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Necessary condition for the existence of algebraic first integrals. I: Kowalevski’s exponents. (English) Zbl 0556.70014

The author investigates singularities of the solutions which are characterized by a set of exponents called the Kowalevski’s exponents for a type of dynamical systems with a special invariance (similarity invariance) and proves the following theorems by introducing the notions of ”weighted homogeneous integrals” and ”weighted degree”:
Theorem 1. Let \(\Phi (x)=\text{const.}\) be a weighted homogeneous first integral of weighted degree \(M\) for the similarity invariant system and suppose that the elements of the vector \(\text{grad}\,\Phi(c)\) are finite and not identically zero for a fixed choice of the set \(c_ 1,...,c_ n\). Then \(\rho =M\) becomes a Kowalevski’s exponent.
Theorem 2. Let \(\Phi (x)=\text{const.}\) and \(\Phi '(x)=\text{const.}\) be two independent weighted homogeneous first integrals of the same weighted degree \(M\). Suppose that the two vectors \(\text{grad}\,\Phi\) (c) and \(\text{grad}\,\Phi'(c)\) are both finite, not identically zero, and that these two vectors are linearly independent for a fixed choice of \(c_ 1,...,c_ n\). Then \(\rho =M\) becomes a Kowalevski’s exponent with multiplicity at least 2.
Theorem 3. Let \(\Phi (p,q)=\text{const.}\) be a weighted homogeneous first integral of weighted degree \(M\) for the similarity invariant Hamiltonian system, where the weighted degree of the Hamiltonian is assumed to be \(h\). Suppose that for a choice of \(c_ 1,...,c_ n\) the vectors \(\text{grad}\,\Phi (c)\) and \(\text{grad}\,\Phi'(c)\) are both finite and not identically zero. Then \(\rho =h-1-M\) becomes a Kowalevski’s exponent.
Furthermore, let \(\Phi (q,p)=\text{const.}\) and \(\Phi'(q,p)=\text{const.}\) be two independent weighted homogeneous first integrals of the same weighted degree \(M\). If the two vectors \(\text{grad}\,\Phi(c)\) and \(\text{grad}\,\Phi'(c)\) are both finite, not identically zero, and are linearly independent, \(\rho =h-1-M\) becomes Kowalevski’s exponent with multiplicity at least 2.

MSC:

70H05 Hamilton’s equations
70F15 Celestial mechanics

Citations:

Zbl 0556.70015
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References:

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