Explore the intricacies of rigid body kinematics and dynamics in this comprehensive lecture on analytical dynamics. Delve into the rotational kinematic differential equations for Euler angles and rotation matrices, with a focus on the 3-2-1 Euler angle convention (yaw, pitch, roll). Examine the singularity issue in Euler angle equations and discover alternative methods for describing orientation, including Euler parameters and quaternions. Investigate the fundamental differential equation for rotation matrices and the Euler principal axis and angle formulation. Learn about Euler's rotational dynamics for torque-free rigid bodies, accompanied by animations demonstrating free rigid body motion. Gain insights into analyzing the dynamics of symmetric free rigid bodies, enhancing your understanding of 3D rigid body dynamics and its applications in engineering and physics.