According to established tradition, courses on analytical mechanics include general equations of motion of holonomic and non-holonomic systems, vari- ational principles, theory of canonical transformations, canonical equations and theory of their integration (the Hamilton-Jacobi theorem), integral in- variants, theory of last multiplier and others. The fundamental laws of mechanics are taken for granted and are not subject to discussion. The present book is concerned with those issues of the above listed sub- jects which, in the author's opinion, are most closely related to engineering problems. Application of the methods of analytical mechanics to non-trivial prob- lems at the very stage of constructing the equations requires detailed knowl- edge of the issues that are normally only briefly touched upon. With this perspective considerable attention is paid to ways of introducing the gener- alised coordinates, the theory of finite rotation, methods of calculating the kinetic energy, the energy of accelerations, the potential energy of forces of various nature, and the resisting forces. These introductory chapters, which have to some extent independent significance, are followed by those on methods of constructing differential equations of motion for holonomic and non-holonomic systems in various forms. In these chapters the issues of their interrelations, determination of the constraint forces and some prob- lems of analytical statics are discussed as well. It is thought useful to include geometric considerations of the motion of a material system as motion of the representative point in Riemannian space.