Week 5 Notes: Higher Dimensional Dynamical Systems
We continue our
study of dynamical systems by studying systems with additional degrees
of freedom.
Keplerian Orbits
Johannes Kepler empirically characterized the orbits of plants using
the following three laws:
- Each
planet moves on a closed elliptical orbit in a plane containing the
Sun, with the Sun at one focus of the ellipse.
- The
area swept out per unit time by a line drawn between a planet and the
Sun is a constant throughout the planet's orbit.
- The
ratio of the cube of the semimajor axis—half the length along the long
axis of the ellipse—of a planet's orbit to the square of the orbital
period is the same for all
planets.
Use the Central Force Model to test the validity of
these laws for a particle attracted to the origin by an inverse square
law. Modify this model to test other central force laws.
Related Models
The following differential equation models will be discussed in class.
- Coupled Oscillators
- Coupled Oscillator Modes
- Wilberforce Pendulum
- Kepler Orbits
- Classical Helium
- Oscillator Chain (advanced)
Additional models may be be posted for self-study.
Credits:
The Central Force Model was created by Wolfgang
Christian using the Easy Java Simulations (EJS) version 4.1 authoring
and modeling tool. You can examine and modify a compiled EJS
model if you run the model (double click on the model's jar file),
right-click within a plot, and select "Open Ejs Model" from the pop-up
menu. You must, of course, have EJS installed on your
computer.
Information about Ejs is available at: <http://www.um.es/fem/Ejs/>
and in the OSP comPADRE collection <http://www.compadre.org/OSP/>.