Vector field decomposition and first integrals with applications to non-linear systems

**Authors:**Scholes, Michael Timothy**Date:**2012-08-20**Subjects:**Vector fields. , Integrals , Nonlinear systems.**Type:**Thesis**Identifier:**uj:2821 , http://hdl.handle.net/10210/6258**Description:**M.Sc. , Roels [1] showed that on a two dimensional symplectic manifold, an arbitrary vector field can be locally decomposed into the sum of a gradient vector field and a Hamilton vector field. The Roels decomposition was extended to be applicable to compact even dimensional manifolds by Mendes and Duarte [2]. Some of the limitations of local decomposition are overcome by incorporating modern work on Hodge decomposition. This leads to a theorem which, in some cases, allows an arbitrary vector field on an even m-dimensional non-compact manifold to be decomposed into one gradient vector field and up to m-1 Hamiltonian vector fields. The method of decomposition is condensed into an algorithm which can be implemented using computer algebra. This decomposition is then applied to chaotic vector fields on non-compact manifolds [3]. This extended Roels decomposition is also compared to Helmholz decomposition in R 3 . The thesis shows how Legendre polynomials can be used to simplify the Helmholz decomposition in non-trivial cases. Finally, integral preserving iterators for both autonomous and non-autonomous first integrals are discussed [4]. The Hamilton vector fields which result from Roels' decomposition have their Hamiltonians as first integrals.**Full Text:**

**Authors:**Scholes, Michael Timothy**Date:**2012-08-20**Subjects:**Vector fields. , Integrals , Nonlinear systems.**Type:**Thesis**Identifier:**uj:2821 , http://hdl.handle.net/10210/6258**Description:**M.Sc. , Roels [1] showed that on a two dimensional symplectic manifold, an arbitrary vector field can be locally decomposed into the sum of a gradient vector field and a Hamilton vector field. The Roels decomposition was extended to be applicable to compact even dimensional manifolds by Mendes and Duarte [2]. Some of the limitations of local decomposition are overcome by incorporating modern work on Hodge decomposition. This leads to a theorem which, in some cases, allows an arbitrary vector field on an even m-dimensional non-compact manifold to be decomposed into one gradient vector field and up to m-1 Hamiltonian vector fields. The method of decomposition is condensed into an algorithm which can be implemented using computer algebra. This decomposition is then applied to chaotic vector fields on non-compact manifolds [3]. This extended Roels decomposition is also compared to Helmholz decomposition in R 3 . The thesis shows how Legendre polynomials can be used to simplify the Helmholz decomposition in non-trivial cases. Finally, integral preserving iterators for both autonomous and non-autonomous first integrals are discussed [4]. The Hamilton vector fields which result from Roels' decomposition have their Hamiltonians as first integrals.**Full Text:**

Vector product and an integrable dynamical system

- Steeb, Willi-Hans, Tanski, Igor, Hardy, Yorick

**Authors:**Steeb, Willi-Hans , Tanski, Igor , Hardy, Yorick**Date:**2011**Subjects:**Vector product , Nambu mechanics , Differential equations , Integrals**Type:**Article**Identifier:**uj:5808 , ISSN 0253-6102 , http://hdl.handle.net/10210/7816**Description:**We study an autonomous system of first order ordinary differential equations based on the vector product. We show that the system is completely integrable by constructing the first integrals. The connection with Nambu mechanics is established. The extension to higher dimensions is also discussed.**Full Text:**

**Authors:**Steeb, Willi-Hans , Tanski, Igor , Hardy, Yorick**Date:**2011**Subjects:**Vector product , Nambu mechanics , Differential equations , Integrals**Type:**Article**Identifier:**uj:5808 , ISSN 0253-6102 , http://hdl.handle.net/10210/7816**Description:**We study an autonomous system of first order ordinary differential equations based on the vector product. We show that the system is completely integrable by constructing the first integrals. The connection with Nambu mechanics is established. The extension to higher dimensions is also discussed.**Full Text:**

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