AI 中文总结
研究关于无穷多个粒子的“金鱼”方程，核心方法是通过特定变换求解，主要贡献是探讨将该方程有限粒子解扩展到无穷粒子情况的可能性，虽有难点但为后续研究提供方向。
AI 中文摘要
“金鱼”方程因形式优美而得名，是由\(N\)个非线性常微分方程组成的系统，\(N\)为任意整数。通过一种涉及多项式系数集到零点集转换的变换将其转化为自由运动，可非常简单地精确求解。本文旨在探讨将此解法扩展到\(N\)为无穷的情况，主要难点在于从多项式过渡到整函数，另一种使用非标准分析的方法留待未来研究。
英文摘要
The ``goldfish'' equations, so named because of their striking beauty, are a system of $N$ nonlinear ODE's, where $N$ is an arbitrary integer. They can be solved exactly in a very simple manner, by transforming them to free motion using a transformation involving the transition from the set of {\em coefficients\/} of a polynomial to that of its {\em zeroes}. This paper aims to explore the possibility of extending this solution to the case in which $N$ is infinite. The main difficulty involves the transition from polynomials to entire functions. Another approach using non-standard analysis, is left to future work.
Comments14 pages, 1 figure
