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Dynamics of Motion in a Central Potential: Deriving Kepler's Laws

Conserved Quantities

The equation of motion is:

http://galileoandeinstein.physics.virginia.edu/7010/CM_15_files/image019.png. ... Read more »

Category: Education | Views: 353 | Added by: farrel | Date: 2017-09-01 | Comments (0)

Keplerian Orbits

Michael Fowler ... Read more »

Category: Education | Views: 378 | Added by: farrel | Date: 2017-09-01 | Comments (0)

Mathematics for Orbits: Ellipses, Parabolas, Hyperbolas

Michael Fowler

Preliminaries: Conic Sections

 

Ellipses, parabolas and hyperbolas can all be generated by cutting a cone with a plane (see diagrams, from Wikimedia Commons). Taking the cone to be  and substituting the in that equation from the planar equation where is the vector perpendicular to the plane from the origin to the plane, gives a quadratic equation in  This translates into a quadratic equation in the plane -- take the line of intersection of the cutting plane with the  ... Read more »

Category: Education | Views: 391 | Added by: farrel | Date: 2017-09-01 | Comments (0)

Adiabatic Invariants and Action-Angle Variables

Michael Fowler

Adiabatic Invariants

Imagine a particle in one dimension oscillating back and forth in some potential. The potential doesn't have to be harmonic, but it must be such as to trap the particle, which is executing periodic motion with period. Now suppose we gradually change the potential, but keeping the particle trapped. That is, the potential depends on some parameter, which we change gradually, meaning over a time much greater than the time of oscillation: 

A crude demonstration is a simple pendulum with a string of variable length, for example (see figure) one hanging from a fixed support, but the string passing through a small loop that can be moved vertically to change the effective length.

If were fixed, the system would have constant energy  ... Read more »

Category: Education | Views: 363 | Added by: farrel | Date: 2017-09-01 | Comments (0)

The Hamilton-Jacobi Equation

Michael Fowler

Back to Configuration Space

We've established that the action, regarded as a function of its coordinate endpoints and time, satisfies

and at the same time so obeys the first-order differential equation

This is the Hamilton-Jacobi equation.

Notice that we're now back in configuration space!

For example, the Hamilton-Jacobi equation for the simple harmonic oscillator in one dimension is

(Notice that this has some resemblance to the Schrödinger equation for the same system.)

If the Hamiltonian has no explicit time dependence  becomes just  ... Read more »

Category: Education | Views: 374 | Added by: farrel | Date: 2017-09-01 | Comments (0)