Lecture 1: Hamiltonian systems - UNIGE

1 Geometric Numerical IntegrationTU M unchenErnst HairerJanuary February 2010 Lecture 1: Hamiltonian systemsTable of contents1 Derivation from Lagrange s equation12 Energy conservation and first integrals, examples33 Symplectic transformations54 Theorem of Poincar e75 Generating functions96 Hamilton Jacobi partial differential equation117 Exercises13 The main topic of this lecture1is a deeper understanding of Hamiltonian systems p= qH(p,q), q= pH(p,q).(1)Here,pandqare vectors inRd, andH(p,q)is a scalar sufficiently differentiablefunction.

2 It is called the Hamiltonian or the total energy .1 Derivation from Lagrange s equationSuppose that the position of a mechanical system withddegrees of freedom isdescribed byq= (q1,..,qd)Tasgeneralized coordinates(this can be for exam-ple Cartesian coordinates, angles, arc lengths along a curve, etc.). Consider theLagrangianL=T U,(2)whereT=T(q, q)denotes the kinetic energy andU=U(q)the potential motion of the system is described by Lagrange s equation2ddt( L q)= L q,(3)1 Most parts of this manuscript are taken from the monographGeometric Numerical Integrationby Hairer, Lubich & Wanner (2nd edition, Springer Verlag 2006).

3 2 Lagrange,Applications de la m ethode expos ee dans le m emoire pr ec edent a la solution dediff erents probl`emes de dynamique, 1760, Oeuvres Vol. 1, 365 are just the Euler Lagrange equations of the variational problemS(q) = baL(q(t), q(t))dt the structure of Lagrange s equations and turnedtheminto a form that has remarkable symmetry, by introducing Poisson s variables, the conjugatemomentapk= L qk(q, q)fork= 1,..,d,(4) considering theHamiltonianH:=pT q L(q, q)(5)as a function ofpandq, , takingH=H(p,q)obtained by expressing qas a function ofpandqvia (4).

4 Here it is, of course, required that (4) defines, for everyq, a continuously differ-entiable bijection q p. This map is called theLegendre 1 Lagrange s equations (3) are equivalent to Hamilton s equations pk= H qk(p,q), qk= H pk(p,q), k= 1,..,d.(6) definitions (4) and (5) for the momentapand for the HamiltonianHimply that H p= qT+pT q p L q q p= qT, H q=pT q q L q L q q q= L Lagrange equations (3) are therefore equivalent to (6).3 Sir Hamilton,On a general method in dynamics; by which the study of the motions ofall free systems of attracting or repelling points is reduced to the search and differentiation ofone central relation, or characteristic function, Phil.

5 Trans. Roy. Soc. Part II for 1834, 247 308;Math. Papers, Vol. II, 103 of quadratic kinetic (q, q) =12 qTM(q) q, whereM(q)isa symmetric and positive definite matrix, we havep=M(q) q. Replacing thevariable qbyM(q) 1pin the definition (5) ofH(p,q), we obtainH(p,q) =pTM(q) 1p L(q,M(q) 1p)=pTM(q) 1p 12pTM(q) 1p+U(q) =12pTM(q) 1p+U(q)and the Hamiltonian isH=T+U, which is thetotal Energy conservation and first integrals, examplesDefinition 1A non-constant functionI(y)is afirst integralof y=f(y)ifI (y)f(y) = 0for ally.

6 (7)This is equivalent to the property thateverysolutiony(t)of y=f(y)satisfiesI(y(t))= 1 (Conservation of the total energy)For Hamiltonian systems (1) theHamiltonian functionH(p,q)is a first 2 (Conservation of the total linear and angular momentum)We con-sider a system ofNparticles interacting pairwise with potential forces dependingon the distances of the particles. This is a Hamiltonian system with total energyH(p,q) =12N i=11mipTipi+N i=2i 1 j=1 Vij(kqi qjk).Hereqi,pi R3represent the position and momentum of theith particle of massmi, andVij(r)(i > j) is the interaction potential between theith andjth equations of motion read qi=1mipi, pi=N j=1 ij(qi qj)where, fori > j, we have ij= ji= V ij(rij)/rijwithrij=kqi qjk.

7 Theconservation of the total linear and angular momentumP=N i=1pi, L=N i=1qi piis a consequence of the symmetry relation ij= ji:3mcosq1qExample 3 (Mathematical pendulum)The mathemati-cal pendulum (massm= 1, massless rod of length = 1,gravitational accelerationg= 1) is a system with one de-gree of freedom having the HamiltonianH(p,q) =12p2 cosq,so that the equations of motion (1) become p= sinq, q=p.(8)Figure 3 below shows some level curves ofH(p,q). By Example 1, the solutioncurves of the problem (8) lie on such level EExample 4 (Two-body problem or Kepler problem)For computing the motion of two bodies (planet andsun) which attract each other, we choose one of thebodies (sun) as the centre of our coordinate system;the motion will then stay in a plane and we can usetwo-dimensional coordinatesq= (q1,q2)for the posi-tion of the second body.

8 Newton s laws, with a suitablenormalization, then yield the following differential equations q1= q1(q21+q22)3/2, q2= q2(q21+q22)3/2.(9)This is equivalent to a Hamiltonian system with the HamiltonianH(p1,p2,q1,q2) =12(p21+p22) 1 q21+q22, pi= qi.(10)The planet moves inelliptic orbitswith the sun at one of the foci (Kepler s4firstlaw). With initial valuesq1(0) = 1 e, q2(0) = 0, q1(0) = 0, q2(0) = 1 +e1 e(11)the solution is an ellipse with eccentricitye(0 e <1),a= 1,b= 1 e2,d= 1 e2, and period2 . The total energy isH0= 1/2, and the angularmomentum isL0= 1 Kepler,Astronomia nova o o o seu Physica celestis, traditia commentariis demotibus stellae Martis, ex observationibus G.

9 V. Tychonis Brahe, Prague 5 (H enon Heiles problem)The polynomial Hamiltonian in two de-grees of freedom5H(p,q) =12(p21+p22) +12(q21+q22) +q21q2 13q32(12)is a Hamiltonian differential equation that can have chaotic solutions. Figure 1shows a regular behaviour of solutions when the value of the Hamiltonian is small,and a chaotic behaviour for large Hamiltonian ..3P0P1P2H=112q2p2 ..4P0P1P2H=18q2p2 Figure 1: Poincar e cuts forq1= 0,p1>0of the H enon Heiles model forH=112(6 orbits, left) andH=18(1 orbit, right).

10 3 Symplectic transformationsThe basic objects to be studied are two-dimensional parallelograms lying suppose the parallelogram to be spanned by two vectors =( p q), =( p q)in the(p,q)space ( p, q, p, q Rd) asP={t +s |0 t 1,0 s 1}.In the cased= 1we consider theoriented (P) = det( p p q q)= p q q p(13)5M. H enon & C. Heiles,The applicability of the third integral of motion : some numericalexperiments, Astron. J. 69 (1964) 73 (left picture of Fig. 2). In higher dimensions, we replace this by thesum of theoriented areas of the projections ofPonto the coordinate planes(pi,qi), , by ( , ) :=d i=1det( pi pi qi qi)=d i=1( pi qi qi pi).