Sat Jan 23 20:27:19 EST 2016

Hamiltonian signals

- Chaotic
- Simplectic update through first order ODEs derived from Hamiltonian
- Also use Hamiltonian for re-normalization using constant energy constraint
- For oscillators: tune simulation parameters based on frequency estimator / regulator.

How to generate interesting systems?

Swinging Atwood's machine:

Pendulum's reactive centrifugal force counteracts the counterweight's
weight.  The force tends go get quite high corresponding to the high
angular velocity (slingshot), and a smooth "bounce" on the

The ODEs have singularities for points passing through r=0.  

I wonder if it is possible to modify the geometry to remove those

E.g. instead of the 1/r^2 in the Hamiltonian, to use a 1/x+r^2

Phase-space is 4D, but due to energy preservation, the points are
constrained on a 3D manifold.

For the 3-body problem, as e.g. in these two fixed masses:

It is possible to guarantee boundedness based on total energy.
Because the singularities are points, the chance that a randomly
chosen initial condition passes through a singularity is zero.
However, due to finite approximation it is not.  Regularization is
probably a good idea.  ( Regularize as a bounce?  Or smooth out the
hamiltonian. )

For integrated simulation, the singularity actually doesn't matter as
the infinite fource only happens for an instance.  What actually
happens when a point passes through the singularity?

Large-scale n-body: