Examensarbete Separation of variables for ordinary differential equations Anna M˚ahl LiTH - MAT - EX - - 06 / 01 - - SE Separation of variables for ordinary differential equations Department of Applied Mathematics, Link¨opings Universitet Anna M˚ahl LiTH - MAT - EX - - 06 / 01 - - SE Examensarbete: 20 p Level: D Supervisor: Stefan Rauch, Department of Applied Mathematics, Link¨opings Universitet Examiner: Stefan Rauch, Department of Applied Mathematics, Link¨opings Universitet Link¨oping: February 2006 Datum Avdelning, Institution Date Division, Department Matematiska Institutionen February 2006 581 83 LINKOPING¨ SWEDEN Spr˚ak Rapporttyp ISBN Language Report category ISRN Svenska/Swedish Licentiatavhandling LiTH - MAT - EX - - 06 / 01 - - SE x Engelska/English x Examensarbete C-uppsats Serietitel och serienummer ISSN D-uppsats Title of series, numbering 0348-2960 Ovrig¨ rapport URL f¨or elektronisk version http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva- 5620 Titel Separation of variables for ordinary differential equations Title F¨orfattare Anna M˚ahl Author Sammanfattning Abstract In case of the PDE’s the concept of solving by separation of variables has a well defined meaning. One seeks a solution in a form of a product or sum and tries to build the general solution out of these particular solutions. There are also known systems of second order ODE’s describing potential motions and certain rigid bodies that are considered to be separable. However, in those cases, the concept of separation of variables is more elusive; no general definition is given. In this thesis we study how these systems of equations separate and find that their separation usually can be reduced to sequential separation of single first order ODE´s. However, it appears that other mechanisms of separability are possible. Nyckelord Keyword ODE, Separation of variables, Potential motion, Heavy symmetric top, Cofactor sys- tems, Direct separability. vi Abstract In case of the PDE’s the concept of solving by separation of variables has a well defined meaning. One seeks a solution in a form of a product or sum and tries to build the general solution out of these particular solutions. There are also known systems of second order ODE’s describing potential motions and certain rigid bodies that are considered to be separable. However, in those cases, the concept of separation of variables is more elusive; no general definition is given. In this thesis we study how these systems of equations separate and find that their separation usually can be reduced to sequential separation of single first order ODE´s. However, it appears that other mechanisms of separability are possible. Keywords: ODE, Separation of variables, Potential motion, Heavy symmetric top, Cofactor systems, Direct separability. M˚ahl, 2006. vii viii Acknowledgements I would like to thank my supervisor and examiner Stefan Rauch. You have been a dedicated, and skillful, educationalist. By dividing problems into pieces big enough for me to calculate and comprehend, you have taught me not to be afraid of approaching non familiar problems and you have encouraged me to try new ways. These things I will treasure and bring with me in the future. I would like to thank my dad Per M˚ahl. All my life you have read and given critique on my writing, but this must be the first time that you have helped me by knowing less! It is because you have not understood majority of the mathematical preliminaries that you have been an invaluable partner for discussion. I would like to thank my opponent Peter Brommesson, and also Mattias Hansson, for reading this report and giving critique. I would also like to thank Zebastian Zaar for creating my pictures of HST and rolling disk. Moreover, I would like to thank Daniel Petersson, in the office next to mine, for helping me to understand some mechanics. Finally, I would like to thank the rest of my family and my friends. You all mean a lot to me. M˚ahl, 2006. ix x Contents 1 Introduction 1 2 Preliminaries 3 2.1 Separation of variables for one dimensional potential motion . 4 3 Potential Newton Equations 7 3.1 Two dimensional potential Newton equations . 7 3.1.1 Separation of variables for two dimensional potential motion 9 3.2 Three dimensional potential motion . 10 3.2.1 Separation of variables for three dimensional potential motion............................. 12 4 Vector equations of a rigid body 15 4.1 Motion of a heavy symmetric top . 15 4.1.1 Separation of variables for the HST equations . 19 4.2 The equations of motion of the rolling disk . 20 4.2.1 Separation of variables for the RD equations . 22 5 Triangular Newton equations 25 5.1 Two dimensional triangular cofactor system . 25 5.1.1 Separation of variables for two dimensional triangular New- tonequations......................... 28 6 The H´enon-Heiles Hamiltonian with cubic potential 31 6.1 The Kaup-Kuperschmidt case of the H´enon-Heiles Hamiltonian withcubicpotential ......................... 32 6.1.1 Separation of variables for the Kaup-Kuperschmidt case of the H´enon-Heiles Hamiltonian . 33 7 Direct separability 35 7.1 An example of second order . 35 7.1.1 Separation of variables for this example . 36 8 Conclusion and discussion 39 M˚ahl, 2006. xi xii Contents Chapter 1 Introduction dy An ordinary differential equation (ODE) is an algebraic equation f(x,y, dx ) = 0 involving derivatives of some unknown function with respect to one independent variable. A separable ordinary differential equation is an ODE which can be written in such a way that the dependent variable and its differential appear on one side of the equals sign and the independent variable and its differential appear on the other side. The method of separation of variables is best known for partial differential equations of mathematical physics, for which it is the main method of solv- ing these equations. The basic idea behind the method of separating variables in the theory of partial differential equations (PDE) is to consider an ansatz for a solution (additive or multiplicative) that allows reducing the problem to a system of uncoupled ordinary differential equations for functions of one variable. Example 1. The heat equation. Consider ∂u = u (1.1) ∂t xx for an initial boundary value problem on the interval 0 <x<L and with the boundary conditions ∂u ∂t = uxx, 0 < t, 0 <x<L u(0,t) = u(L, t) = 0, 0 <t u(x, 0) = f(x), 0 <x<L One usually makes an ansatz assuming the solution to be a product of two functions. Substitution of u(x,t) = X(x)T (t) into (1.1) reduces this partial differential equation to two uncoupled ordinary differential equations 2 ∂tT = k T − 2 ∂xxX = k X ( − These two ordinary differential equations have solutions −k2t Tk(t) = Ae (Xk(x) = B cos kx + C sin kx M˚ahl, 2006. 1 2 Chapter 1. Introduction and due to linearity of (1.1) any formal linear combination u(x,t) = k ckXk(x)Tt(t) is also a solution. This solution can be further specified according to the bound- nπ P ary conditions u(0,t) = u(L, t) = 0. If we put kn = L , where n = 1, 2,..., the complete solution reads ∞ 2 −knt u(x,t) = cne sin knx (1.2) n=1 X 2 L where the coefficients cn are given by the integrals cn = L 0 f(x) sin knxdx ([3],[6]). R Example 2. The Hamilton-Jacobi equation of natural Hamiltonian Another example of an equation known to be solvable by separating variables is the Hamilton-Jacobi equation for the natural Hamiltonian ([4],[6]). One method of solving the canonical Hamilton equations ∂H(x,y) ∂H(x,y) x˙ = , y˙ = , x = (x1,...,xn), y = (y1,...,yn) (1.3) ∂y − ∂x where H(x,y) is a Hamiltonian, is to find a function W (x, α), given by the ∂W (x,α) ∂W (x,α) canonical transformation y = ∂x , β = ∂α , for which (1.3) is trans- formed into a set of simple linear equations for new variables β, α. This function W (x, α) satisfies the first order nonlinear PDE ∂W (x, α) H x, = α (1.4) ∂x 1 which is known as the Hamilton-Jacobi equation. In order to find it, one makes n k k an additive ansatz W (x, α) = k=1 Wk(x , α), where the n functions Wk(x , α) each depend on a single variable xk. This ansatz works well for the Hamiltoni- P 1 ii 2 ans of St¨ackel class with H = G + V = 2 i g (x)yi . It simplifies the problem; instead of integrating the nontrivial Hamilton-Jacobi equation the problem re- P k duces to integrating n uncoupled first order ODE’s for functions Wk(x , α). The purpose of this thesis is to investigate how variables can be separated in case of ODE’s. In order to do so we put together known examples of Newton equations solved in mechanics and describe the features of their separability. In chapter 2 we present basic concepts needed for this work; we give a definition of separability in general and we explain why we consider the Newton equation of one dimensional potential motion to be separable. In the following chapters we study a variety of mechanical problems, mainly of Newtonian form, to see how these equations are solved by separating variables. In chapter 3 we describe the problem of potential motion. In chapter 4 we look at the equations of motion of rigid bodies and we show the problems of the heavy symmetric top and of the rolling disk. In chapter 5 we describe how separation of variables takes place when solving triangular systems of Newton equations and in chapter 6 we consider the Kaup-Kuperschmidt case of the H´enon-Heiles Hamiltonian with cubic potential.