Department of Physics

Wake Forest University

PHYS 337: Analytical Mechanics


Instructor: Dr. David Carroll

Class Location: 103 Olin Hall

Time: 12:30 - 1:45 T/TH

Ave. Out of Class Prep Time: 2 hours/class

Recitation: TBA

Office hours: Tuesday and Thursday, 10:00 - 11:00 (am) or by appointment

Office: 214 Olin Physical Laboratory, Reynolda Campus
ph: 336 727 1804


Prologue: 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 are among the oldest and most direct, making 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 energy and 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 the Analytical Mechanics / Dynamics formulation of the laws of motion?  The simple answer is that it extends the range of problems we can address.  Moreover, it supplies a perspective to motion in more complex systems that can touch upon all areas of modern physics research.  It forms the basis of quantum mechanics formulations.  It provides the underpinnings of many astrophysical models, it 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 and buried within it is a more sophisticated view of the fundamental aspects of physical determinism.

Welcome to PHYS 337/637

This course will provide a mathematical introduction 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 and is an excellent refresher for those preparing for the graduate qualifier. This course runs 1/2 a semester and is evaluated midterm (October).


The text varies from year to year in the course.  The text you used for the previous mechanics class will be important as a refresher for certain principles.  But most of this material will be dealt with in lectures - so attendance is important.  The text for the course will be announced at the beginning of the first class. if it is not already in the bookstore.


The course is divided up into a set of seven lectures.  We may not get to all seven of them - they take more than one class period each to give.  Within the lectures are embedded HW problems and helpful exercises.  These are to be turned in for credit.  The notes from the lectures will be provided.

Essential Topics

  1. I.  Introduction to Coordinates, Constraints, and Virtual work 

This section explores how to break a problem down, describe it in terms of generalized coordinates, constraints and minimum coordinate sets.

It introduces Maupertuis' principle and the action functional, then describes how to solve such a functional using Variational Calculus,


Problem assignment 1.

Quiz 1.

II.  The Lagrangian 

In this section we derive the Lagrangian to achieve the equations of motion. We will focus on the use of these techniques to solve simple problems of motion and use Lagrangian multipliers to solve for forces.

Problem assignment 2.

Quiz 2. 

III.  The Hamiltonian

This section transforms the Lagrangian into the Hamiltonian to achieve Hamilton’s equations of motion. We will observe conservation principles and describe our solutions in global phase space diagrams. This covers up through 1/2 of lecture 6.


Problem assignment 3.

Quiz 3.

Grading policy

The HW assignments and attendance are 10% of the grade. Three, short, in-class quizzes make up the remaining 90% of the grade. Each student is expected to schedule at least one meeting with me to discuss your progress in the class and your grading. This should take place well before the end of the class.   


Attendance and Examination Policy

Attendance is required for the class.  It is important to realize that not all topics covered will be found in the text.  Students must work independently on all assignments; including HW and exams. 

Structure of Exercises

The HW will be assigned problems that pertain to the material being covered in class as well as the material that has been covered previously - so they are comprehensive.  These assignments should take no more than an hour or two to complete for any given lecture.  The quizzes will have 5 problems on them.  They will be taken from modified parts of the HW and written so as not to make this too obvious.  The quizzes reflect GRE type questions and should help with preparing you for such exams.  They are taken in class.