Mathematical expressions for the motion of solid objects in space and time, can be formulated using a number of fundamental principles.  "Newtons Laws" are the best known and simplest of these.  They make direct reference to an applied force vector which results in the vector quantities of displacement, velocity and acceleration.  With this we may accurately predict in time, the motion of objects given that the applied forces and initial conditions are well defined and well behaved.   However, there are other basic and universal principles upon which a formal expression of motion might be derived.  These are principles of conservation, minimization, and symmetry.  They are based on the formal parameterization of time evolution in dynamical systems.   Analytical Mechanics / Dynamics seeks to formulate the prediction of motion using such principles, thereby extending the range of problems accessible to mathematical understanding.     

Why does the physicist need Analytical Mechanics / Dynamics?  All areas of modern physics research are touch by it.  It forms the basis of quantum mechanics formulations.  It provides the underpinnings of many astrophysical models, is used in the description of chaos, and forms a bridge to statistical mechanics through classical ensemble theory.  It gives us a more complete vision of how the symmetries of space and time are connected to the motion of objects.

Welcome to PHYS 337/637

This course will provide a mathematical introduction of to variational methods of mechanics (Lagrangian and Hamiltonian formulations) and the geometries of global behaviors. The course is highly theoretical developing an advanced formalism for dynamics. The 639 section will be assigned extra reading and homework assignments.

This course runs 1/2 a semester and is evaluated midterm (October).

Syllabus

I.  Variational Calculus

Variational calculus,

Introduction to Virtual work,

Maupertuis' principle and the action functional, 

Homework assignment 1.

II.  The Lagrangian

Generalized coordinates, constraints and minimum coordinate sets. 

Derivation of the Lagrangian

The resulting equations of motion.

Homework assignment 2. 

Exam 1

III.  The Hamiltonian

Derivation of Hamilton's equations

Conservation principles

Phase Space  

Homework assignment 3.

Exam 2

Grading policy

Grades are derived from the two exams and the three hw assignments in the proportions: hw - 30%, Exam 1 - 30%, Exam 2 - 30%.  HW will NOT be accepted late. Exams may be made up ONLY for excused absences.  

 

Attendance and Examination Policy

Attendance is required for the class.  Students are responsible for information covered in class including class notes, scheduling and hw discussions, changes in class topics etc.  Students must work independently on all assignments; including hw and exams.