Skew-symmetric matrices in classical mechanics

D.A. Overdijk

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    The results in this paper belong to classical mechanics and can for instance be found in Goldstein [1], Pars [2], Rosenberg [3]. This paper is an attempt to illustrate the use of the exponential function with skew-symmetric matrix argument in three-dimensional kinematics. The orthogonal matrix in the transfonnation fonnula of the coordinates of a point relative to two cartesian coordinate systems, can be written as the product of three exponential functions with matrix argument. The argument of these exponential functions is a fixed skew-symmetric matrix multiplied by an Eulerian angle. Since the time derivatives of these exponential functions can easily be calculated, they are pre-eminently appropriate for the description of the three-dimensional kinematics of systems of particles. The components of the generalized force on a system of particles in the Lagrangian fonnulation of the equations of motion, can also be calculated conveniently by use of the exponential function with matrix argument. The representation of the orthogonal matrix corresponding to a rotation as the product of three exponentials is presented in Section 1. In Section 2 we briefly discuss the concept of generalized coordinates and generalized velocities of a system of particles. The inertia tensor of rigid systems of particles is treated in Section 3. In Section 4 we introduce the definitions of inertial coordinate systems and inertial forces in connection with Newton's second law of motion. The Lagrangian fonnulation of the equations of motion for discrete systems of particles is described in Section 5. Finally, in Section 6 we derive by way of illustration Lagrange's equations of motion for rigid systems of particles. Keywords: skew-symmetric matrix, exponential function, Lagrangian equations of motion, Eulerian angles, inertial force.
    Original languageEnglish
    Place of PublicationEindhoven
    PublisherTechnische Universiteit Eindhoven
    Number of pages16
    Publication statusPublished - 1989

    Publication series

    NameMemorandum COSOR
    ISSN (Print)0926-4493


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