×

Time-independent invariants of motion for the quadratic system. (English) Zbl 0822.34008

Summary: A Hamiltonian method is developed to obtain first integrals of the form \(PQ^ \mu\) and \(PQ^ \mu R^ \nu\) for a general system of two- dimensional autonomous ordinary differential equations with quadratic terms, where \(P\), \(Q\) and \(R\) are linear or quadratic polynomials, and \(\mu\), \(\nu\) are two real numbers. It is found that there is an intimate relationship between the polynomials and the equilibrium points of the system, which is useful for determining the existence of periodic orbits and asymptotic behaviour.

MSC:

34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
70H25 Hamilton’s principle
34C25 Periodic solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
34A05 Explicit solutions, first integrals of ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI