particle moving on the astroid at constant frequency

In parametric Cartesian equations, the astroid can be represented by

$x=a\cos^{3}\omega t,\quad y=a\sin^{3}\omega t,$

where $a>0$ is a known constant, $\omega>0$ is the constant angular frequency, and $t\in[0,\infty)$ is the time parameter. Thus the position vector of a particle, moving over the astroid, is

$\mathbf{r}=a\cos^{3}\omega t\,\mathbf{i}+a\sin^{3}\omega t\,\mathbf{j},$

and its velocity

$\mathbf{v}=-3a\omega\sin\omega t\cos^{2}\omega t\,\mathbf{i}+3a\omega\sin^{2}% \omega t\cos\omega t\,\mathbf{j},$

where $\{\mathbf{i},\mathbf{j}\}$ is a reference basis. Hence for the particle speed we have

$v=3a\omega\sin\omega t\cos\omega t.$

From the last two equations we get the tangent vector

$\mathbf{T}=-\sin\omega t\,\mathbf{i}+\cos\omega t\,\mathbf{j},$

and by using the well known formula ¹¹By applying the chain rule, $\bigg{\|}\frac{d\mathbf{T}}{dt}\bigg{\|}=\bigg{\|}\frac{d\mathbf{T}}{ds}\bigg{% \|}\bigg{|}\frac{ds}{dt}\bigg{|}=\bigg{\|}\frac{\mathbf{N}}{\rho}\bigg{\|}v=% \frac{v}{\rho},$ by Frenet-Serret. $\mathbf{N}$ is the normal vector.

$\bigg{\|}\frac{d\mathbf{T}}{dt}\bigg{\|}=\frac{v}{\rho},$

$\rho>0$ being the radius of curvature at any instant $t$ , we arrive to the useful equation

$v=\omega\rho.$

Title	particle moving on the astroid at constant frequency
Canonical name	ParticleMovingOnTheAstroidAtConstantFrequency
Date of creation	2013-03-22 17:14:09
Last modified on	2013-03-22 17:14:09
Owner	perucho (2192)
Last modified by	perucho (2192)
Numerical id	9
Author	perucho (2192)
Entry type	Topic
Classification	msc 70B05