View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by University of Groningen University of Groningen Kinematical Constraints and Algebroids Clemente-Gallardo, J.; Maschke, B.; Schaft, A.J. van der Published in: Reports on mathematical physics IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2001 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Clemente-Gallardo, J., Maschke, B., & Schaft, A. J. V. D. (2001). Kinematical Constraints and Algebroids. Reports on mathematical physics, 47(3), 413-429. Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date: 12-11-2019 Vol. 47 (2001) REPORTS ON MATHEMATKAL.PHYSICS No. 3 KINEMATICAL CONSTRAINTS AND ALGEBROIDS J. CLEMENTE-GAJURDO’,B. IViASCHKE2 and A. J. VANDER SCHAFT’ lFaculty of Mathematical Sciences, Department of Systems, Signals and Control University of Wente, PO. Box 217, 7500 AE Enschede, The Netherlands 2Labocatoire d’Automatique et de Genie des Procedes UCB Lyon 1 UFR Genie Electrique et des Procedes - CNRS UMR 5007 Universite Claude Bernard- Lyon 1, Villeurbanne, France (e-mails: [email protected]; [email protected]; [email protected]) (Received December 5, 2000 - Revised March 19, 2ooO) This note intends to establish a link between the description of dynamics on a Lie alge- broid defined by an integrable subbundle of the tangent bundle, and the dynamics associated to the Hamiltonian structure defined by [14] in the case of holonomic constraints. Keyworrkx Lie algebroids, constrained dynamics. 1. Introduction The study of mechanical systems with constraints has been a topic of continuing interest over the last century. Though the most important and open subject in this field is the nonholonomic case, the tools developed for holonomic systems may be indirectly important and useful in the study of nonholonomic constraints. The present paper establishes the geometrical structure which defines a Poisson structure on the constrained phase space of a system with kinematical constraints proposed by two of us [14]. The construction in [14] is valid both for holonomic and nonholonomic constraints, and a theorem is presented proving that the almost Poisson structure of the general case is a true Poisson structure (that is satisfying the Jacobi identity) if and only if the constraints are holonomic. This Poisson structure is obtained as a suitable restriction of the canonical structure (defined by the symplectic form of a cotangent bundle) on the submanifold defined by the constraints on that cotangent bundle (see [6]). The mathematical approach here is inspired by [8] (see also references therein) and combines it with the recently developed Lagrangian and Hamiltonian formalisms for Lie algebroids [ 15, 9, 71. The Lie algebroid is the bundle defined by the sub- bundle of admissible velocities. In order to emphasize the algebroid properties, our notation for the distribution defined by the admissible velocities, denoted by D, will be different from the notation for the Lie algebroid itself, which we will denote by dn. Of course, dD is canonically isomorphic to the distribution ID. [4131 414 J.CLEMENTEbGALLARDo. B. MASCHKE and A. I. VAN DER SCHAFT The structure of the paper is as follows. Sections 2 and 3 provide a brief sketch of the Poisson structure proposed in [14], a summary of the definition and main properties of Lie algebroids and the particular example of an integrable subbundle of a tangent bundle. Section 4 contains the main result of the paper: the Hamiltonian dynamics on the dual of the Lie algebroid defined by an integrable subbundle of a tangent bundle is equivalent to the Hamiltonian dynamics defined by van der Schaft and Maschke. Finally, in Section 5 we recall, for completeness, Martinez’s definition of a symplectic structure on the Lie algebroid which provides the Poisson algebra we are dealing with. We also establish the relation between the Poisson tensor of the constrained phase space and that of the dual of the Lie algebroid. 2. Van der Schaf-m construction for constrained dynamics The main idea of the construction is that it is possible to define a dynamical description of a constrained system without making explicit use of the constraint equations, just by defining a suitable Poisson structure adopted to the constrained phase space. Let us consider a dynamical system defined on a given manifold M and let us assume that it is constrained by a set of conditions on its velocities: A*(q)4= 0, q E M, 4 E T,M where we consider AT as a p x n matrix of rank p < n everywhere (n is the dimen- sion of M) defined on T,M, or in more intrinsic terms, as a set of p independent l-forms on M. From the geometrical point of view, this means that the set of admissible veloci- ties defines a subbundle D of the tangent bundle TM. In the present paper, we only consider the case where 2) is an integrable subbundle of TM, though the original paper [14] is devoted to the general case of nonintegrable subbundles. We want to define a Hamiltonian dynamics for this system, and hence we must define the corresponding submanifold of T*M, endowed with a Poisson structure. We restrict ourselves to the case of regular Hamiltonians, for simplicity. The definition was given in [14], from the geometric point of view, by specifying the submanifold of T*M defined by Legendre transformation of the integrable subbundle defined by the admissible velocities. This definition depends on the Hamiltonian of the system, and not only on the constraints. In any case, we define the constrained submanifold XC on T* M as follows. DEFINITION 2.1. The constrained phase space XC c T*M is defined as the fol- lowing set of points of the cotangent bundle: xc = (q, p) E T*M : A(q)aHt ‘) = 0 (1) KINEMATICAL CONSTRAINTS AND ALGEBROIJX 415 Throughout we require that tire Hamiltonian satisfies the regularity property, det(AT$A) +O, (2) which implies that the Legendre transformation of any admissible velocity is never contained in the annihilator of the distribution 27. An interesting property of the constrained system is that we can provide a system of coordinates for Xc which does not depend of the Hamiltonian, contrary to the manifold itself, and hence is the same coordinate system for any Hamiltonian. The set of coordinates is obtained from a mapping from the vector bundle defined by tbe admissible velocities and the distribution on M corresponding to the admissible vector fields. In order to understand better the Lie algebroid issues of the following sections, we consider this vector bundle as a separate bundle, not as a subbundle of TM, and we denote it as dp. In Section 3 we present its Lie algebroid structure. Thus, we define S : AD + 2) c TM, of maximal rank on D, satisfying AToS=O. We will see how this mapping coincides with the anchor mapping defined in the next section. If we take the dual S* of this mapping we can define a set of coordinates for XC, $1) = s*(q)& p E T;M. NOTE 2.1. Later on it will become clear why these coordinates are the same for all possible regular Hamiltonians we define. The main idea is that the Legendre transformation of the subbundle of integrable velocities defines a choice of a repre- sentative in a certain quotient bundle (i.e., it defines an element of the fiber of this bundle, by choosing an element of the manifold with respect to which we factor out). These coordinates will be defined for this quotient bundle, and any particular representation of the fibers (i.e., the choice of any regular Hamiltonian to define the Legendre transformation) will be admissible for them. With this set of coordinates for XC we can proceed to define a Poisson structure on it, as a suitable restriction of the natural Poisson structure on T*M which includes the constraints. The process can be summart‘zed as follows: l We complement the set of coordinates above to define a set of coordinates adopted to the constrained phase space (see the following sections for a geo- metrical version). p p = s*(q)p, = AT(qh p E T,*M. This set of local coordinates {q, p-cl), $2)} defines a system which will be not, in general, canonical. In any case it is a valid coordinate system on the total phase space, and particularly useful since it is adopted to the constraints. 416 J. CLEMENTE-GALLARDo, B. MASCHKE and A. J. VAN DER SCHAFT l We take the natural Poisson structure on T*M, and we write it in the coor- dinates above: (3) l Finally, we define the restriction of this Poisson bracket to the constrained phase space, just by removing the set of brackets of the $*) coordinates: (4) We have used Greek indices for denoting the coordinates of the fibers of the con- strained phase space, as we will do in the case of algebroids, in order to make the relation between the two formalisms more evident.