoia) of the system. Equatios 1 are called Hamilto s equatios. Defiitio 2 The umber of degrees of freedom of a Hamiltoia system is the umber of (x i, p i ) pairs i Hamilto s equatios, i.e. the value of. The phase space is therefore 2-dimesioal. I mechaics, the vector x represets the geeralized coordiates of the compoets of the system (positios, agles, etc.), while p is a set of geeralized mometa. There is a elaborate theory for costructig geeralized mometa ad Hamiltoias which we wo t go ito here. Note that the geeralized mometa are ot always ordiary liear mometa (mv i ), although that is ofte the case. Note that the Hamiltoia fuctio is a cotat of the motio: Ḣ = = H ẋ i + x i H H x i p i + H p i ṗ i 1 H p i ( H ) =. x i (1b)

2 E (J) p (kg m/s) x (m) -1 Figure 1: Harmoic oscillator orbits for k = 2 N/m, m =.1 kg, ad various values of the total eergy E. Maple code which ca be used to draw such a figure is give i Appedix A. (I actually used guplot because I thik the output looks a little icer, but the priciple is the same.) Note that these closed orbits are ot limit cycles sice differet iitial coditios (which i geeral lead to differet values of E) will produce differet orbits. I fact, the Hamiltoia is ofte just the total eergy i mechaical systems, although this is t always the case. Let us for the momet specialize the discussio to plaar systems, i.e. systems for which = 1. The fact that H is costat is meas that the motio is costraied to the curve H(x, p) = h, where h is the value of the Hamiltoia fuctio implied by the iitial coditios. This curve is of course just the orbit of the system i phase space. If we chage the total eergy, we get a differet orbit. Example 1.1 A harmoic oscillator is a mass-sprig system with potetial eergy 1 2 kx2, where x is the displacemet of the sprig from equilibrium. For simple systems like this oe i which the potetial eergy simply depeds o the positio, the Hamiltoia is just the total eergy: H(x, p) = 1 2 kx2 + p2 2m, (2) where p is the mometum. Because H is a costat, the orbits are just the family of ellipses 1 2 kx2 + p2 2m = E. The value of E is fixed by the iitial coditios. Differet values of E correspod to ellipses of differet sizes. Figure 1 shows a example. 2

3 If we are iterested i the equatios of motio, we ca recover them from Hamilto s equatios: ẋ = H p = p/m. ṗ = H x = kx. Sice v = p/m, the first equatio just says that ẋ is the velocity of the particle. Recall that F = ṗ. (We usually write this as F = ma i elemetary physics courses.) Thus, the secod equatio gives us the force law correspodig to the potetial eergy 1 2 kx2, which you may recogize from your earlier studies of physics. As you well kow from mechaics, mechaical systems are subject to other coservatio laws: mometum, agular mometum, ad so o. If we ca fid eough coservatio laws for a system the, as i the example above, the busiess of determiig the behavior of the system becomes rather simple. O the other had, if we have more degrees of freedom tha costats, the behavior ca be complex. These issues are discussed briefly i the followig sectios. (3a) (3b) 2 Itegrable systems We start with some specialized machiery ad defiitios pertaiig to Hamiltoia systems. Defiitio 3 Let H(x,p) ad L(x,p) be differetiable fuctios of their argumets for x ad p R. The Poisso bracket of H with L, {H,L}, is defied by {H,L} = ( H L H ) L. p i x i x i p i Defiitio 4 A quatity L is called a first itegral of a Hamiltoia system if it is a costat of the motio, i.e. if L = uder the flow implied by Hamilto s equatios. Note that the Hamiltoia itself is a first itegral accordig to this defiitio. Theorem 1 The quatity L is a first itegral of a Hamiltoia system with Hamiltoia H if {H,L} =. The proof of this propositio is similar to the proof that H is a costat, ad is left as a exercise to the reader. Defiitio 5 A Hamiltoia system is said to be completely itegrable if it has first itegrals (icludig the Hamiltoia itself), where is the umber of degrees of freedom. I mechaical systems, the first itegrals are ofte familiar quatities. 3

4 Example 2.1 Cosider the two-body orbital problem with m 1 m 2. I this case, we ca treat the body of mass m 1 as beig essetially motioless. 1 Moreover, with just two bodies, the coordiate system ca always be chose i such a way that m 1 is at the origi, ad that z =, i.e. the motio is cofied to the (x,y) plae. The Hamiltoia is H = p2 x + p2 y 2m 2 Gm 1m 2 x 2 + y 2. Based o our kowledge of mechaics, we expect the agular mometum L = r p = xp y yp x to be a costat of the motio. If this is so, the Poisso bracket of H with L should be zero: {H,L} = H L p x x H L + H L x p x p y y H L y p y = p x p y Gm 1m 2 x m 2 (x 2 + y 2 ) 3/2 ( y) + p y ( p x ) Gm 1m 2 y m 2 (x 2 + y 2 ) 3/2 x =. The agular mometum is therefore, as expected, a costat of the motio. This meas that the system is cofied to the itersectio of the hyperplaes H = E ad L = l, where E ad l are costats determied by the iitial coditios. This itersectio is a ivariat maifold of the differetial equatios sice all possible trajectories for a give E ad l lie withi it. What does this maifold look like? We have two equatios coectig our four variables, so these equatios implicitly defie a two-dimesioal surface i the four-dimesioal phase space. I this case, this surface is easy to plot usig Maple. See Appedix B for sample code. We start by solvig for y (say) from the agular mometum: y = (xp y l)/p x. We the substitute this expressio ito the eergy equatio. Depedig o the total eergy, we get differet kids of results: 1. The surface is a torus if E < (Fig. 2). The trajectories o the torus ca either be periodic (which they are for this simple problem) or quasiperiodic, i.e. they wid aroud ad aroud the torus without repeatig themselves. I the latter case, a trajectory evetually covers the whole torus ad is said to be ergodic. I the former case, slightly differet iitial coditios give differet orbits. 2. The surface is essetially a cylider if E > (Fig. 3) or if E = (Fig. 4). This correspods to a case where the lighter particle has eough eergy to escape the gravitatioal attractio of its heavier eighbor. The trajectories wid aroud this cylider, either toward x = or toward x =. 1 A more careful treatmet would trasform to cetre-of-mass ad relative coordiates. We would obtai essetially the same Hamiltoia, with a small adjustmet i the costats appearig i the equatio. 4

5 4 2 x px.5.5 py Figure 2: Ivariat torus arisig from the sigle-plaet orbital problem. This case correspods to m 1 = 1kg, m 2 = 1kg, E = J, ad l = 1kgm 2 /s. The foregoig example brigs out importat features of itegrable Hamiltoia problems: Ivariat tori or cyliders are geerically see. The study of these ivariat maifolds ad how they chage whe we chage the Hamiltoia is a importat part of the theory of Hamiltoia dyamical systems. The study of trajectories o these maifolds is of course also iterestig ad importat. I the case of itegrable Hamiltoias, the behavior ca t be much more complicated tha a quasiperiodic trajectory. 3 Numerical itegratio We wo t go ito great detail, but there is a importat issue which we must discuss i relatio to the umerical itegratio of Hamiltoia systems: Most umerical methods do ot preserve the costacy of H. May umerical methods do a good job of holdig H reasoably costat for simple problems, but ot for more complex problems, especially if log trajectories are wated. Numerical itegratio methods which boud the variatio of H are said to be symplectic. We will discuss this problem ad the costructio of symplectic itegrators i this sectio. However, we will ot udertake a full implemetatio, the details beig rather messy. As a itroductio to this topic, let us discuss the simplest umerical itegratio method, amely the forward Euler method. The idea behid this algorithm is very simple: The differetial 5

6 1 5 x py px Figure 3: Ivariat cylider arisig from the sigle-plaet orbital problem. All parameters are as i Fig. 2, except E = J ad l = 12kgm 2 /s. equatio dz = f(z,t), (4) dt where z is the vector of phase-space coordiates, ca be iterpreted as follows, accordig to the basic defiitio of the derivative: z(t + ) z(t) lim = f(z(t),t). Euler s method simply cosists i takig small values of ad doig away with the limit. Sice is fixed, we will be calculatig z at followig sequece of times:,2,3,..., j,... We therefore label the values of z by j, the umber of multiples of the step size which have passed. Euler s method ca thus be summarized by the formula z j+1 = z j + f(z j, j). (5) Equatio 5 defies a dyamical system with discrete time (idexed by j) i which the state at the ext time icremet is calculated from a simple formula. This is called a map. I the limit, the Euler map coverges o the solutios of the differetial equatio 4. Hopefully, for small values of, Euler s method will give results which are at least reasoably represetative of those of the origial differetial equatio. 6

7 1 5 x px py Figure 4: Ivariat surface arisig from the sigle-plaet orbital problem with E = ad l = 115kgm 2 /s, all other parameters beig set as i Fig. 2. As you ca imagie, Euler s method is a very aïve techique, ad teds ot to be very accurate. Nevertheless, it has its uses, amog them pedagogical applicatios. We will use it here to illustrate the problem you ru ito with ordiary umerical methods ad Hamiltoia systems. Example 3.1 Suppose that we wat to solve the differetial equatios 3 which arise from the Hamiltoia 2 by Euler s method. The we have x j+1 = x j + p j /m, p j+1 = p j kx j. The value of the Hamiltoia at time idex j + 1 is H(x j+1, p j+1 ) = 1 2 kx2 j+1 + p2 j+1 2m = k ( x j + p j /m ) 2 1 ( ) 2 + p j kx j 2 2m ( ) = 1 2 kx2 j + p2 j 2m + k 1 m ()2 2 kx2 j + p2 j 2m = H(x j, p j ) (1 + km ) ()2. 7

8 The value of the Hamiltoia therefore grows by a factor of 1 + k m ()2 at every step. The problem is less serious the smaller we make, but it remais that it is ofte just ot acceptable to have H grow (or decrease) with time. Most of the commo umerical methods you may have heard of (Ruge-Kutta, Gear, etc.) are ot symplectic. For some problems, some of these methods give reasoable results. However, if you rely o these methods for Hamiltoia systems, you will sooer or later get bured. Euler s method is a explicit method: Give the curret poit i phase space, we ca calculate the ext poit just by pluggig umbers ito a formula. Ufortuately, all geeral-purpose symplectic itegratio methods are implicit: We ca t i geeral write dow a formula for the ext poit. Rather, we have to solve a equatio (usually umerically) to calculate (x j+1,p j+1 ). There are however some explicit methods for special forms of the Hamiltoia which we will ot discuss here. The key to maitaiig the costacy of the Hamiltoia is to derive a umerical method directly from Hamilto s equatios 1. Let s start with equatio 1a. We ca estimate the time derivative o the left-had side as we did i Euler s method: dx i dt x i, j+1 x i, j. I this equatio, x i, j+1 is the value of x i at time step j. The right-had side of equatio 1a is also a derivative. I each time step, we will icremet the variables, for istace icremetig p i from p i, j to p i, j+1. Sice we oly take small time steps, p i should oly chage by a small amout. This gives us a opportuity to evaluate H/ p i by usig aother fiite-differece approximatio. You ca probably imagie what we eed to do, but it s a bit of a mess to write dow without some special otatio. Let p j p i, j+1 be the vector obtaied by replacig p i, j by p i, j+1 i the vector of mometa at time step k, p j. The, Puttig the two sides together, we have H H(x j,p j p i, j+1 ) H(x j,p j ). p i p i, j+1 p i, j x i, j+1 x i, j = H(x j,p j p i, j+1 ) H(x j,p j ) p i, j+1 p i, j. (6a) We ca repeat this procedure for equatio 1b, with oe mior variatio: p i, j+1 p i, j = H(x j x i, j+1,p j p i, j+1 ) H(x j,p j p i, j+1 ) x i, j+1 x i, j. (6b) The reaso that we used the updated mometum i equatio 6b is that this decisio makes the scheme symplectic. To see this, rearrage equatios 6a ad 6b to ad 1 ( )( ) xi, j+1 x i, j pi, j+1 p i, j = H(x j,p j p i, j+1 ) H(x j,p j ), 1 ( )( ) [ xi, j+1 x i, j pi, j+1 p i, j = H(x j x i, j+1,p j p i, j+1 ) H(x j,p j p i, j+1 ) ]. 8

9 Now subtract these two equatios: or = H(x j x i, j+1,p j p i, j+1 ) H(x j,p j ), H(x j x i, j+1,p j p i, j+1 ) = H(x j,p j ). I other words, the update rule cosistig of equatios 6 keeps the value of H costat for each pair of cojugate variables (x i, p i ) to which it is applied. Thus, the method is symplectic. The tradeoff is that we have a implicit method: We eed values of the Hamiltoia with some of the coordiates evaluated at the ukow ext time poit. We therefore have 2 equatios i 2 ukows. Uless the Hamiltoia is of a particularly simple form, we wo t be able to reduce this problem to a simple mappig. Not all symplectic itegratio methods hold the value of the Hamiltoia exactly costat. Rather, their effect is to limit the variatio i the value of the Hamiltoia. The drift i the Hamiltoia ecoutered with ordiary umerical methods ca be disastrous. Holdig the Hamiltoia withi tight bouds is ofte eough to give good results, ad oe is ofte cotet to trade off exact costacy of the Hamiltoia agaist step size i order to accelerate log itegratios. Appedices A Maple code for drawig the orbits of the harmoic oscillator k := 2; m :=.1; H := (x,p) -> pˆ2/(2*m) + k*xˆ2/2; with(plots): cotourplot(h(x,p),x= ,p= ,cotours=[$1..6]); B Maple code for drawig the ivariat tori or cyliders of Hamiltoias with two degrees of freedom y := (x*py - ell)/px; # Calculate y from agular mometum H := (x,y,px,py) -> (pxˆ2+pyˆ2)/(2*m2) - G*m1*m2/sqrt(xˆ2+yˆ2); m1 := 1e11; # A slightly differet set of masses tha i the otes m2 := 1/1e6; G := 6.67e-11; x := 1; # Iitial poit (x,y,px,py) = (x,,,p) p :=.1/sqrt(1e6); E := H(x,,,p); # Set the eergy ell := x*p; # Set the agular mometum with(plots): implicitplot3d(h(x,y,px,py)=e,px=-p..p,py=-p..p,x=-5*x..5*x, axes=boxed,grid=[1,4,4],orietatio=[-35,5],shadig=none); 9