On the Non-Integrability of the Stark-Zeeman Hamiltonian System

Non-Integrability of the Generalized Two Fixed Centres Problem

“…Proceeding as in the proof of Lemma 2, we conclude that under our assumption equation (21) is solvable if and only if it has a solution of the form y = P/(z − 1) m+2 with P a polynomial of degree m. Following the method of [23], we make the change of variables y = Y /(z − 1) m+2 in (21) and compute the recurrence relation satisfied by the coefficients of a power series solution Y = u n z n at zero. The recurrence is:…”

“…In Section 6, we study the exceptional case a = −k and conclude that the MoralesRamis method yields no obstruction to integrability, whereas dynamical analysis seems to indicate that the system is not completely integrable. The Morales-Ramis theory was applied to study the integrability of many Hamiltonian systems, see examples in book [17] and in papers [19,20,21,22,23,24,25,26,27,28]. The differential Galois approach was used also for proving non-integrability of non-Hamiltonian systems, see [29,30,31].…”

Non-integrability of a class of Painlevé IV equations as Hamiltonian systems

“…If (u) is meromorphic function of u in U and satisfies (7), then (u) is called a meromorphic first integral of system (6). Generally speaking, system (6) is integrable if it has a sufficiently rich set of first integrals such that its solutions can be expressed by these first integrals.…”

Non-integrability of the generalized spring-pendulum problem

“…Proof. Local computation shows that equation (21) has logarithms in its formal solutions at zero and infinity whenever k = 0. Thus, as we know from Remark 2, if the equation has a Liouvillian solution, then it must be an exponential one, i.e.…”

“…In Section 6, we study the exceptional case a = −k and conclude that the Morales-Ramis method yields no obstruction to integrability, whereas dynamical analysis seems to indicate that the system is not completely integrable. The Morales-Ramis theory was applied to study the integrability of many Hamiltonian systems, see examples in book [17] and in papers [19,20,21,22,23,24,25,26,27,28]. The differential Galois approach was used also for proving non-integrability of non-Hamiltonian systems, see [29,30,31].…”