Non-integrability of the generalized spring-pendulum problem

We investigate a generalization of the three-dimensional spring-pendulum system. The problem depends on two real parameters (k, a), where k is the Young modulus of the spring and a describes the nonlinearity of elastic forces. We show that this system is not integrable when k ≠ −a. We carefully investigated the case k = −a when the necessary condition for integrability given by the Morales-Ruiz–Ramis theory is satisfied. We discuss an application of the higher order variational equations for proving the non-integrability in this case.

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