Classical Mechanics: Hamiltonian and Lagrangian Formalism (Google eBook)
Formalism of classical mechanics underlies a number of powerful mathematical methods that are widely used in theoretical and mathematical physics. This book considers the basics facts of Lagrangian and Hamiltonian mechanics, as well as related topics, such as canonical transformations, integral invariants, potential motion in geometric setting, symmetries, the Noether theorem and systems with constraints. While in some cases the formalism is developed beyond the traditional level adopted in the standard textbooks on classical mechanics, only elementary mathematical methods are used in the exposition of the material. The mathematical constructions involved are explicitly described and explained, so the book can be a good starting point for the undergraduate student new to this field. At the same time and where possible, intuitive motivations are replaced by explicit proofs and direct computations, preserving the level of rigor that makes the book useful for the graduate students intending to work in one of the branches of the vast field of theoretical physics. To illustrate how classical-mechanics formalism works in other branches of theoretical physics, examples related to electrodynamics, as well as to relativistic and quantum mechanics, are included.
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Chapter 2 Hamiltonian Formalism
Chapter 3 Canonical Transformations of TwoDimensional Phase Space
Chapter 4 Properties of Canonical Transformations
Chapter 5 Integral Invariants
Chapter 6 Potential Motion in a Geometric Setting
according to Eq acquires the form action functional affine connection algebraic canonical transformation classical mechanics compute configuration space conservation Consider const construct coordinate transformations corresponding covariant curve defined denoted Dirac bracket Dirac procedure discuss dynamics energy equations of motion equivalent Euclidean evolution example Exercise expression extended Lagrangian first-class constraints first-order gauge geodesic line geometric given Hamilton–Jacobi equation Hamiltonian action Hamiltonian equations Hamiltonian formulation Hamiltonian system Hence identity implies infinitesimal symmetry initial conditions integral invariant inverse Lagrangian action Lagrangian equations linear Lorentz matrix metric Noether charge Noether theorem notation obeys Eq obtain parallel transport particle phase space Poincaré transformations Poisson bracket primary constraints properties quantity relativistic represents result Riemann space scalar second-class constraints singular theory solution substitution symplectic matrix tensor tion total derivative trajectory vanishes variables vector field velocities write