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Some variational principles associated with ODEs of maximal symmetry. Part 1: Equations in canonical form

Ndogmo J.-C.

Journal of Applied Analysis

2018

Vol. 24, nr 1

17--26

Variational and divergence symmetries are studied in this paper for linear equations of maximal symmetry in canonical form, and the associated first integrals are given in explicit form. All the main results obtained are formulated as theorems or conjectures for equations of a general order. Some of these results apply to linear equations of a general form and of arbitrary orders or having a symmetry algebra of arbitrary dimension.

Some variational principles associated with ODEs of maximal symmetry. Part 2: The general case

Ndogmo J.-C.

Journal of Applied Analysis

2018

Vol. 24, nr 2

175--183

Variational and divergence symmetries are studied in this paper for the whole class of linear and nonlinear equations of maximal symmetry, and the associated first integrals are given in explicit form. All the main results obtained are formulated as theorems or conjectures for equations of a general order. Adiscussion of the existence of variational symmetries with respect to a different Lagrangian, which turns out to be the most common and most readily available one, is also carried out. This leads to significantly different results when compared with the former case of the transformed Lagrangian. The latter analysis also gives rise to more general results concerning the variational symmetry algebra of any linear or nonlinear equations.

Symbolic computation of variational symmetries in optimal control

Gouveia P. D., Torres D. F., Rocha E. A.

Control and Cybernetics

2006

Vol. 35, no 4

831-849

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 a sub-Riemannian nilpotent problem (2,3,5,8).