Skew-symmetric matrices in classical mechanics Citation for published version (APA): Overdijk, D. A. (1989). Skew-symmetric matrices in classical mechanics. (Memorandum COSOR; Vol. 8923). Technische Universiteit Eindhoven. Document status and date: Published: 01/01/1989 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. 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Box 513 5600 MB Eindhoven The Netherlands Keywords: skew-symmetric matrix, exponential function, Lagrangian equations of motion, Eulerian angles, inertial force. Skew-symmetric matrices in classical mechanics O. Introduction and summary The results in this paper belong to classical mechanics and can for instance be found in Goldstein [ll, Pars [2], Rosenberg [3]. This paper is an attempt to illustrate the use of the exponential func­ tion with skew-symmetric matrix argument in three-dimensional kinematics. The orthogonal matrix in the transfonnation fonnula of the coordinates ofa point relative to two cartesian coordi­ nate 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 ofthese exponential functions can easily be calculated, they are pre-eminently appropriate for the description ofthe three-dimensional kinematics of sys­ tems of particles. The components of the generalized force on a system ofparticles in the Lagran­ gian 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 ofparticles. 1. Rotations and skew-symmetric matrices Choose a right-handed cartesian xyz-coordinate system in three-dimensional space. A rotation of space such that the origin ofthe coordinate system is a fixed point, is specified by the components 4 = (dx ' ely, dz) of a unit vector along the axis of rotation and the angle of rotation OS q>S 1t. The orientation of the rotation and the direction of the vector d correspond in the sense of the right­ hand screw motion. Let D be the matrix representation of the rotation relative to the coordinate system. The matrix D is orthogonal having detenninant one. Furthennore, the vector 4. is an - 2- eigenvector ofD with eigenvalue one. The angle of rotation <I> can be calculated from (1.1) <I> = arccos«trace(D) -1)/2) . We shall prove (1.2) D =exp(A) , where the skew-symmetric matrix A is given as 0 -dz dy (1.3) A =<1> dz 0 -dx -dy dx 0 The exponential function exp(A) in (1.2) with matrix-argument A is defined as (1.4) exp(A) = L A "In! , n=O where the convergence is elementwise. The vector::! occurring in a formula, is always to be understood as a (3 x I)-column matrix. We now prove (1.2). It is an easy calculation to verify, that Let !! be a unit vector perpendicular to f!: From (1.5) we infer A!! = <I>~ X!!) , (1.6) A 3!! = A (A 2!!) = -<1>2A!! =-<1>3 (4 X!!) , Hence, 00 (1.7) exp(A)!! = LAn!!In! = !! cos(4» + (4 x !!) sine<1» • n=O Furthermore, from A4 = Qwe get (1.8) exp(A) 4=4 . The relations (1.7) and (1.8) imply D = exp(A), which proves (1.2). - 3 - Now consider two right-handed cartesian coordinate systems C and C' in space. The coordinates of a point with respect to the systems C and C' are denoted by:! and:!' respectively. We discuss the transformation formula for the coordinates:! and:!' of a point relative to C and C'. Let f! be the coordinates ofthe origin of the system C' with respect to the system C. Furthermore, consider the rotation of the reference frame of C such that its orientation coincides with the orientation of (d~. the reference frame of C'. Let f!. = dy • dz) be the components relative to C of the unit vector along the axis of rotation, and let cI> be the angle of rotation. It follows from (1.2) that the coordi­ nates -x and -x' of a point with respect to the coordinate systems C and C' respectively, satisfy the transformation formula (1.9) :! = f! + exp(A):!' , where the skew-symmetric matrix A is given in (1.3). We now write the matrix exp(A) in (1.9) as the product of three exponentials involving the so called Eulerian angles. For ease of calculation we introduce a third right-handed cartesian coordi­ nate system C". The origins of the systems C' and C" coincide and the reference frames of the systems C and C" have the same orientation in space. Let n be the line of intersection of the x'y'-coordinate plane of C' and the x"y"-coordinate plane of C". The line n is referred to as the line of nodes. Rotate the reference frame of C" through an angle cI> about the Z"-axis such that the x"-axis is along the line of nodes. The rotated reference frame corresponds to the cartesian ~T\'­ coordinate system. The coordinates :!" and ~ of a point with respect to C" and the ~T\'-system respectively, satisfy the transformation formula (compare (1.9» (1.10) :!" =exp(cI>A 3) ~ , where 0 -1 0 A 3 = 1 0 0 0 0 0 The ~-axis of the ~T\'-system is perpendicular to both the z'-axis of C' and the ,-axis of the ~T\'­ system. Rotate the reference frame of the ~T\'-system through an angle e about the ~-axis such that the C-axis coincides with the z'-axis of the system C'. The rotated reference frame corresponds to the cartesian ~'T\'"-system. The coordinates ~ and ~' of a point with respect to the ~T\C-system and the ~'T\"'-system respectively, satisfy the transfonnation formula (compare (1.9» (1.11) where -4- o 0 0 Al = 0 0 -1 o 1 0 The C'-axis of the ~'TlT-system and the z'-axis of C' coincide. Rotate the reference frame of the ~'TlT-system through an angle 'If about the ~' -axis such that it coincides with the reference frame of the system C'. The coordinates ~' and !' of a point with respect to the ~'TlT-system and C' respectively, satisfy the transformation formula (compare (1.9» (1.12) where the skew-symmetric matrix A 3 is given in (1.10). From (1.10, 11, 12) we conclude that the coordinates! and !' of a point with respect to the coordinate systems C and C' respectively, satisfy the transformation formula (1.13) ! =f! +!"=f! + D!.' , where D =exp(A) =exp«j)A3) exp(eA I ) exp('lfA 3), 0 0 0 0 -1 0 A I = 0 0 -1 and A 3 = 1 0 0 0 1 0 0 0 0 The angles <1>, e, 'If are referred to as the Eulerian angles describing the orientation of the reference frame of C' relative to the reference frame of C. Now suppose that the reference frames of C and C' are in relative motion, i.e.