Mathematical Methods of Lagrangian and Hamiltonian Mechanics

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 appliedmathematical methods of Lagrangian and Hamiltonian mechanics. The bookhas the benet of learning, in addition to the physical processes of classicalmechanics, with focus on Lagrangian and Hamiltonian mechanics, the math-ematical methods that are equally needed in other branches of physics. Theseinclude: 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'slaws are a by-product of this.Chapter 2 explains the origins of the variation technique with its historicalorigin in the brachistochrone problem. After introducing generalised coordin-ates and applying Newton's principle of determinacy, the Lagrangian approachfor mechanical systems is derived. The conservation laws play an importantrole in this context. Applications are shown for motions in a central eld. TheLagrangian dynamics for oscillations with the various modes is discussed indepth. The application of linear algebra (eigenvectors, normal coordinates) istreated in great detail.Chapter 3 develops the Hamiltonian dynamics for mechanical systems. Thetransition from the conguration space of Lagrangian mechanics to the sym-plectic phase space of Hamiltonian mechanics (Legendre transformation) isdiscussed. An additional section deals with Routh's procedure, which can bedescribed as a mixture of Lagrangian and Hamiltonian mechanics.The extension of the permissible transformations of the variables (qi; pi) ofHamiltonian mechanics in comparison to Lagrangian approach leads us to thecanonical transformations, Chapter