Differential equation of line tangent to caustics - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T10:44:19Z http://mathoverflow.net/feeds/question/35017 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/35017/differential-equation-of-line-tangent-to-caustics Differential equation of line tangent to caustics A B 2010-08-09T15:31:23Z 2011-01-23T12:47:25Z <p>This problem (or rather, statement that I cannot understand) has arisen in a paper I have been reading <a href="http://arxiv.org/abs/math-ph/0512049" rel="nofollow">"Geometry of Integrable Billiards and Pencils of Quadrics"</a> by Dragovic and Radnovic. I'd be most grateful for any explanations of it (it may be a simple fact, but I'm not sure).</p> <p>Let $\Omega \subset \mathbb{R}^d$ be a bounded domain such that its boundary $\partial \Omega$ lies in the union of several quadrics from the (confocal) family <code>$\mathcal{Q}_{\lambda}: Q_{\lambda}(x)=1$</code> where $$Q_{\lambda}(x)=\sum_{i=1}^{d}\frac{x_{i}^2}{a_{i}-\lambda}.$$ Then in elliptic coordinates, $\Omega$ is given by: $$\beta_{1}'\leq\lambda_{1}\leq\beta_{1}'', \ldots, \beta_{d}'\leq\lambda_{d}\leq\beta_{d}''$$ where $a_{s+1}\leq \beta_{s}'\leq\beta_{s}''\leq a_{s}$ for $1\leq s \leq d-1$ and $- \infty &lt; \beta_{d}'&lt;\beta_{d}''\leq a_{d}.$</p> <p>Define $P(x):= (a_1 -x)\ldots(a_d -x)(\alpha_{1} -x)\ldots(\alpha_{d} - x).$</p> <p>Now, we consider a billiard system inside $\Omega$ with caustics $\mathcal{Q}_{\alpha_1}, \ldots, \mathcal{Q}_{\alpha_d-1}.$ </p> <p>Why does the system of equations: $$\sum_{s=1}^{d}\frac{d\lambda_s}{\sqrt{P(\lambda_s)}}=0, \sum_{s=1}^{d}\frac{\lambda_{s}d\lambda_{s}}{\sqrt{P(\lambda_s)}}=0, \ldots, \sum_{s=1}^{d}\frac{\lambda_{s}^{d-2}d\lambda_{s}}{\sqrt{P(\lambda_s)}}=0,$$</p> <p>(which are apparently due to Jacobi and Darboux - I'd appreciate a modern reference because the only version I can find is scanned page-by-page in German), where $\sqrt{P(\lambda_s)}$ is taken with the same sign in all expressions, represent a system of differential equations of a line tangent to all the caustics $\mathcal{Q}_{\alpha_1}, \ldots, \mathcal{Q}_{\alpha_d-1}$? Moreover, why does: $$\sum_{s=1}^{d}\frac{\lambda_{s}^{d-1}d\lambda_{s}}{\sqrt{P(\lambda_s)}}=2dl$$ where $dl$ is an element of the" line length?</p> <p>I found similar looking equations on page 4 of <a href="http://www.math.univ-montp2.fr/~rs/geohypfi.ps" rel="nofollow">another paper</a> (by Buser and Silhol), but cannot understand them either.</p>