Department of Physics

Wake Forest University

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


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 touched 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 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).


  1. I.  Introduction to 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.

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. 

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.


Problem assignment 3.

Exit Exam

Grading policy

There are three HW assignments and one “exit” exam in this class.  Each part will present a specific number of points that you can earn toward a total of 400 points.  Roughly the breakdown will be 100 points on each parts. Grades are assigned based on the total number of points accumulated by the end of the class. 

Typically grades are assigned as follows:

350 - 400 points    - A range

300 - 350 points    +/- B range

250 - 300 points    +/- C range

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.

IMPORTANT: Over the course of the lectures you will have worked 30 problems and taken one final test.  It is important to keep up as we go.    


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. 


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 pretty important.

The text for the course will be announced at the beginning of the first class. if it is not already in the bookstore.

Office Hours

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

214 Olin Physical Laboratory
Reynolda Campus
Wake Forest University
Winston-Salem NC 27109

ph: 336 727 1804