Richard Schwartz, the Chancellor’s Professor of Mathematics, and Director of Undergraduate Studies in Mathematics at Brown University, spoke at the Math Colloquium Talk on Thursday.
Schwartz’s lecture centered around his research with outer billiards, which have major applications in determining whether it is possible for objects to have unbounded orbits. His overarching question regarded “asking if the solar system is stable”.
Outer billiards refers to a dynamical system, which is based on a convex shape in the plane. The field of outer billiards is different from the larger field of dynamical billiards in that outer billards deals with the sequence of moves outside a given shape rather than inside of it. The more general field of dynamical billiards refers to a system in which a particle changes between straight-line motion and reflections from a defined boundary.
Schwartz described how regular polygons produce a fractal-looking tiled structure when their orbits are mapped, but these orbits are always bounded. The term bounded here refers to the orbits being of a finite size.
Schwartz recently proved, however, that a kite’s orbit is unbounded, which essentially means that the kite’s “orbit goes out to infinity, but it does so in a complicated way”. This involves the orbit as “dense,” meaning that the object starts to move toward infinity, but then returns again and again to retrace its path.
Schwartz spoke about how the kite is one of only two shapes (the other being a half-disk) which has been shown to follow an unbounded orbit; all other shapes maintain periodic orbits like that of the Earth around the sun.
Consequently, Schwartz assured the crowd that there is no need to worry about planets shooting off towards infinity, given their shape. As Schwartz himself said during the lecture, “This is hard to say in words”, but is an interesting field of inquiry nonetheless.