Semester: VII (2+0), VIII (2+0)

**INTRODUCTION.** Problems in Celestial Mechanics and the Theory of Earth's
Artificial Satellites Motion. Kepler's Laws. Newton's Law of Gravity.

**TWO BODIES PROBLEM AND THE STABLE ORBITING OF PLANETS AROUND THE SUN.**
Determine the Problem. Both Vectorial Integrals of Differential Equation
of Stable Heliocentric Planetary Motion. Geometric Meaning of these Vectorial Integrals.
Sixth Scalar Integral of Vectorial Differential Equation. Basic Integrals
of Differential Equation of Stable Planetary Motion around the Sun in their Final Form.
Calculus of Vectorial Elements from Initial Conditions. The Satellites Problem Reduced to
the Two Bodies Problem. The Connection between Vectorial and Elliptical Elements of
Planetary Motion.

**THE PROBLEM OF N BODIES.** Determine the Problem. Basic Integrals of the Problem
of N Bodies. Translational Motion of the Solar System. Laplace's Invariable Plane.

**FUNDAMENTAL THEORY OF PLANETARY PERTURBATIONS.** Differential Equations of
Heliocentric Planetary Motion. Basic Vectorial Equations of Perturbations.
Lagrange's Brackets and their Properties.

**INTRODUCING VECTORIAL ELEMENTS INTO THE CALCULUS OF PERTURBATIONS.**
Computation of Lagrange's Brackets of Vectorial Elements. Differential
Equations of Vectorial Elements.

**SURVEY OF CLASSICAL PERTURBATIONS THEORY.** Differential Equations of Elliptical
Elements. Integration of Differential Equations of Planetary Perturbations.
Classification of Perturbations.

**SECULAR PERTURBATIONS.** Importance of Secular Perturbations. Differential Equations of
Secular Perturbations. Secular Perturbations of Planets. Some Basic
Conclusions from those Integrals.

**LAPLACE'S EQUATION. SPHERICAL FUNCTIONS.** Laplace's Operator in Orthogonal Coordinates.
Gravitational Potential. Laplace's Equation. Spherical Functions.
Legendre's Polynomials. Representation of Functions at Spherical Surface
by their Series on Spherical Functions. Additional Theorem of Spherical Functions.

**REPRESENTATION OF GRAVITATIONAL POTENTIAL.** Representation of Rigid Body Potential
by Series on Spherical Functions. Gravitational Potential of Spheroid.
Potential of the Earth's Attraction. Earth's Ellipsoid. Normal Gravitational
Potential of the Earth.

**PERTURBATIVE ACCELERATION BECAUSE OF EARTH'S ELLIPTICITY.** Coordinates
of Perturbative Acceleration in Representation of the Earth's Normal
Potential.

**DIFFERENTIAL EQUATIONS OF PERTURBATIVE MOTION.** Differential Equation
of Perturbative Motion of the Earth's Artificial Satellite. Differential
Equations of Elliptical Orbital Elements of the Earth's Artificial
Satellite with Coordinates of Perturbative Acceleration. Equations of
Longitude of the Ascending Node, Parameter and Inclination. Equations of
Eccentricity, True Anomaly and Argument of Pericentre. Equations of
Lambda1 and Lambda2. Equation of Latitude Argument. Solving the System
of Equations by Using the Method of Successive Approximations.

**SECULAR PERTURBATIONS OF SATELITES.** Secular Perturbations of the
Earth's
Artificial Satellite Elliptical Orbit. Secular Perturbations Caused by
Noncentral Earth's Attraction. Perturbations of Orbiting Period
of an Elliptical Orbit. Draconic Period. Siderial Period at Ascending Node.