This book is intended to help students of physics and other branches of sci- ence in the rst semesters of their studies to better understand the applied mathematical methods of Lagrangian and Hamiltonian mechanics. The book has the benet of learning, in addition to the physical processes of classical mechanics, with focus on Lagrangian and Hamiltonian mechanics, the math- ematical methods that are equally needed in other branches of physics. These include: Vector calculus, matrix calculus, tensor calculus, dierential equa- tions, derivative chain rule, Taylor series, dierential geometry, implicit func- tion theorem, coordinate transformation (Jacobian), curvilinear coordinates, Legendre transformation, and much more. Chapter 1 describes the basics of Newtonian mechanics in a review. In addi- tion 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 coordin- ates and applying Newton's principle of determinacy, the Lagrangian approach for mechanical systems is derived. The conservation laws play an important role in this context. Applications are shown for motions in a central eld. 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 conguration space of Lagrangian mechanics to the sym- plectic phase space of Hamiltonian mechanics (Legendre transformation) is discussed. 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 (qi; pi) of Hamiltonian mechanics in comparison to Lagrangian approach leads us to the canonical transformations, Chapter 4. Here the generating functions of the canonical transformations are derived with the help of the Legendre trans- formation. The symplectic relationship of canonical transformations is clearly worked out. In Chapter 5, the Hamiltonian equations of motion are described using the Poisson formalism, which provides the equations of motion with a symmetrical form. Further topics such as constants of motion, Jacobi identity, canonical invariance, Liouville's theorem, etc. are treated in detail. 1 Hamilton-Jacobi theory, Chapter 6, considers the interesting approach of nding a canonical transformation in which the phase space coordinates and the new Hamiltonian are all constant. This is discussed in depth and the stu- dent is given a procedure for solving a mechanical system. A canonical transformation, the so-called action-angle variable, which is dis- cussed in Chapter 7, is suitable for periodic phase orbits. The important eld of adiabatic invariants with reference to quantum mechanics is also discussed. The texts are supported with many graphics and help the student to grasp the current topic more intuitively. All chapters contain many exercises. The student is encouraged to rst try to solve the exercises independently before consulting the solutions provided.weiterlesen