Explore Lagrange's equations with conservative and non-conservative forces in this comprehensive lecture on analytical dynamics. Delve into the decomposition of applied forces, the introduction of the Lagrangian function, and the application of Lagrange's equations to systems of rigid bodies. Work through practical examples to solidify understanding. Discover the concept of phase space and phase portraits as methods for analyzing motion characteristics and solving equations of motion. Learn about non-dimensionalization techniques and their importance in dynamics analysis. Gain valuable insights into advanced topics in engineering dynamics, including generalized coordinates, degrees of freedom, and the interplay between kinetic and potential energy in mechanical systems.