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First integrals for two linearly coupled nonlinear Duffing oscillators. (English) Zbl 1210.34055
Summary: We investigate Noether and partial Noether operators of point type corresponding to a Lagrangian and a partial Lagrangian for a system of two linearly coupled nonlinear Duffing oscillators. Then, the first integrals with respect to Noether and partial Noether operators of point type are obtained explicitly by utilizing Noether and partial Noether theorems for the system under consideration. Moreover, if the partial Euler-Lagrange equations are independent of the derivatives, then the partial Noether operators become Noether point symmetry generators for such equations. The difference arises in the gauge terms due to Lagrangians being different for respective approaches. This study points to new ways of constructing first integrals for nonlinear equations without regard to a Lagrangian. We illustrate it here for nonlinear Duffing oscillators.
MSC:
34C15Nonlinear oscillations, coupled oscillators (ODE)
34C14Symmetries, invariants (ODE)
34A05Methods of solution of ODE
70H33Symmetries and conservation laws, reverse symmetries, invariant manifolds, etc.
70H03Lagrange’s equations