# Math 21a Summer 2020

## Multivariable Calculus

# Double pendulum

An older Javascript animation (where your browser actually integrates the differential equations). Here, Mathematica was solving the system numerically with NDSolve. The big open question here is "Is this system chaotic on a set of positive measure". The common believe is yet, but we are mathematically just in an infantile stage and have currently no idea how to prove this mathematically. The flow is a differential equation on a three dimensional energy surface in a four dimensional manifold (the product of two two dimensional cylinders). While chaos can not happen in two dimensions by Poincaré-Bendixon, this is possible in three dimensions. If dA(t) is the 4x4 Jacobean matrix of the time t map, the conjecture is that some matrix entries grow exponentially fast on a set of initial conditions which have positive volume. This is a problem belongs to**smooth ergodic theory**which is a field of mathematics which mixes

**differential equations**,

**linear algebra**,

**calculus**and

**probability theory**. The problem is equivalent to the statement

**"Does the system have positive metric entropy?"**. The entropy notion is a dynamical spin-off of the Shannon entropy S we have seen in class. "Chaos" essentially means that the entropy of finite partitions grows exponentially with time. It means that the system actually behaves like a true random number generator in some parts of the phase space. I worked on this in my thesis. To the background music: "Liberators" from album "Vengeance" (2015) by Epic Score.