Normal Forms and Integrability of ODE Systems
We consider a special case of the Euler–Poisson system describing the motion of a rigid body with a fixed point. It is the autonomous ODE system of sixth order with one parameter. Among the stationary points of the system we select two one-parameter families with resonance (0,0,λ,–λ,2λ,–2λ) of eigenvalues of the matrix of the linear part. For the stationary points, we compute the resonant normal form of the system using a program based on the MATHEMATICA package. Our results show that in cases of the existence of an additional first integral of the system its normal form is degenerate. So we assume that the integrability of a system can be checked through its normal form.
Unable to display preview. Download preview PDF.
- 1.Bruno, A.D.: The normal form of differential equations. Doklady Akad. Nauk. SSSR 157, 1276–1279 (1964) (in Russian); Soviet Math. Doklady 5, 1105–1108 (1964) (in English)Google Scholar
- 2.Bruno, A.D.: Analytical form of differential equations. Trudy Mosk. Mat. Obshch. 25, 119–262 (1971); 26, 199–239 (1972)(in Russian); Trans. Moscow Math. Soc. 25, 131–288 (1971); 26, 199–239 (1972) (in English)Google Scholar
- 6.Golubev, V.V.: Lectures on Integration of Equations of Motion of a Rigid Body Around a Fixed Point. Moscow, GITTL (1953) (in Russian)Google Scholar
- 9.Starzhinsky, V.M.: Applied Methods in Nonlinear Oscillation, ch. IX, Nauka, Moscow (1977) (in Russian)Google Scholar
- 10.Ziglin, S.L.: Branching solutions and nonexistence of integrals in the Hamiltonian mechanics I, II. Functional Analysis and its Applications 16(3), 30–41 (1982); 17(1), 8–23 (1983)Google Scholar