Irrelevant Topics in Physics

Volume I

Topics:

Dissipative Lagrangians

Physics

Supershapes

Geometry

Random Walks On Regular Lattices

Physics

Dissipative Lagrangians

Forbidden couplings in phase space

Normally we work with "nice" potentials

However, there are some systems that depend on the path taken. The simplest is the damped harmonic oscillator

Forbidden couplings in phase space


One way to formalize this system was done by Morse and Feshbach

When we apply Euler's equations of motion

We get the following differential equations

This system conserves energy! For every bit lost by the original damped harmonic oscillator, it's gained by it's mirror.
  • Lest you think that dissipative Lagrangians can only be constructed with a "dual", one can also use
  • The exponential multiplier does not induce any extraneous solutions.
  • Euler's equations of motion would produce the original differential equation.

Supershapes

almost squaring the circle

Not so super


Circles:

Ellipses:

Superellipses

The last one really is cheerfully named the squircle.

Superellipses

Generalizable in the extreme. A superellipse, also known as a Lame curve after Gabriel Lame, is a closed curve resembling the ellipse, retaining the geometric features of semi-major axis and semi-minor axis, and symmetry about them, but a different overall shape.

Random Walks on Regular Lattices

or the best dimension to get drunk

A random walk on a lattice takes an integer step with equal probability in one of the possible directions.

A walk is recurrent if the probability of returning to the starting point approaches one as the number of steps go to infinity.

One-Dimension

Let

be the number of possible ways from starting at i and ending at j in exactly n moves. From the binomial theorem:

With some approximations:

Thus a random walk is recurrent in 1D.

Two-Dimensions

Xn+ and Xn- are simple symmetric independent random walks.

Thus a random walk is recurrent in 2D

Three-Dimensions

Thus a random walk in 3D is not recurrent !

Do not get drunk in 3D!

If you lose your keys in space, odds are you will never find them again...

Thankfully, Earth has a 2D periodic surface!