Mathematical Methods of Lagrangian and Hamiltonian Mechanics

This book is intended to help students of physics and other branches of science in the first semesters of their studies to better understand the applied mathematical methods of Lagrangian and Hamiltonian mechanics. The book has the benefit of learning, in addition to the physical processes of classical mechanics, with focus on Lagrangian and Hamiltonian mechanics, the mathematical methods that are equally needed in other branches of physics. These include: Vector calculus, matrix calculus, tensor calculus, differential equations, derivative chain rule, Taylor series, differential geometry, implicit function theorem, coordinate transformation (Jacobian), curvilinear coordinates, Legendre transformation, and much more.Chapter 1 describes the basics of Newtonian mechanics in a review. In addition to Newton's laws, the two-body problem is dealt with in detail. Kepler's laws are a by-product of this.Chapter 2 explains the origins of the variation technique with its historical origin in the brachistochrone problem. After introducing generalised coordinates and applying Newton's principle of determinacy, the Lagrangian approachfor mechanical systems is derived. The conservation laws play an important role in this context. Applications are shown for motions in a central field. The Lagrangian dynamics for oscillations with the various modes is discussed in depth. The application of linear algebra (eigenvectors, normal coordinates) is treated in great detail.Chapter 3 develops the Hamiltonian dynamics for mechanical systems. The transition from the configuration space of Lagrangian mechanics to the symplectic phase space of Hamiltonian mechanics (Legendre transformation) isdiscussed. An additional section deals with Routh's procedure, which can be described as a mixture of Lagrangian and Hamiltonian mechanics.The extension of the permissible transformations of the variables of Hamiltonian mechanics in comparison to Lagrangian approach leads us to the canonical