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- Computing ODE symmetries as abnormal variational symmetriesPublication . Gouveia, Paulo D.F.; Torres, Delfim F.M.We give a new computational method to obtain symmetries of ordinary differential equations. The proposed approach appears as an extension of a recent algorithm to compute variational symmetries of optimal control problems [P.D.F. Gouveia, D.F.M. Torres, Automatic computation of conservation laws in the calculus of variations and optimal control, Comput. Methods Appl. Math. 5 (4) (2005) 387-409], and is based on the resolution of a first order linear PDE that arises as a necessary and sufficient condition of invariance for abnormal optimal control problems. A computer algebra procedure is developed, which permits one to obtain ODE symmetries by the proposed method. Examples are given, and results compared with those obtained by previous available methods.
- Smarandache sequences: explorations and discoveries with a computer algebra systemPublication . Gouveia, Paulo D.F.; Torres, Delfim F.M.We study Smarandache sequences of numbers, and related problems, via a Computer Algebra System. Solutions are discovered, and some conjectures presented.
- Computation of conservation laws in optimal controlPublication . Gouveia, Paulo D.F.; Torres, Delfim F.M.Making use of a computer algebra system, we define computational tools to identify symmetries and conservation laws in optimal control.
- Computing ODE symmetries as abnormal variational symmetriesPublication . Gouveia, Paulo D.F.; Torres, Delfim F.M.We give a new computational method to obtain symmetries of ordinary differential equations. The proposed approach appears as an extension of a recent algorithm to compute variational symmetries of optimal control problems [P.D.F. Gouveia, D.F.M. Torres, Automatic computation of conservation laws in the calculus of variations and optimal control, Comput. Methods Appl. Math. 5 (4) (2005) 387 409], and is based on the resolution of a first order linear PDE that arises as a necessary and sufficient condition of invariance for abnormal optimal control problems. A computer algebra procedure is developed, which permits one to obtain ODE symmetries by the proposed method. Examples are given, and results compared with those obtained by previous available methods.
- Symbolic computation of variational symmetries in optimal controlPublication . Gouveia, Paulo D.F.; Torres, Delfim F.M.; Rocha, Eugénio A.M.We use a computer algebra system to compute, in an efficient way, optimal control variational symmetries up to a gauge term. The symmetries are then used to obtain families of Noether’s first integrals, possibly in the presence of nonconservative external forces. As an application, we obtain eight independent first integrals for the sub-Riemannian nilpotent problem (2, 3, 5, 8).