On an example of a system of differential equations that are integrated in Abelian functions
| Autor: | L A Sevastianov, M D Malykh |
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| Rok vydání: | 2017 |
| Předmět: |
History
Pure mathematics Dynamical systems theory Differential equation 010102 general mathematics Symbolic computation Dynamical system 01 natural sciences 010305 fluids & plasmas Computer Science Applications Education 0103 physical sciences Order (group theory) 0101 mathematics Abelian group Hyperelliptic curve Analytic function Mathematics |
| Zdroj: | Journal of Physics: Conference Series. 937:012027 |
| ISSN: | 1742-6596 1742-6588 |
| DOI: | 10.1088/1742-6596/937/1/012027 |
| Popis: | The short review of the theory of Abelian functions and its applications in mechanics and analytical theory of differential equations is given. We think that Abelian functions are the natural generalization of commonly used functions because if the general solution of the 2nd order differential equation depends algebraically on the constants of integration, then integrating this equation does not lead out of the realm of commonly used functions complemented by the Abelian functions (Painleve theorem). We present a relatively simple example of a dynamical system that is integrated in Abelian integrals by "pairing" two copies of a hyperelliptic curve. Unfortunately, initially simple formulas unfold into very long ones. Apparently the theory of Abelian functions hasn't been finished in the last century because without computer algebra systems it was impossible to complete the calculations to the end. All calculations presented in our report are performed in Sage. |
| Databáze: | OpenAIRE |
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