Summary
In this work we show that a number of well known nonlinear second order ODE appearing in theoretical physics provide the necessary condition for the minimum of the functional\(I = \int\limits_a^b {L(x,\mathop x\limits^{..} ,t) dt} \) with the Lagrangian\(L = ( - \lambda F(t)\frac{x}{{\mathop x\limits^{..} }})^\alpha \). Also we prove that those second-order differential equations may be viewied as conservation laws for the corresponding Euler-Lagrange equations that are the fourth-order ODE. Several special cases that have importance in physics, mechanics and optimal rod theory are studied in detail.
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Atanackovic, T.M., Vujanovic, B.D. & Baclic, B.S. A variational principle motivated by the optimal rod theory. Acta Mechanica 139, 57–71 (2000). https://doi.org/10.1007/BF01170182
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DOI: https://doi.org/10.1007/BF01170182