Equivalent lagrangians and the solution of some classes of non-linear equations (1K)

The second-order ordinary differential equation

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, where μ ≠ 1 is linearizable(sl(3, R) algebra) via a point transformation if and only if n = μ or n = 1. We construct a quadratic Lagrangian , which determines the point transformation Q = F(t,q) and = G(t,q) that maps the Lagrangian to the simple completely integrable Lagrangian . For n = 4μ − 3 the equation admits the sl(2, R) algebra. In this case we again construct a quadratic Lagrangian and then obtain the corresponding point transformation that reduces the original Lagrangian to the representative Lagrangian . For both cases, sl(2,R) and sl(3,R), we obtain complete solutions (cf. 1,2]).