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Lagrangian Mechanics and Variational Integrators on Two-Spheres 3 LAGRANGIAN MECHANICS AND VARIATIONAL INTEGRATORS ON TWO-SPHERES∗ TAEYOUNG LEE†, MELVIN LEOK‡, AND N. HARRIS MCCLAMROCH§ Abstract. Euler-Lagrange equations and variational integrators are developed for Lagrangian mechanical systems evolving on a product of two-spheres. The geometric structure of a product of two-spheres is carefully considered in order to obtain global equations of motion. Both continuous equations of motion and variational integrators completely avoid the singularities and complexities introduced by local parameterizations or explicit constraints. We derive global expressions for the Euler-Lagrange equations on two-spheres which are more compact than existing equations written in terms of angles. Since the variational integrators are derived from Hamilton’s principle, they preserve the geometric features of the dynamics such as symplecticity, momentum maps, or total energy, as well as the structure of the configuration manifold. Computational properties of the variational integrators are illustrated for several mechanical systems. Key words. Lagrangian mechanics, geometric integrator, variational integrator, two sphere, homogeneous manifold AMS subject classifications. 70H03, 65P10, 37M15 1. Introduction. The two-sphere S2 is the set of all points in the Euclidean space R3 which are a unit distance from the origin. It is a two dimensional manifold that is locally diffeomorphic to R2. Many classical and interesting mechanical systems, such as a spherical pendulum, a double spherical pendulum, and magnetic models, evolve on the two-sphere or on a product of two-spheres. In this paper, we derive Euler-Lagrange equations on configuration spaces of the form (S2)n, for a positive integer n. We also develop geometric numerical integrators referred to as discrete Euler-Lagrange equations or variational integrators on (S2)n. In most of the literature that treats dynamic systems on (S2)n, either 2n angles or n explicit equality constraints enforcing unit length are used to describe the configuration of the system [2, 15]. These descrip- tions involve complicated trigonometric expressions and introduce additional complexity in analysis and computations. In this paper, we focus on developing continuous equations of motion and discrete equations of motion directly on (S2)n, without need of local parameterizations, constraints, or reprojections. This provides a remarkably compact form of the equations of motion. Geometric numerical integrators are numerical integration algorithms that preserve the geometric struc- ture of the continuous dynamics, such as invariants, symplecticity, and the configuration manifold [5]. Con- ventional numerical integrators construct a discrete approximation to the flow using only information about arXiv:0707.0022v1 [math.NA] 30 Jun 2007 the vector field, and ignore the physical laws and the geometric properties inherent in the differential equa- tions [17]. Consequently, they do not preserve important characteristics of the dynamics of the continuous equations of motion. In contrast, variational integrators are constructed by discretizing Hamilton’s principle, rather than discretizing the continuous Euler-Lagrange equation [18, 16]. Since they are developed by using a discrete version of a physical principle, the resulting integrators have the desirable property that they are symplectic and momentum preserving, and they exhibit good energy behavior for exponentially long times. Geometric numerical integration on S2 has been studied in [19, 14, 13]. The two-sphere is a homogeneous ∗TL and ML have been supported in part by NSF Grant DMS-0504747 and DMS-0726263. TL and NHM have been supported in part by NSF Grant ECS-0244977 and CMS-0555797. †Department of Aerospace Engineering, The University of Michigan, Ann Arbor, MI 48109 ([email protected]). ‡Department of Mathematics, Purdue University, West Lafayette, IN 47907 ([email protected]). §Department of Aerospace Engineering, The University of Michigan, Ann Arbor, MI 48109 ([email protected]). 1 2 T. Lee, M. Leok, and N. H. McClamroch manifold; the special orthogonal group SO(3) acts transitively on S2, and Lie group methods [7] can be adapted to generate numerical flows on S2. In this paper, we study Lagrangian mechanical systems on (S2)n. Thus, it is desirable to preserve the geometric properties of the dynamics, such as momentum map, symplecticity, and total energy, in addition to the structure of the configuration manifold [11]. We combine the approaches of geometric integrators on homogeneous manifolds and variational integrators to obtain variational integrators on (S2)n that preserve the geometric properties of the dynamics as well as the homogeneous structure of the configuration manifold (S2)n concurrently. The contributions of this paper can be summarized in two aspects. In the continuous time setting, the global Euler-Lagrange equations on (S2)n are developed in a compact form without local parameterization or constraints. This provides insight into the global dynamics on (S2)n, which is desirable for theoretical studies. As a geometric numerical integrator, the discrete Euler-Lagrange equations on (S2)n are unique in the sense that they conserve both the geometric properties of the dynamics and the manifold structure of (S2)n simultaneously. The exact geometric properties of the discrete flow not only generate improved qualitative behavior, but they also provide accurate and reliable computational results in long-time simulation. This paper is organized as follows. Lagrangian mechanics on (S2)n is described in Section 2. Variational integrators on (S2)n are developed in Section 3. Computational properties are illustrated for several me- chanical systems, namely a double spherical pendulum, an n-body problem on a sphere, an interconnected system of spherical pendula, pure bending of an elastic rod, a spatial array of magnetic dipoles, and molecular dynamics that evolves on a sphere. 2. Lagrangian mechanics on (S2)n. In this section, continuous equations of motion for a mechanical system defined on (S2)n are developed in the context of Lagrangian mechanics. It is common in the published literature that the equations of motion are developed by using either two angles or a unit length constraint to characterize S2. Any description with two angles has singularities, and any trajectory near a singularity experiences numerical ill-conditioning. The unit length constraint leads to additional complexity in numerical computations. We develop global continuous equations of motion without resorting to local parameterizations or constraints. To achieve this, it is critical to understand the global characteristics of a mechanical system on (S2)n. This section provides a good background for understanding the theory of discrete Lagrangian mechanics on (S2)n to be introduced in the next section. The two-sphere is the set of points that have the unit length from the origin of R3, i.e. S2 = {q ∈ 3 2 2 R | q · q =1}. The tangent space TqS for q ∈ S is a plane tangent to the two-sphere at the point q. Thus, a curve q : R → S2 and its time derivative satisfy q · q˙ = 0. The time-derivative of a curve can be written as q˙ = ω × q, (2.1) where the angular velocity ω ∈ R3 is constrained to be orthogonal to q, i.e. q · ω = 0. The time derivative of the angular velocity is also orthogonal to q, i.e. q · ω˙ = 0. 2.1. Euler-Lagrange equations on (S2)n. We consider a mechanical system on the configuration manifold S2 ×···× S2 = (S2)n. We assume that the Lagrangian L : T (S2)n → R is given by the difference between a quadratic kinetic energy and a configuration-dependent potential energy as follows. 1 n L(q ,...,q , q˙ ,..., q˙ )= M q˙ · q˙ − V (q ,...,q ), (2.2) 1 n 1 n 2 ij i j 1 n i,j=1 X Lagrangian Mechanics and Variational Integrators on Two-Spheres 3 2 where (qi, q˙i) ∈ T S for i ∈ {1,...,n}, and Mij ∈ R is the i, j-th element of a symmetric positive definite inertia matrix M ∈ Rn×n for i, j ∈ {1,...,n}. The configuration dependent potential is denoted by V : (S2)n → R. The action integral is defined as the time integral of the Lagrangian, and the variation of the action integral leads to continuous equations of motion by applying Hamilton’s principle. These are standard 2 procedures to derive the Euler-Lagrange equations. The expression for the infinitesimal variation of qi ∈ S should be carefully developed, since the configuration manifold is not a linear vector space. As in (2.1), the infinitesimal variation of qi can be written as a vector cross product, δqi = ξi × qi, (2.3) 3 where ξi ∈ R is constrained to be orthogonal to qi, i.e. ξi · qi = 0. From this, the expression for the infinitesimal variation ofq ˙i is given by δq˙i = ξ˙i × qi + ξi × q˙i. (2.4) These expressions are the key elements to obtaining global equations of motion on (S2)n. The variation of the Lagrangian can be written as n n ∂V δL = δq˙ · M q˙ − δq · , i ij j i ∂q i,j=1 i=1 i X X where the symmetric property Mij = Mji is used. Substituting (2.3) and (2.4) into this, and using the vector identity (a × b) · c = a · (b × c) for any a,b,c ∈ R3, we obtain n n ∂V δL = ξ˙ · (q × M q˙ )+ ξ · (q ˙ × M q˙ ) − ξ · q × . i i ij j i i ij j i i ∂q i,j=1 i=1 i X X G G T Let be the action integral defined as = 0 L(q1,...,qn, q˙1,..., q˙n) dt. Using the above equation and integrating by parts, the variation of the actionR integral is given by n T n T n ∂V δG = ξ · (q ˙ × M q˙ + q × M q¨ ) − ξ · (q × M q¨ )+ q × . i i ij j i ij j i i ij j i ∂q i,j=1 0 i=1 Z0 j=1 i X X X From Hamilton’s principle, δG = 0 for any ξi vanishing at t = 0,T .
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