at Oxford University, Oct-Dec 2010.
Lecture times: Monday 12, Wednesday 12.
Newton's laws. Free and forced linear oscillations. Simple oscillatory systems with two degrees of freedom, natural frequencies. Two dimensional motion, projectiles. Use of polar coordinates, circular motion. Central forces, differential equation for the particle path. Inverse square law, planetary orbits. Energy and potential for one dimensional motion. Equivalent ideas for central force problems and three dimensional problems with axial symmetry. Examples of stability and instability in physical situations, via linearized equations. Simple ideas of phase space, stable and unstable fixed points, periodic orbits. Informal introduction to chaos.
David Acheson From Calculus to Chaos; an
Dynamics OUP 1997.
D W Jordan & P Smith Mathematical Techniques (3rd edn) OUP 2002
M Lunn A First Course in Mechanics OUP 1991
(Sheet n is usually based on
the lectures given in week n.)
The following programs, written in QuickBasic to accompany the course textbook From Calculus to Chaos,
may not run on the most modern versions of Windows; for help please click here
motion of planets and
The gravitational three-body problem
An elementary example of chaos
Chaos and catastrophe for a vibrated pendulum
Some other similar
software can be found by
Here is a ready-to-run sample:
The particle starts at x=1, y=0 with initial velocity v in the y-direction. The force is
c multiplied by (r to the power p) + d multiplied by (r to the power q)
(thus for an
inverse-square law set, say, c = 1 , p = -2 , d = 0 , q = 1
The user enters the Window Size, the constants c,p,d,q , the initial velocity v, and the time step h (for which .01 is usually a fair choice). The equations are "solved" by the Runge-Kutta numerical method, and any "results" can only be trusted if essentially the same motion occurs, over a given time interval, when the time step is doubled or halved.
To EXIT this program, press CTRL-BREAK.
The following programs illustrate three of the phase plane representations that we encounter in the course:
Sheet 7 Question 3
Sheet 8 Question 1