Solving Differential Equations on Manifolds Ernst Hairer Universit´ede Gen`eve June 2011 Section de math´ematiques 2-4 rue du Li`evre,CP 64 CH-1211 Gen`eve 4 Acknowledgement. These notes have been distributed during the lecture “Equations´ diff´erentielles sur des sous-vari´et´es”(2 hours per week) given in the spring term 2011. At several places, the text and figures are taken from one of the monographs by the author. Contents I Introduction by Examples . 1 I.1 Differential equation on a sphere – the rigid body . 1 I.2 Problems in control theory . 3 I.3 Constrained mechanical systems . 4 I.4 Exercises . 6 II Submanifolds of Rn ................................ 7 II.1 Definition and characterization of submanifolds . 7 II.2 Tangent space . 9 II.3 Differential equations on submanifolds . 11 II.4 Differential equations on Lie groups . 14 II.5 Exercises . 16 III Integrators on Manifolds . 19 III.1 Projection methods . 19 III.2 Numerical methods based on local coordinates . 21 III.3 Derivative of the exponential and its inverse . 23 III.4 Methods based on the Magnus series expansion . 24 III.5 Convergence of methods on submanifolds . 26 III.6 Exercises . 27 IV Differential-Algebraic Equations . 29 IV.1 Linear equations with constant coefficients . 29 IV.2 Differentiation index . 30 IV.3 Control problems . 32 IV.4 Mechanical systems . 34 IV.5 Exercises . 36 V Numerical Methods for DAEs . 39 V.1 Runge–Kutta and multistep methods . 39 V.2 Index 1 problems . 42 V.3 Index 2 problems . 45 V.4 Constrained mechanical systems . 47 V.5 Shake and Rattle . 50 V.6 Exercises . 51 3 Recommended Literature There are many monographs treating manifolds and submanifolds. Many of them can be found under the numbers 53 and 57 in the mathematics library. Books specially devoted to the numerical treatment of differential equations on manifolds (differential-algebraic equa- tions) are listed under number 65 in the library. We give here an incomplete list for further reading: the numbers in brackets (e.g. [MA 65/403]) allow one to find the book without computer search. R. Abraham, J.E. Marsden and T. Ratiu, Manifolds, Tensor Analysis, and Applications, 2nd edition, Applied Mathematical Sciences 75, Springer-Verlag, 1988. [MA 57/266] V. Arnold, Equations Diff´erentielles Ordinaires, Editions Mir (traduction fran¸caise),Moscou, 1974. [MA 34/102] U.M. Ascher and L.R. Petzold, Computer Methods for Ordinary Differential Equations and Differential- Algebraic Equations, Society for Industrial and Applied Mathematics (SIAM), Philadel- phia, PA, 1998. K.E. Brenan, S.L. Campbell and L.R. Petzold, Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations. Revised and corrected reprint of the 1989 original, Classics in Applied Mathematics 14, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. [MA 65/294] E. Eich-Soellner and C. F¨uhrer, Numerical methods in multibody dynamics, European Consortium for Mathematics in Industry. B.G. Teubner, Stuttgart, 1998. E. Griepentrog and R. M¨arz, Differential-Algebraic Equations and Their Numerical Treatment, Teubner-Texte zur Mathematik 88, Teubner Verlagsgesellschaft, Leipzig, 1986. [MA 65/256] E. Hairer, C. Lubich and M. Roche, The Numerical Solution of Differential-Algebraic Systems by Runge-Kutta Methods, Lecture Notes in Mathematics 1409, Springer Berlin, 1989. [MA 00.04/3 1409] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Al- gorithms for Ordinary Differential Equations, 2nd edition, Springer Series in Compu- tational Mathematics 31, Springer Berlin, 2006. [MA 65/448] E. Hairer, S.P. Nørsett and G. Wanner, Solving Ordinary Differential Equations I. Nonstiff Prob- lems, 2nd edition, Springer Series in Computational Mathematics 8, Springer Berlin, 1993. [MA 65/245] E. Hairer and G. Wanner, Solving Ordinary Differential Equations II. Stiff and Differential-Agebraic Problems, 2nd edition, Springer Series in Computational Mathematics 14, Springer Berlin, 1996. [MA 65/245] E. Hairer and G. Wanner, Analysis by Its History, Undergraduate Texts in Mathematics, Springer New York, 1995. [MA 27/256] P. Kunkel and V. Mehrmann, Differential-Algebraic Equations. Analysis and Numerical Solution, EMS Textbooks in Mathematics. European Mathematical Society (EMS), Z¨urich, 2006. [MA 34/325] S. Lang, Introduction to Differentiable Manifolds, 2nd edition, Universitext, Springer New York, 2002. [MA 57/15] J.M. Lee, Introduction to Smooth Manifolds, Graduate Texts in Mathematics, Springer New York, 2003. [MA 53/302] Chapter I Introduction by Examples Systems of ordinary differential equations in the Euclidean space Rn are given by y˙ = f(y), (0.1) where f : U → Rn with an open set U ⊂ Rn. If f is sufficiently smooth and an initial value n y(0) = y0 is prescribed, it is known that the problem has a unique solution y :(−α, α) → R for some α > 0. This solution can be extended until it approaches the border of U. In the present lecture we are interested in differential equations, where the solution is known to evolve on a submanifold of Rn, and the vector field f(y) is often only defined on this submanifold. We start with presenting a few typical examples, which serve as motivation for the topic. Later, we shall give a precise definition of differential equations on submanifolds, we shall discuss their numerical treatment, and we shall analyze them rigorously. I.1 Differential equation on a sphere – the rigid body Let I1,I2,I3 be the principal moments of inertia of a rigid body. The angular momentum T vector y = (y1, y2, y3) then satisfies Euler’s equations of motion −1 −1 y˙1 = (I3 − I2 ) y3 y2 y˙1 0 −y3 y2 y1/I1 −1 −1 y˙2 = (I − I ) y1 y3 or y˙2 = y3 0 −y1 y2/I2 . (1.1) 1 3 −1 −1 y˙3 = (I2 − I1 ) y2 y1 y˙3 −y2 y1 0 y3/I3 Fig. I.1: Euler’s equation of motion for I1 = 1.6, I2 = 1, I3 = 2/3; left picture: vector field on the sphere; right picture: some solution curves. 2 Introduction by Examples This differential equation has the property that the function 1 C(y) = y2 + y2 + y2 (1.2) 2 1 2 3 is exactly preserved along solutions, a property that can be checked by differentiation: d dt C y(t) = ... = 0. As a consequence, the solution remains forever on the sphere with radius that is determined by the initial values. The left picture of Figure I.1 shows the vector f(y) attached to selected points y of the unit sphere. To study further properties of the solution we write the differential equation as 0 −y y 3 2 2 2 2 1 y1 y2 y3 y˙ = B(y)∇H(y) with B(y) = y3 0 −y1 ,H(y) = + + . 2 I1 I2 I3 −y2 y1 0 The function H(y) is called Hamiltonian of the system, whereas C(y) of (1.2) is called d Casimir function. Exploiting the skew-symmetry of the matrix B(y), we obtain dt H(y(t)) = ∇H(y(t))TB(y)∇H(y(t)) = 0, which implies the preservation of the Hamiltonian H(y) along solutions of (1.1). Consequently, solutions lie on the intersection of a sphere C(y) = Const with an ellipsoid H(y) = Const, and give rise to the closed curves of the right picture in Figure I.1. Solutions are therefore typically periodic. Numerical solutions are displayed in Figure I.2. The top picture shows the numerical result, when the explicit Euler method yn+1 = yn + hf(yn) is applied with step size h = 0.025 and with the initial value y0 = (cos(0.9), 0, sin(0.9)). The numerical solution drifts away from the manifold. The bottom left picture shows the result of the trapezoidal rule h yn+1 = yn + 2 (f(yn+1) + f(yn)) with h = 1, where the numerical solution is orthogonally Fig. I.2: Top picture: integration with explicit Euler; bottom left picture: trapezoidal rule with projection onto the sphere; bottom right picture: implicit midpoint rule. Introduction by Examples 3 projected onto the sphere after every step. The bottom right picture considers the implicit 1 mid-point rule yn+1 = yn + hf( 2 (yn+1 + yn)) with h = 1. Even without any projection, the solution agrees extremely well with the exact solution. All these behaviours will be explained in later chapters. I.2 Problems in control theory In control theory one often encounters problems of the form y˙ = f(y, u) (2.1) 0 = g(y), where u(t) is a control function that permits to steer the motion y(t) of a mechanical system. Differentiating the algebraic equation g(y(t)) = 0 with respect to time yields g0(y)f(y, u) = 0. Under suitable regularity assumptions this relation permits us to express u as a function of y (using the implicit function theorem). Inserting u = G(y) into (2.1) gives a differential equation for y on the manifold M = {y ; g(y) = 0}. θ2 θ1 θ4 θ3 Fig. I.3: Sketch of an articulated robot arm. Example 2.1 (articulated robot arm). Consider n > 2 segments of fixed length 1 that are connected with joints as illustrated in Figure I.3. We assume that the starting point of the first segment is fixed at the origin. Denoting by θj the angle of the jth segment with respect to the horizontal axis, the endpoint of the last segment is given by g(θ), where for θ = (θ1, . , θn) we have cos θ + cos θ + ... + cos θ ! g(θ) = 1 2 n . (2.2) sin θ1 + sin θ2 + ... + sin θn The problem consists in finding the motion θ(t) of the articulated robot arm such that the endpoint of the last segment follows a given parametrized curve γ(t) in the plane and kθ˙(t)k → min subject to g(θ(t)) = γ(t).