On the first integrals in the center problem (Q388129)
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scientific article; zbMATH DE number 6239376
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On the first integrals in the center problem |
scientific article; zbMATH DE number 6239376 |
Statements
On the first integrals in the center problem (English)
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19 December 2013
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analytic differential systems
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center problem
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first integral
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0.9217352
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0.89325285
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0.88681924
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0.87555754
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0.87330794
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The following theorem is the main result of the article.NEWLINENEWLINE Theorem. Let \(H(x,y;\varepsilon)\) be a first integral of the system NEWLINE\[NEWLINE\dot x= P(x,y;\varepsilon), \, \dot y= Q(x,y;\varepsilon).NEWLINE\]NEWLINE Assume that the first integral has a critical value for \(\varepsilon= \varepsilon_0\), namely \(H(x, y;\varepsilon_0)\) becomes a constant. Let \(k\) be the smallest integer number such that NEWLINE\[NEWLINE{\partial^k H(x,y; \varepsilon)\over \partial\varepsilon^k}\Biggl|_{\varepsilon= \varepsilon_0}\tag{1}NEWLINE\]NEWLINE is not constant. Then (1) is a first integral of the system NEWLINE\[NEWLINE\dot x= P(x,y; \varepsilon_0), \, \dot y= Q(x,y; \varepsilon_0).NEWLINE\]
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