Linder Jacob, Brevik Iver H. Introduction to Lagrangian & Hamiltonian Mechanics

Bookboon, 2016. — 112 p. — ISBN: 978-87-403-1249-2.In both classical and quantum mechanics, the Lagrangian and Hamiltonian formalisms play a central role. They are powerful tools that can be used to analyze the behavior of a vast class of physical systems. The aim of this book is to provide an introduction to the Lagrangian and Hamiltonian formalisms in classical systems, covering both non-relativistic and relativistic systems. The lectures given in this course have been recorded on video and uploaded on YouTube. At the beginning of each chapter we provide a link to the YouTube-videos covering that particular chapter. It is our goal that students who study this material afterwards will find themselves well prepared to dig deeper into the remarkable world of theoretical physics at a more advanced level. We have carefully chosen the topics of this book to make students proficient in using and understanding important concepts such as symmetries and conservation laws, the special theory of relativity, and the Lagrange/Hamilton equations.Fundamental principles
Notation and brief repetition
Many-particle systems
Constraints and generalized coordinates
D’Alembert’s principle and Lagrange’s equations
Levi-Civita symbol
Friction and other velocity-dependent potentials
Examples
Lagrange’s equations and the variational principle
Hamilton’s principle
Derivation of Lagrange’s equations from Hamilton’s principle
Variational calculus
Hamilton’s principle for non-holonomic systems
Conservation laws and symmetries
Hamilton’s equations
Legendre transformations
Going from Lagrangian to Hamiltonian formalism
The two-body problem: central forces
Reduction to equivalent one-body problem
Equations of motion
Equivalent one-dimensional problem
The virial theorem
The Kepler problem
Scattering cross section
Kinematics and equations of motion for rigid bodies
Orthogonal transformations and independent coordinates
Transformation matrix and its mathematical properties
Formal properties of the transformation matrix
Euler angles
Infinitesimal transformations
The rate of change of time-dependent vectors
Components of ! along the body axes
The Coriolis force
Angular momentum and kinetic energy
The Euler equations
Free rotation of rigid body; precession
Heavy symmetric top with one point fixed
Small-scale, coupled oscillations
Coupled oscillators
Application to a triatomic linear symmetric molecule (CO2)
The theory of special relativity
Introductory remarks
Lorentz transformations
Choices of metric
Covariant 3+1 dimensional formulation
Maxwell’s equation, 4-potential, and electromagnetic field tensor
Relativistic mechanics and kinematics
The relativity of simultaneity
Canonical transformations
Transformation of phase space
Poisson brackets
Hamilton-Jacobi theory