Aspects of the planetary Birkhoff normal form

Pinzari, Gabriella

doi:10.1134/s1560354713060178

“…The canonical character of K follows from [13]. Indeed, in [13], we considered a set of coordinates for the three-body problem 4 , that here 5 we denote as P = (Z, C, R, R, Θ, Φ, z, g, r, r, ϑ, ϕ), that are related to K above via the canonical change where ξ = ξ(Λ, G, ), ν = ν(Λ, G, ) are, respectively, the eccentric and the true anomaly, defined below (see (27), (29)).…”

Section: K Coordinatesmentioning

confidence: 99%

“…The canonical character of K follows from [13]. Indeed, in [13], we considered a set of coordinates for the three-body problem 4 , that here 5 we denote as P = (Z, C, R, R, Θ, Φ, z, g, r, r, ϑ, ϕ), that are related to K above via the canonical change where ξ = ξ(Λ, G, ), ν = ν(Λ, G, ) are, respectively, the eccentric and the true anomaly, defined below (see (27), (29)). Since the map D e ,pl in (24) and the coordinates P of [13] are canonical, so is K. Observe, incidentally, the unusual π 2 -shift in (25), due to the fact that, according to the definitions in (23), g is the longitude of M × P in the plane of i 3 , j 3 , relatively to i 3 .…”

Section: K Coordinatesmentioning

confidence: 99%

Euler integral and perihelion librations

Pinzari¹

2020

Discrete &Amp; Continuous Dynamical Systems - A

Self Cite

View full text Add to dashboard Cite

show abstract

“…3/2 (α) with α, a 2 as in (7), (35), and, as usual, b (j) s (α)'s being the Laplace coefficients, defined via the Fourier expansion…”

Section: Birkhoff Theory In the Partially Reduced Retrograde Problemmentioning

confidence: 99%

“…The existence of a positive measure set of Lagrangian tori with maximal number (see the next section) of frequencies for the general planetary problem, in the regime of well spaced orbits, small eccentricities and small inclinations, has been established in the papers Refs. 3,9,12,22,[33][34][35] . We refer to such technical papers for details, to Refs.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the co-existence of maximal and whiskered tori in the planetary three-body problem

Pinzari

¹

2018

Journal of Mathematical Physics

Self Cite

View full text Add to dashboard Cite

In this paper we discuss about the possibility of coexistence of stable and unstable quasi-periodic kam tori in a region of phase space of the three-body problem. The argument of proof goes along kam theory and, especially, the production of two non smoothly related systems of canonical coordinates in the same region of the phase space, the possibility of which is foreseen, for "properly-degenerate" systems, by a theorem of Nekhorossev and Miščenko and Fomenko. The two coordinate systems are alternative to the classical reduction of the nodes by Jacobi, described, e.g., in [V.I. Arnold, Small denominators and problems of stability of motion in classical and celestial mechanics, 18, 85 (1963); p. 141]. PACS numbers: 02. Mathematical methods in physics a) gabriella.pinzari@math.unipd.it 1963's paper Ref. 3 -on the study of the dynamics of the so-called planetary problem, i.e.,the problem of (1 + N) point masses, one of which ("sun") is of "order one", while the remaining N ("planets") are of much smaller size, interacting through gravity. Indeed, when the reciprocal attraction among the planets, which is of much smaller order compared to the attraction between any planet and the sun, is set to zero, the planetary problem reduces to N uncoupled two-body problems ("unperturbed problem"). He considered the case of N planets in prograde configuration; i.e., revolving in the same verse, even though the question of the sense of rotation, at his time, was definitely of secondary importance, compared to the difficulties that had to be overcome and that we are going to recall. The lack of frequencies(a translationally invariant system with 1 + N bodies possesses 3N degrees of freedom. In the case of the planetary problem, it exhibits, as mentioned, only N < 3N frequencies) in the unperturbed problem was named by Arnold proper degeneracy. It represented a serious difficulty, if one wanted (as he was aiming to do) to apply Kolmogorov's theorem, Ref. 21 to the planetary problem.At a technical level, the appearance of the proper degeneracy consists, we might say, of a "loss of frequencies", caused by the "too many" (or, better Poisson non commuting 45 , see below) first integrals of motion. For such abundance, this kind of systems is often called super-integrable. Despite of the fact that the solutions of the two-body problem are known since Newton's times, a general, theoretical setting clearly explaining the phenomenon has been given only recently, thanks to the works by Nekhorossev and Miščenko and Fomenko, Refs. 27,30 (hereafter, nmf). The three authors proved a generalization of the best known Liouville-Arnold theorem, Ref.

show abstract

The Steep Nekhoroshev’s Theorem

Guzzo

¹

,

Chierchia

²

,

Benettin

³

2016

Commun. Math. Phys.

View full text Add to dashboard Cite

Revising Nekhoroshev’s geometry of resonances, we provide a fully constructive and quantitative proof of Nekhoroshev’s theorem for steep Hamiltonian systems proving, in particular, that the exponential stability exponent can be taken to be (Formula presented.) ((Formula presented.) ’s being Nekhoroshev’s steepness indices and (Formula presented.) the number of degrees of freedom). On the base of a heuristic argument, we conjecture that the new stability exponent is optimal

show abstract

Aspects of the planetary Birkhoff normal form

Cited by 19 publications

References 47 publications

Euler integral and perihelion librations

Euler integral and perihelion librations

On the co-existence of maximal and whiskered tori in the planetary three-body problem

The Steep Nekhoroshev’s Theorem

Contact Info

Product

Resources

About