May 01, 2018 / / In news

Affine Nonholonomic Systems

Lagrangian mechanics is an important tool to understanding mechanical systems, but often must incoporate nonholonomic constraints. When the constraints are affine, energy may not be conserved. By using a time dependent change of coordinates, a related conserved quantity arises that faciliates analysis of the system's dynamics.

April 15, 2018 / / In news

Differentiation under the integral sign

In calculus, Leibniz's rule for differentiation under the integral sign states that, modulo precise regularity assumptions, $$\frac{d}{dx} \int_{a(x)}^{b(x)} f(x,t)\, dt = f(x,b(x))b'(x) - f(x,a(x))a'(x) + \int_{a(x)}^{b(x)} \frac{\partial}{\partial x}f(x,t)\, dt.$$ There is a nice generalization of this result to the case of integrating a time-dependent differential $k$-form over a time-varying $k$-submanifold with boundary.

March 20, 2018 / / In booklist

The Twisting Tennis Racket

This paper describes, analyzes, and explains a novel twisting phenomenon which occurs in a triaxial rigid body (such as a tennis racket) when it is rotating about an axis initially near its unstable intermediate principal axis.