The existence of Liapunov functions for some non-conservative positional mechanical systems

Cesar, Mauro de Oliveira; Barone-Netto, Angelo

doi:10.1016/0022-0396(91)90140-5

Cited by 8 publications

(5 citation statements)

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“…For all these functions which give stable equilibria, at least whenever g ∈ C 2 , the system (3.1) admits a Lyapunov function which is a further first integral and it is positive definite near the origin. This additional first integral is smooth in a neighborhood of the origin in R 4 and at least continuous at the origin as proved by Barone and Cesar in [4]. In the following statement C ⊆ R 2 is the set defined below formula (3.2).…”

Section: The Dynamics In the Lagrangian Frameworkmentioning

confidence: 70%

“…In this case all orbits of (1.1) in R 4 , with (q 1 , p 2 ) near (0, 0), are periodic and have the same period. Moreover, a further first integral appears as we shall see using a result of Barone and Cesar [4]. This happens for instance for the following two functions (see (5.7) and (5.10)) (1.5) g(x) = 1 − 1 √ 1 + x , g(x) = 1 + x − 1 (1 + x) 3 .…”

Section: Introductionmentioning

confidence: 73%

“…In the following statement C ⊆ R 2 is the set defined below formula (3.2). The first integral is obtained by means of the functions in (3.5) and a suitable inverse function, see [4] for details. Proposition 3.5 (Eigenvalues).…”

Section: The Dynamics In the Lagrangian Frameworkmentioning

confidence: 99%

“…Suppose that all orbits of ẍ = −g(x) which intersect the J interval of the x-axis in the x, ẋ-plane, are periodic and have the same period 2π/ g (0), so the origin in R 4 is a stable equilibrium for the system (3.1). Then there exists The first integral is obtained by means of the functions in (3.5) and a suitable inverse function, see [4] for details. Proposition 3.5 (Eigenvalues).…”

Section: The Dynamics In the Lagrangian Frameworkmentioning

confidence: 99%

“…Let us have a quick look of our main results, the details and the proofs are in the paper. We introduce the Hamiltonian system in R 4 (1.1)…”

Section: Introductionmentioning

confidence: 99%

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Completely Integrable Hamiltonian Systems with Weak Lyapunov Instability or Isochrony

Zampieri

2011

Commun. Math. Phys.

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The aim of this paper is to introduce a class of Hamiltonian autonomous systems in dimension 4 which are completely integrable and their dynamics is described in all details. They have an equilibrium point which is stable for some rare elements of the class, and unstable in most cases. Anyhow, it is linearly stable (all orbits of the linearized system are bounded) and no motion is asymptotic in the past, namely no non-constant solution has the equilibrium as limit point as time goes to minus infinity. In the unstable cases, there is a sequence of initial data which converges to the equilibrium point whose corresponding solutions are unbounded and the motion is slow. So instability is quite weak and perhaps no such explicit examples of instability are known in the literature. The stable cases are also interesting since the level sets of the 2 first integrals independent and in involution keep being non-compact and stability is related to the isochronous periodicity of all orbits near the equilibrium point and the existence of a further first integral. Hopefully, these superintegrable Hamiltonian systems will deserve further research.

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Section: The Dynamics In the Lagrangian Frameworkmentioning

confidence: 70%

Section: Introductionmentioning

confidence: 73%

Section: The Dynamics In the Lagrangian Frameworkmentioning

confidence: 99%

Section: The Dynamics In the Lagrangian Frameworkmentioning

confidence: 99%

“…Let us have a quick look of our main results, the details and the proofs are in the paper. We introduce the Hamiltonian system in R 4 (1.1)…”

Section: Introductionmentioning

confidence: 99%