Euler’s Equation via Lagrangian Dynamics with Generalized Coordinates Dennis S. Bernstein, University of Michigan, Ann Arbor, MI * , Ankit Goel, University of Maryland, Baltimore County, Baltimore, MD † , Omran Kouba, Higher Institute for Applied Sciences and Technology, Damascus, Syria ‡ Euler’s equation relates the change in angular momentum of a rigid body to the applied torque. This paper fills a gap in the literature by using Lagrangian dynamics to derive Euler’s equation in terms of generalized coordinates. This is done by parameterizing the angular velocity vector in terms of 3-2-1 and 3-1-3 Euler angles as well as Euler parameters, that is, unit quaternions. I. Introduction The rotational dynamics of a rigid spacecraft are modeled by Euler’s equation [1, p. 59], which relates the rate of change of the spacecraft angular momentum to the net torque. Let ω ∈ R 3 denote the angular velocity of the spacecraft relative to an inertial frame, let J ∈ R 3×3 denote the inertia matrix of the spacecraft relative to its center of mass, and let τ denote the net torque applied to the spacecraft. All of these quantities are expressed in the body frame. Applying Newton-Euler dynamics yields Euler’s equation J ˙ ω + ω × Jω = τ. (1) An alternative approach to obtaining the dynamics of a mechanical system is to apply Hamilton’s principle in the form of Lagrangian dynamics given by d t ∂ ˙ q T − ∂ q T = Q, (2) where T is the kinetic energy of the system, q is the vector of generalized coordinates, and Q is the vector of generalized forces arising from all external and dissipative forces and torques, including those arising from * Professor, Aerospace Engineering Department, Corresponding Author, dsbaero@umich.edu † Assistant Professor, Mechanical Engineering Department, ankggoel@umbc.edu ‡ Professor, Department of Mathematics, omran kouba@hiast.edu.sy 1 arXiv:2212.11789v1 [math.DS] 22 Dec 2022