Normally we work with "nice" potentials

However, there are some systems that depend on the path taken.

The simplest is the damped harmonic oscillator

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.

Circles

Ellipses

The last one is cheerfully named the **squircle**.

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.

and ending at j in exactly n moves. From the binomial theorem:

With some approximations:

Thus a random walk is recurrent in 1D.

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

Thus a random walk is recurrent in 2D

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!