# SYLLABUS

Philosophical approaches to classical mechanics

Introduction, comparison, and the philosophical implications, of the contrasting Newtonian approach, which is based on momentum and forces, and the analytical approaches of Lagrange and Hamilton which are based on energy and variational principles.

Newtonian formulation

**Mathematics of vector fields:**

Review of matrix and vector algebra. Orthogonal coordinate systems. Coordinate transformations, rotation matrix, finite, infinitesimal, proper, and improper rotations, spatial inversion, and time reversal, the Jacobian. Tensor algebra. Transformation properties of common observables. Scalar and vector differential calculus, vector integral calculus.

**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, and variable mass. Rigid-body rotation about a fixed axis. Solution of equations of motion by analytic, perturbation, successive approximation, and numerical methods

**Gravitation***

Gravitational field, potential, vector differential form of Newton's law of gravitation

**Linear Oscillators**

Equilibrium and stability. Oscillations, phase diagram, two-dimensional oscillator, damped free oscillators, forced oscillation, resonance. Linearity and superposition. Solutions of the wave equation. Fourier analysis and impulsive driving forces of a periodic driven oscillator. Periodic sampling, analog and digital signal processing. Wave packets, phase, group and signal velocities, uncertainty principle. **
Non-linear dynamics and chaos:
** Non-linear dynamics and superposition. Attractors and the van der Pol oscillator. The driven damped plane pendulum. Phase diagrams, bifurcation, Lyapunov exponent, approach to chaos, Poincaré sections. Order to chaos transition in nuclei. Wave propagation and soliton waves.

Lagrangian formulation

**Calculus of variations **

Euler's Equation, maximum/minimum problems, Brachistochrone, geodesic. Functions of several variables and selection of the independent variable. Equations of constraint, generalized coordinates, Euler's equations with holonomic and non-holonomic constraints. Lagrange multipliers. Catenary and isoperimetric problems. Connections to classical mechanics.

**Lagrangian Mechanics ** Use of both d'Alembert's Principle and Hamilton's Principle to derive the Euler-Lagrange equations. Lagrange equations using Lagrange multipliers. Applications to systems with holonomic and non-holonomic constraints. Non-conservative forces, and velocity dependent potentials. Noether's theorem, symmetries, invariance, cyclic coordinates, and conservation laws.

Hamiltonian formulation

**Hamiltonian Mechanics
**Legendre transformation. Kinetic energy in generalized coordinates, canonical momenta. Hamiltonian, relation to energy conservation, total energy, and cyclic coordinates. Generalized Hamilton's principle and Hamiltonian mechanics. Hamilton's equations in cylindrical and spherical coordinates. State and phase space. Applications of Hamiltonian mechanics. Routhian reduction.

Applications

**The central force problem**

Reduced mass, two-body central force, equations of motion, differential equation of orbit, inverse square law, Kepler's laws, isotropic harmonic two-body central force. Orbit stability. The three-body problem. Two-body scattering and two-body kinematics.

**Motion in Non-inertial Frames**

Accelerating translational systems. Rotating coordinate systems, centrifugal and Coriolis effects. Lagrangian and Hamiltonian in rotating frame. Routhian reduction. Projectile motion, weather systems on Earth. Foucault pendulum.

** Rigid-body** **Dynamics**

Rigid-body coordinates. Angular momentum for rotation about any body-fixed point, Inertia tensor, parallel-axis theorem, kinetic energy, principal axes, Euler angles, Euler equations for rigid body, rotation of free and fixed symmetric and asymmetric tops, stability of rigid-body motion, rolling wheel, static and dynamic balancing. Rotating deformable bodies.

**Coupled linear oscillations**

Two coupled oscillators, weak coupling. General analytic theory, normal modes, degeneracy for discrete few-body coupled linear oscillators. Damped coupled oscillators. Applications to molecules and nuclei. Discrete lattice chain. Collective synchronization.

Advanced topics in classical mechanics

**Non-standard Lagrangian mechanics*
**Equivalent standard Lagrangians, gauge invariance, non-standard Lagrangians. inverse variational calculus, dissipative Lagrangians.

**Advanced Hamiltonian mechanics**

Hamilton's principle of least action. Poisson brackets representation of Hamiltonian mechanics, Liouville's theorem. Canonical transformations. Hamilton-Jacobi theory. Action-angle variables. Comparison of Lagrangian and Hamiltonian mechanics.

**Lagrangian and Hamiltonian mechanics in the continua***

Lagrangian and Hamiltonian density formulations. Linear elastic solids, stress and strain tensors. Ideal fluid dynamics. Viscous fluid dynamics.

**Relativistic mechanics**

Lorentz transformation, geometry of space-time, relativistic kinematics, the Extended Lagrangian and Hamiltonian formalisms. Elements of General Relativity.

**Transition to quantum physics**

Correspondence between classical and quantal physics. Connections to Heisenberg's matrix mechanics, Schrodinger's wave mechanics, and the Lagrangian in quantum mechanics.

* Reading assignments.