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On Yoshida's Method for Constructing Explicit Symplectic Integrators For faculty of science mathematics and applied and engineering mathematics On Yoshida’s Method For Constructing Explicit Symplectic Integrators For Separable Hamiltonian Systems Bachelor’s Project Mathematics July 2019 Student: J.C. Pim First supervisor: Dr. M. Seri Second assessor: Dr. A.E. Sterk Abstract We describe Yoshida's method for separable Hamiltonians H = T (p)+ V (q). Hamiltonians of this form occur frequently in classical mechanics, for example the n-body problem, as well as in other fields. The class of symplectic integrators constructed are explicit, reversible, of arbitrary even order, and have bounded energy growth. We give an introduction to Hamiltonian mechanics, symplectic geometry, and Lie theory. We com- pare the performance of these integrators to more commonly used methods such as Runge-Kutta, using the ideal pendulum and Kepler problem as examples. 2 Contents 1 Introduction 4 2 Preliminaries And Prerequisites 5 2.1 Manifolds, Vector Fields, And Differential Forms . .5 2.2 The Matrix Exponential . .6 2.3 The Vector Field Exponential . .7 3 Hamiltonian Mechanics 9 3.1 Motivating Examples . 10 4 Symplectic Geometry 15 4.1 Symplectic Vector Spaces . 15 4.2 Symplectic Manifolds . 17 5 Lie Theory And BCH Formula 21 6 Symplectic Integrators 28 6.1 Separable Hamiltonians . 28 6.2 Reversible Hamiltonians . 29 6.3 Yoshida's Method For Separable Hamiltonians . 30 6.4 Properties Of The Yoshida Symplectic Integrators . 38 6.5 Backward Error Analysis . 38 7 Numerical Simulations 43 7.1 The Ideal Pendulum . 43 7.2 The Kepler Problem . 46 8 Conclusion 50 References 51 3 1 Introduction Many interesting phenomena in science can be modelled by Hamiltonian sys- tems, for example the n-body problem, oscillators, problems in molecular dy- namics, and models of electric circuits [11]. In general these are defined by systems of differential equations that cannot be solved analytically and very of- ten numerical methods are needed in order to study their behaviour. This gives rise to some issues: many of these systems have invariants such as conservation of energy or angular momentum which numerical methods do not necessarily preserve, sometimes with drastic consequences on the evolution of the computed solution. In the case of chaotic systems, like the n-body problem, this is wors- ened by the fact that the perturbations introduced by numerical methods may lead to wildly different solutions over long time scales. This is also undesirable in many applications such as modelling the trajectory of spacecraft or other small bodies, Hamiltonian Monte-Carlo methods, and physics-based animation used in the production of video games, and movies [2]. A way to mitigate these problems is to use high order numerical methods with a sufficiently small step size, but this can be extremely computationally expensive. Also, techniques such as adaptive time-stepping may not preserve important features of the system's dynamics. For example, in neuronal dy- namics [3] the model's limit cycles have an important role in neuronal spiking. However, these are not well preserved by commonly used \Euler-like" methods. Hence, we would like to have numerical methods which preserve at least some of the system's invariants. We can do this by exploiting the underling geometry of Hamiltonian systems, called symplectic geometry. The natural space for the Hamiltonian dynamics, the so-called phase space, is endowed with a canonical differential 2-form, called the symplectic form, which the flow of the Hamilto- nian system preserves. This can be used to derive numerical methods, called symplectic integrators, which themselves preserve the symplectic form. In this bachelor project, we aim to describe Yoshida's method for construct- ing explicit symplectic integrators for separable Hamiltonian systems. The class of integrators we construct are reversible, of arbitrary even order and have bounded energy growth. The construction makes use of Lie algebras and the Baker-Campbell-Hausdorff (BCH) formula. However, we will first describe the basic theory of Hamiltonian systems and their geometric properties using sym- plectic geometry. We will study some examples of Hamiltonian systems includ- ing the ideal pendulum and the Kepler problem, which will act as our main motivation for developing symplectic integrators. Then, we will give an intro- duction to Lie theory, before explaining Yoshida's construction, and showing some of its important properties. This will include a backward error analysis to show that this class of integrators has bounded energy growth. Finally, we will implement and compare some integrators from this class to more commonly used methods, such as Runge-Kutta, by applying them to our motivating examples. 4 2 Preliminaries And Prerequisites We will briefly describe and recall some of the theory used in later sections. We discuss some concepts from differential geometry such as manifolds, vector fields and differential forms, which will be useful later. We then review some facts about the matrix exponential, and give some intuition for the vector field exponential. 2.1 Manifolds, Vector Fields, And Differential Forms In later sections of this thesis we will make use of many concepts from differential geometry, especially in Section 4 and Section 5. The details about any of what follows, can be found in any book on differential geometry such as [14]. We will mainly use, though not discuss in detail: manifolds, smooth maps between manifolds, vector fields and differential forms on manifolds. In general, it will be sufficient to consider manifolds as open subsets of, or as surfaces in Rn. When we say smooth it is safe to read this as: of class C1, though many results also hold for Ck with k ≥ 1. We will write x1; : : : ; xm for the local coordinates on a smooth manifold M @ @ of dimension m. We use @x1 ;:::; @xm as the basis of the tangent space TpM 1 m ∗ for p 2 M, and dx ;:::; dx as the basis of the cotangent space Tp M. We denote the differential of a smooth map F : M ! N between M and another manifold N, as F∗ : TpM ! TF (p)N at p 2 M. For a tangent vector Xp 2 TpM we define it pointwise as F∗(Xp)f = Xp(f ◦ F ) where f is a smooth function on M. It can also be computed using a smooth curve c(t) on M with c(0) = p 0 d and c (0) = Xp as F∗(Xp) = dt t=0(F ◦ c)(t). Pm @ A vector field X = i=1 ai @xi is called smooth if its coefficients ai are smooth functions on M. Smooth vector fields have a flow Φ: I × M ! M, where I is an interval. Definition 1 A smooth vector field on a manifold is called complete if its flow is defined for all time. That is, if the vector field’s flow Φt has I = R. We will often make use of complete vector fields as they simplify the theory in places. We interpret vector fields as differential operators, and thus they can be applied to smooth functions. Pm i Similarly a differential 1-form or covector field on M is given by η = i=1 aidx and is again called smooth if the ai are smooth. Differential k-forms for k > 1 can be written as a sum of wedge products of the basis 1-forms: dx1;:::; dxm, 1 2 2 3 for example dx ^dx +5dx ^dx . The 0-forms are functions on M. We use ηp ∗ for p 2 M to denote it as a covector in Tp M. We can compute the application 1 k of a k-form dx ^ · · · ^ dx to k tangent vectors v1; : : : ; vk as 2 1 1 3 dx (v1) ··· dx (vk) 2 2 6dx (v1) ··· dx (vk)7 dx1 ^ · · · ^ dxk(v ; : : : ; v ) = det 6 7 : 1 k 6 . 7 4 . 5 k k dx (v1) ··· dx (vk) 5 The exterior derivative d transforms k-forms into (k+1)-forms. The exterior @f 1 @f m derivative of a smooth function f is given by df = @x1 dx + ··· + @xm dx . Since covectors \consume" tangent vectors, we can see that the differential f∗ and exterior derivative df of a function f are in fact the same. A k-form η is called closed if dη = 0, while it is called exact if there exists a (k − 1)-form µ s.t. dµ = η. Finally, the pullback of a k-form η at p 2 M by a smooth function ∗ F is given by F ηp(v1; : : : ; vk) = ηF (p)(F (v1);:::;F (vn)) for v1; : : : ; vn 2 TpM. We will also make use of the following results. Theorem 1 (Regular Level Set Theorem[14, Theorem 9.9, pg. 105]) Let N and M be smooth manifolds of dimension n and m respectively. Let F : N ! M be a smooth map. If c 2 M is a regular value of F s.t. F −1(c) 6= ;, then F −1(c) is a regular submanifold of N of dimension n − m. Theorem 2 ([14, Theorem 11.15, pg. 124]) Let f : N ! M be a smooth map of manifolds, and let f(N) ⊂ S ⊂ M. If S is a regular submanifold of M, then the map i ◦ f = f~: N ! S induced by the inclusion map i: S ! M, is smooth. Theorem 3 ([14, Proposition 14.3, pg. 151]) A vector field X on a man- ifold M is smooth if and only Xf 2 C1(M) for all f 2 C1(M). 2.2 The Matrix Exponential We will briefly review some facts about the matrix exponential which will be useful when we discuss Lie theory and the BCH formula. Proofs of these facts can be found in books on ordinary differential equations such as [1, Chapter 3] or [14, Section 15.3 and 15.4].
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