Physics 43 (Dynamics) Home Page, Fall 2008

Physics 43: Dynamics


Welcome to the new semester!


Course Information

Course Catalog Description:

This course begins with the foundation of classical mechanics as formulated in Newton's Laws of Motion. We then use Hamilton's Principle of Least Action to arrive at an alternative formulation of mechanics in which the equations of motion are derived from energies rather than forces. This Lagrangian formulation has many virtues, among them a deeper insight into the connection between symmetries and conservation laws. From the Lagrangian formulation we will move to the Hamiltonian formulation and the discussion of dynamics in phase space, exploring various avenues for the transition from the classical to the quantum theory. We will study motion in a central force field, the derivation of Kepler's laws of planetary motion from Newton's law of gravity two-body collisions, and physics in non-inertial reference frames. Other topics may include the dynamics of driven, damped oscillators, and non-linear dynamics of chaotic systems. Three class hours per week. Requisite: Physics 27 or consent of the instructor.


Times and places:


Physics 27 or consent of the instructor

Course requirements

Statement of Intellectual Responsibility: particulars for this course

How to get the most from this class:



Required (I've confirmed that it is available at the Jeffrey Amherst bookstore): Additional useful references:

On mechanics in general: Math books:

Mathematica Tutorials

We will use Mathematica fairly extensively in the homework, to obtain numerical solutions to problems that are not analytically solvable and to simplify plotting of results. If you've never used Mathematica before, or haven't used it much, the tutorials will help you get started. They were written by Professor Emeritus Bob Hilborn and revised by Rebecca Erwin '02. If you download the file and save it to the desktop with a .nb suffix in the name, your computer will recognize it as a Mathematica notebook and will start up Mathematica automatically when you double-click on the icon, provided you have Mathematica installed. Mathematica is installed on lots of the college's public machines, including on the computers in the Physics Department computer lab. Alternately, you can pay the $140 or so to buy the student version.

Lecture Schedule
Week Notes Hmwk Other
1. September 1
Space, Time, and Newton's Laws / Vectors

Sept 2: Intro and Logistics / Space, time, and vectors

Scope of classical mechanics, and scope of the course. Space and time: the background manifold we'll use. Geometrical vectors: define vectors as quantities that transform under rotations as the displacement.

[Handouts: Student information sheet ]

Sept 4: More about vectors

Write general vectors in terms of orthonormal basis vectors. Einstein summation convention. In 2D, compare, contrast, and convert between Cartesian and polar basis vectors. Products of vectors and the geometrical interpretation: scalar and vector products. Levi-Civita symbol and Kronecker delta. Differentiation of vector functions: definition, versions of familiar calculus rules. Beware: basis vectors in non-cartesian coordinates have non-trivial derivatives!

[Handouts: Vector analysis with diagrams (pp. 7-11 of Stedman's Diagrammatic Methods in Group Theory)]

Read: Kibble, Chap. 1 and Appendix A

Problems: [Due Tuesday 9/9/08, 11:59 pm]
Chap. 1: 7, 8, 10
Appendix A: 7
Chap. 2: 3, 7
2. September 8
Vectors / Newton's Laws / Linear Motion

Sept 9: Vectors / Newton's Laws

Differentiation of vectors: position, velocity, and acceleration in polar coordinates. Diagrammatic argument that derivatives of radial and polar vectors "make sense." Interpretation of terms in expression for acceleration vector. Finite rotations are not vectors (infinitesimal rotations are).

Reference frames defined (inertial frames particularly important). Mass and force defined rather casually.

Newton I addresses natural state of motion, serves to define inertial frames. Newton II is the "workhorse." Newton III (strong form and weak form). Consequence of Newton III: conservation of linear momentum for composite systems. Failures of NIII (in relativity, in electrodynamics).

Linear motion: conservative forces and energy conservation.

Sept 11: Linear Motion

Conservative forces and conservation of energy: When F=F(x), can always find a first integral of motion (fn of x and v) which is constant in time. Show that T+V=E is constant in time. Show how the potential energy landscape can be used to give qualitative description of motion evey without explicit solution (stable and unstable equilibria, turning points).

Small oscillations about equilibrium: Taylor expand about equilibrium point. Only second order and higher terms in small displacement are nonzero. If equilibrium is stable, displacements may remain small for all time; not so if unstable. In stable equilibrium, leading term is usually quadratic: harmonic oscillator potential.

Harmonic oscillator potential: possible motions for SHO and inverted SHO potentials. Equation of motion. 2nd order linear ODE--general solution has two linearly independent terms and two constants of integration. Linear ODEs satisfy superposition principle. Show that superposition of solutions is also a solution for linear equations, and that it's not true in general for nonlinear equations. Guess solutions of exponential form, solve.

Read: Kibble, Chap. 2

Problems: Kibble, Chap. 2: 8, 9, 15, 20, 26, 30
3. September 15
Energy and Angular Momentum

Sept 16: Title

Sept 18: Title


4. September 22
Central Conservative Forces

Sept 23: Title

Sept 25: Title


5. September 29
Noninertial reference frames

Sept 30: Title

Oct 1: Exam 1 (7-9 pm)

Oct 2: Title


6. October 6
Potential theory

Oct 7: Title

Oct 9: Title


Exam practice: P43F04 exam 1
7. October 13
Two-body and Many-body problems

Oct 14: Fall Break

Oct 16: Title


8. October 20
Rigid body motion

Oct 21: Title

Oct 23: Title


9. October 27
Lagrangian mechanics

Oct 28: Title

Oct 30: Title


10. November 3
Small oscillations and normal models

Nov 4: Title

Nov 6: Title


11. November 10
Hamiltonian mechanics

Nov 11: Title

Nov 13: Title


Exam practice: P43F04 exam 2
12. November 17
Geometry of dynamical systems

Nov 18: Title

Nov 19 (evening): Exam 2

Nov 20: Title



13. December 1
Chaos in Hamiltonian systems

Dec 2: Title

Dec 4: Title


14. December 8
Geometric mechanics

Dec 9:

Dec 11: Reading period