Aspects of the planetary Birkhoff normal form

The discovery in [G. Pinzari. PhD thesis. Univ. Roma Tre. 2009], [L. Chierchia and G. Pinzari, Invent. Math. 2011] of the Birkhoff normal form for the planetary many--body problem opened new insights and hopes for the comprehension of the dynamics of this problem. Remarkably, it allowed to give a {\sl direct} proof of the celebrated Arnold's Theorem [V. I. Arnold. Uspehi Math. Nauk. 1963] on the stability of planetary motions. In this paper, using a "ad hoc" set of symplectic variables, we develop an asymptotic formula for this normal form that may turn to be useful in applications. As an example, we provide two very simple applications to the three-body problem: we prove a conjecture by [V. I. Arnold. cit] on the "Kolmogorov set"of this problem and, using Nehoro{\v{s}}ev Theory [Nehoro{\v{s}}ev. Uspehi Math. Nauk. 1977], we prove, in the planar case, stability of all planetary actions over exponentially-long times, provided mean--motion resonances are excluded. We also briefly discuss perspectives and problems for full generalization of the results in the paper.Comment: 44 pages. Keywords: Averaging Theory, Birkhoff normal form, Nehoro{\v{s}}ev Theory, Planetary many--body problem, Arnold's Theorem on the stability of planetary motions, Properly--degenerate kam Theory, steepness. Revised version, including Reviewer's comments. Typos correcte

Intervenção fisioterapêutica nas sequelas de drenagem linfática manual iatrogênica: relato de caso

Atualmente, a drenagem linfática manual (DLM) é um dos recursos de grande destaque no tratamento de edemas, linfedemas e condições inestéticas, porém, na prática clínica têm-se observado sua aplicação de forma iatrogênica. Os objetivos deste relato de caso foram apontar as sequelas da aplicação iatrogênica da DLM, em uma jovem caucasiana, e verificar os efeitos da intervenção fisioterapêutica no tratamento dessas. Voluntária de 27 anos, apresentava-se com uma equimose importante na região lateral da coxa esquerda (28 x 9,5 cm), telangiectasias, microvaricosidades e petéquias em ambas as coxas e pernas, dor de forte intensidade e edema, após ter se submetido a um procedimento manual iatrogênico com finalidade estética há dois dias. As condutas fisioterapêuticas adotadas foram: Ultrassom 3 MHz, pulsado (2 ms on, 8 ms off), 0,4 W/cm² SATP na região da equimose e DLM (Método Leduc) durante 20 minutos, na coxa esquerda. Foram realizadas nove sessões até a alta no 16º dia pós-lesão, observando-se apenas a redução gradativa da dor e da equimose. Verificou-se a efetividade da intervenção fisioterapêutica em apenas algumas das sequelas provocadas pela iatrogenia, sem desconsiderar que a remissão espontânea dos sinais e sintomas também pode ter contribuído para tal. Destaca-se a importância de conscientizar os fisioterapeutas que atuam com este procedimento para que o aplique de forma adequada, evitando tais iatrogenias.Recently, the manual lymphatic drainage (MLD) is an important resource in the treatment of edemas, lymphedemas, and other unaesthetic conditions. However, problems caused by the incorrect application of this therapy have been observed in the clinical practice. The aims of this case report were to describe sequelae resulting from the negligent application of MLD on a young patient, and to determine the effects of physical therapy on the treatment of sequelae. A 27-year-old Caucasian woman presented with a large ecchymosis (28 x 9.5 cm) on the outer side of the left thigh, telangiectasias, micro varicose veins, and petechiae on the legs and thighs, intense pain, and edema, after undergoing an incorrect manual procedure for aesthetic purposes two days earlier. Physical therapy consisted of ultrasound application (3 MHz, pulsed mode, 20% duty cycle, SATP intensity of 0.4 W/cm²) to the ecchymosis site and MLD (Leduc method) for 20 minutes on the left thigh. The patient underwent nine physical therapy sessions until discharge from the hospital 16 days after injury. The effectiveness of physiotherapy intervention was verified in only some of the consequences caused by iatrogenic, without disregarding that the spontaneous remission of signs symptoms can have contributed to such fact. The study emphasizes the importance of awareness among physical therapists who work with this procedure to apply it appropriately in order to avoid such conditions