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Mass Parabolas

An atomic nucleus consists of protons and neutrons (collectively referred to as nucleons) bound together through the strong nuclear force. Models for the nuclear binding energy were introduced in a couple of previous posts on this blog.

Direct linear least squares fitting of an ellipse

Fitting a set of data points in the $xy$ plane to an ellipse is a suprisingly common problem in image recognition and analysis. In principle, the problem is one that is open to a linear least squares solution, since the general equation of any conic section can be written $$F(x, y) = ax^2 + bxy + cy^2 + dx + ey + f = 0,$$ which is linear in its parameters $a$, $b$, $c$, $d$, $e$ and $f$. The polynomial $F(x,y)$ is called the algebraic distance of any point $(x, y)$ from the conic (and is zero if $(x, y)$ happens to lie on the conic).

Chaotic Balls

Inspired by this recent Numberphile video, here is a demonstration of chaos in a simple dynamical system: two balls, with near-identical starting conditions, bounce around elastically off a circular wall. After a short time, the balls' trajectories diverge completely.

Measurements of the electron charge over time

In his 1986 book, Surely You're Joking, Mr. Feynman! physicist Richard Feynman writes:

Iceberg dynamics

Prompted by this tweet and campaign for icebergs to be depicted in their most stable equilibrium orientation, here is a Python script modelling the dynamics of a two-dimensional iceberg which starts in an arbitrary orientation and position and relaxes under gravitational and buoyant forces to its most stable configuration. A cork floats "on its side": with its longest axis parallel to the water's surface (it doesn't bob around with its longest axis vertical), and an iceberg does the same.