Classical Motion of a Single Particle

Overview

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ABOUT THE COURSE: This undergraduate level course is designed to offer a systematic development of the basic concepts of classical mechanics of a single particle. A special emphasis will be given on the vector formulation of the problems of dynamics which are appropriate for understanding and analyzing the motion of a particle in more than one dimensions. The notion of symmetry and conservation and their importance in classical dynamics will be discussed to pave the way for the motivated students to advance theories of physics. A separate section will be dedicated introducing the generalized coordinates, generalized velocities and the method of Lagrangian to solve constrained motions of a single particle. Finally, the phase space dynamics along with some introductory elements of nonlinear dynamics related to the motion of a single particle will be discussed.INTENDED AUDIENCE: 1st and 2nd year UG students and anyone interested in understanding the basics of classical mechanicsPREREQUISITES: Basics knowledge of calculus, partial differentiation and vector algebra is preferred

Syllabus

Week 1:Objective of classical mechanics, classical mechanics as a dynamical theory, Newton’s laws of motion
Week 2:Conservation of linear momentum, angular momentum and energy, work done, work-energy principle, conservative and non-conservative force fields, concept of potential, Stable and unstable equilibria, analysis of motion using energy diagram;
Week 3:Velocity and acceleration in polar coordinates; Motion under central forces, effective potential energy, finding trajectory for a given force law and vice-versa
Week 4:Kepler’s problem, Laplace Runge-Lenz vector, Rutherford scattering
Week 5:Linear harmonic oscillator with and without forcing, resonance, damped harmonic oscillator without forcing
Week 6:Damped harmonic oscillator with forcing, condition of resonance; Symmetry of Newton’s laws, Galilean transformations;
Week 7:Inertial and non-inertial frames of references, relation between the time derivatives of an inertial and a non-inertial frame, pseudo forces
Week 8:Motion of a particle under constraints, classification of constraints; principle of virtual work
Week 9:Generalized coordinates and generalized velocities;Euler-Lagrange’s equations from D’alembert’s principle, cyclic coordinates
Week 10:Properties of Lagrangian, application to constrained motions of a single particle
Week 11:Concept of phase space, fixed points and linear stability analysis
Week 12:Conservative vs. nonconservative systems, attractors, chaos in conservative and nonconservative systems