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Group theoretic methods for approximate invariants and Lagrangians for some classes of $$y^{\prime\prime} + F(t)y^{\prime}+y=f(y,y^{\prime})$$. (English) Zbl 1116.34318
Summary: Some recent results on the Lie symmetry generators of equations with a small parameter and the relationship between symmetries and conservation laws for such equations are used to construct first integrals and Lagrangians for autonomous weakly non-linear systems, $$y^{\prime\prime} + F(t)y^{\prime}+y=f(y,y^{\prime})$$. An adaptation of a theorem that provides the point symmetry generators that leave the invariant functional involving a Lagrangian for such equations is presented. A detailed example to illustrate the method is given (and other examples are discussed). The (approximate) symmetry generators, invariants and Lagrangians maintain the perturbation order of the ‘small parameter’ stipulated in the equation – first order in this case.

##### MSC:
 34C14 Symmetries, invariants of ordinary differential equations 70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
##### Keywords:
Approximate symmetries; First integrals; Invariants; Lagrangians
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