Spiral motion in classical mechanics

Pages: 191 - 197, Issue Special, September 2009

 Abstract        Full Text (135.7K)              

Jamie Cruz - Departamento de Ciencias Básicas, UAM-Azc., Av. San Pablo 180, Col. Reynosa, México D. F. 02200, Mexico (email)
Miguel Gutiérrez - San Antonio 64, Col. Las Fuentes, Zapopan, Jalisco, 45070, Mexico (email)

Abstract: We present various models in classical mechanics which exhibit 'exotic' orbits. We give an example of a smooth $|\mathbf{r}|$-independent potential
$V$ in dimension three, which exhibits an orbit that spirals as time goes to infinity. This kind of orbits cannot occur for this class of potentials in dimension two [4] or, see below, if ${Cr}=\{\omega\in S^{n-1}:\nabla V(\omega)=0\}$, $n\geq 3$, is totally disconnected. In addition, for each $\mu>2$ we give an example of a potential of the form $V(r,\theta)=O(r^{-\mu})$, in two dimensions, which is not radially symmetric and has a zero-energy orbit that escapes towards infinity in spirals. Zero energy orbits escaping towards infinity in spirals cannot occur for radial potentials with the same rate of decay.

Keywords:  Newton's equation, spiral motion, angular potentials
Mathematics Subject Classification:  Primary: 34C05, 37E35; Secondary: 70F99

Received: October 2008;      Revised: February 2009;      Published: September 2009.