Chapter 1 - Mathematical Mechanical Preliminaries

Abstract

This chapter presents the basics of mathematics and mechanics used in the book. In the first section, the elliptic integrals and elliptical functions are considered. In mechanics, these mathematical objects describe nonlinear oscillations, unlike the trigonometric functions, which describe linear oscillations. In the second section, rigid body kinematics is considered. The section deals with two methods to describe attitude motion of rigid body, which are used in the book: two types of Euler angles (ZYZ and XYZ rotation sequences) and directional cosine matrix. Kinematic equations are written for each type of coordinates. The kinematic equations are the relation between angular velocity of the rigid body and the derivatives of the coordinates described in orientation of the rigid body. In the third section, dynamic equations of the rigid body are presented including equations in Serret-Andoyer canonical variables, which allow reducing torque-free rotational dynamics of a rigid body to two-dimensional Hamiltonian system. In the last section, a chaotic motion is considered. The Poincaré method is described, which is used to investigate the motion of complex systems. In addition, the Melnikov method is described, which is used in the book to prove the existence of chaos in considered mechanical systems.