Yoshida, Haruo Necessary condition for the existence of algebraic first integrals. I: Kowalevski’s exponents. (English) Zbl 0556.70014 Celestial Mech. 31, 363-379 (1983). 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. Reviewer: Yoshihide Kozai (Tokyo) Cited in 5 ReviewsCited in 70 Documents MSC: 70H05 Hamilton’s equations 70F15 Celestial mechanics Keywords:algebraic first integral; Kowalevski’s exponents; similarity invariance; weighted homogeneous first integral; weighted degree Citations:Zbl 0556.70015 PDFBibTeX XMLCite \textit{H. Yoshida}, Celest. Mech. 31, 363--379 (1983; Zbl 0556.70014) Full Text: DOI References: [1] Bountis, T., Segur, H., and Vivaldi, F.: 1982,Phys. Rev. A25, 1257. · doi:10.1103/PhysRevA.25.1257 [2] Bruns, H.: 1887,Acta Math. 11, 25. · doi:10.1007/BF02612319 [3] Forsyth, A. R.: 1900,Theory of Differential Equations, Chap. 17, Cambridge University Press. [4] Hall, L. S.: 1983,Physica 8D, 90. [5] Kowaievski, S.: 1889,Acta Math. 12, 177. · JFM 21.0935.01 · doi:10.1007/BF02592182 [6] Kowalevski, S.: 1890,Acta Math.,14, 81. · JFM 22.0921.02 · doi:10.1007/BF02413316 [7] Liapounov, A. M.: 1896, inCollected Works, Tom. 1, p. 402 (in Russian). [8] Olshanetsky, M. A. and Perelomov, A. M.: 1981,Phys. Rep. 71, 313. · doi:10.1016/0370-1573(81)90023-5 [9] Poincaré, H.: 1892,Les méthods nouvelles de la mécanique céleste, Tom. 1. [10] Whittaker, E. T.: 1936,A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Chap. 14, Cambridge University Press. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.