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In response to the new NATO complex examine Institute "Chaotic Worlds: From Order to illness in Gravitational N-Body Dynamical Systems", this state-of-the-art textbook, written through the world over well known specialists, presents a useful reference quantity for all scholars and researchers in gravitational n-body platforms. The contributions are particularly designed to offer a scientific improvement from the elemental arithmetic which underpin sleek reports of ordered and chaotic behaviour in n-body dynamics to their program to actual movement in planetary structures. This quantity offers an up to date synoptic view of the topic.

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**Example text**

Let us call H (p(r) , q (r) ) the (r) truncated Hamiltonian. Then the canonical equations for H (p(r) , q (r) ) admit the simple solution p(r) = 0, (r) (r) q (r) = ωt + q0 , (16) where q0 are the values of the phases at time zero. If we denote (p, q) = C(p(r) , q (r) ) the canonical transformation (14) truncated at order r, then we can recover the orbit in the original coordinates by simply substituting the solution (16) in the transformation. 22 A. Giorgilli and U. Locatelli The argument above provides us a method for checking the reliability (r) of our calculation: we ﬁrst determine the value of q0 corresponding to the initial conditions (by using (15) with s = r); then we calculate the solution (16) expressed in the original coordinates (through the canonical transformation C); ﬁnally, we compare this “semi-analytical” integration with a numerical one (still in the original coordinates).

The action variables for the two planets are Λ1 , Λ2 , with conjugate angles λ1 , λ2 . We recall that the Poincar´e variables are ⎧ ⎪ ⎪Λ = β √µ a , ⎪ ξj = 2Λj 1 − 1 − e2j cos ωj , j j j ⎪ ⎨ j ⎪ ⎪ ⎪ ⎪ ⎩ λj = lj + ωj , j = 1, 2, ηj = − 2Λj 1− 1− e2j sin ωj , with the usual notations aj , ej , lj and ωj for the semi-major axes, the eccentricities, the mean anomalies and the perihelion arguments, respectively. Using the classical methods of Celestial Mechanics, we can expand the distance ∆ in the Hamiltonian (19) as a function of the Poincar´e variables, and we can calculate the so called secular system at order two in the masses (used, for instance, in Laskar 1988 in a model with 8 planets, to study the long term evolution of the solar system).

A family of symmetric periodic orbits is represented by a continuous curve in the space of initial conditions x10 , x˙ 20 . This curve is called a characteristic curve. 3. Variational equations A periodic orbit is an orbit which repeats itself for inﬁnite time with period T . e, initial conditions in the vicinity of the initial conditions of the periodic orbit. We express the system of diﬀerential equations (1) as a system of four diﬀerential equations of ﬁrst order, x˙ 1 x˙ 2 x˙ 3 x˙ 4 = = = = x3 , x4 , F1 (x1 , x2 , x3 , x4 ), F2 (x1 , x2 , x3 , x4 ), 47 Periodic orbits in gravitational systems or, in general x˙ i = fi (x1 , x2 , x3 , x4 ).