|
|
2.1 Newtonian formulation
Mathematical Review:
Scalar and vector fields, rotation matrix, infinitesimal rotation, angular momentum. Transformation properties of scalars and vectors under rotation, spatial inversion and time reversal. Scalar and vector differential and integral calculus, Gauss and Stokes theorems.
Newtonian Mechanics
Newton's Laws, mass, inertial frames, finite and many-body systems. Center of mass. Linear and angular momentum. Work and energy, conservation laws, virial theorem. Solution of Newton's equations of motion for forces that are: conservative, velocity-dependent, time-dependent, impulsive, position-dependent, systems with variable mass. Perturbation theory
Gravitation
Gravitational field, potential, vector differential form of Newton's law of gravitation
Oscillations
Equilibrium and stability. Oscillations, phase diagram, two-dimensional oscillator, damped free oscillators, forced oscillation. Linearity and superposition. Fourier analysis, impulsive driving forces. Non-linear oscillation, phase diagrams, plane pendulum, approach to chaos, Poincaré sections.
2.2 Lagrangian formulation
Calculus of variations
Euler's Equation, maximum/minimum problems, Brachistochrone, geodesic. Functions of several variables. Equations of constraint, generalized coordinates, Euler's equations with constraints. Lagrange multipliers. Catenary and isoperimetric problems. Connections to classical mechanics.
Lagrangian Mechanics
Hamilton's principle. Lagrange's equations of motion relative to Newton's laws. Euler-Lagrange equations and applications to harmonic oscillator, pendulum, constrained motion. Lagrange equations with undetermined multipliers and applications to problems with constraints. Symmetries and conservation laws.
Hamiltonian
Kinetic energy in generalized coordinates, canonical momenta, Hamiltonian, Hamilton's equations, one-dimensional oscillator and spherical pendulum. Hamiltonian properties, conservation of the Hamiltonian and energy.
2.3 Applications
The central force problem
Reduced mass, two-body central force, equations of motion, differential equation of orbit, inverse square law, Kepler's laws, stability of circular orbits, closure.
Motion in Non-inertial Frames
Accelerating translational systems. Rotating coordinate systems, centrifugal and Coriolis effects. Projectile motion, weather systems on Earth. Foucault pendulum. Lagrangian and Hamiltonian in rotating frame.
Dynamics of Rigid Bodies
Rigid-body coordinates, inertia tensor, kinetic energy, angular momentum , principal axes, parallel-axis theorem, general features of inertia tensor, Euler angles, Euler equations for rigid body, rotation of free and fixed symmetric tops, stability of rigid-body motion.
Coupled Oscillations
Coupled pendula, weak coupling. General solution, normal coordinates, degeneracy for discrete oscillatory systems. Continuous oscillatory systems.
2.4 Modern Classical Mechanics
Hamiltonian Mechanics
Generalized Hamilton's Principle and Hamiltonian mechanics. Hamilton's equations in cylindrical and spherical coordinates. Applications of Hamiltonian mechanics. Canonical transformations, Poisson Brackets, commutators and applications to quantum physics. Hamilton-Jacobi theory, Liouville's theorem, relation between classical and quantum mechanics.
Relativistic mechanics
Lorentz transformation, geometry of space-time, relativistic mechanics, Lagrangian and Hamiltonian in special relativity.
|
|
|
|
|
|
|
|