A new analytic theory of the Moon's motion is deduced from the Lagrangian of the Sun-Earth-Moon system expressed with relative velocities. In its first-order approximation, the first-degree terms of the ratio of distances from Earth to the Moon and to the Sun are taken. The calculated relative variation in the Moon's orbital angular momentum is resolved into components whose integrals yield the inclination of the orbital plane, the ecliptic longitude of the ascending node and the norm of the angular momentum as functions of the angle between the Moon and the ascending node.

The French mathematician and physicist Jean-Marie Souriau studied Gibbs states for the Hamiltonian action of a Lie group on a symplectic manifold and considered their possible applications in Physics and Cosmology. These Gibbs states are presented here with detailed proofs of all the stated results. A companion paper to appear will present examples of Gibbs states on various symplectic manifolds on which a Lie group of symmetries acts by a Hamiltonian action, including the Poincaré disk and the Poincaré half-plane.

Here we derive a bunch of explicit formulas for the circumferences of all types of Cassinian ovals in terms of the complete elliptic integrals of the first kind and their equivalent expressions in terms of the hypergeometric functions.

The problem of separating variables in integrable Hamiltonian systems has been extensively studied in the last decades. A recent approach is based on the so called Kowalewski's Conditions used to characterized a Control Matrix $M$ whose eigenvalues give the desired coordinates. In this paper we calculate directly a second compatible Hamiltonian structure for the cubic Hénon-Heiles systems and in this way we obtain the separation variables as the eigenvalues of a recursion operator $N$. Finally we re-obtain the Control Matrix $M$ from $N$..

We prove that in the general case the number of critical points of the distance function between two ellipses in $\mathbb{R}^3$ equals to 4, 6, 8, 10, 12, 14 or 16. As an example, the distance between the nowadays orbits of Neptune and Pluto has six critical points: one maximum two minima and three saddle points. The global minimum is 2.503 au (astronomical units), while the global maximum is 79.111 au. If we ignore the perturbations, then in year 21103 AD the distance between Neptune and Pluto would be 2.527 au.