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) 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 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 transformation. 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, canonicalinvariance, Liouville's theorem, etc. are treated in detail.

Hamilton-Jacobi theory, Chapter 6, considers the interesting approach of finding a canonical transformation in which the phase space coordinates and the new Hamiltonian are all constant. This is discussed in depth and the student is given a procedure for solving a mechanical system.

A canonical transformation, the so-called action-angle variable, which is discussed in Chapter 7, is suitable for periodic phase orbits. The important field 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 first try to solve the exercises independently before consulting the solutions provided.