May 01, 2018 / by George Council / In news
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.
Read moreApril 15, 2018 / by Matthew Kvalheim / In news
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.
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