3. Also, as , we have:

Where C is an arbitrary constant.

 4. If the rate of change with respect to time is zero, then the above expression must be constant and we shall call it Em.

  = Constant, called "Specific mechanic energy"               2.5

In the above equation we have that the first term is the kinetic energy per unit of mass and that the constant C must be interpreted as the value of the reference point for which the potential energy is rendered to zero

 As a conclusion, we can state that the specific mechanic energy (per unit of mass) of a body in motion around another is the sum of its kinetic energy and its potential energy and that it remains constant along all the orbit without increasing nor decreasing.

 That is, the Specific mechanic energy is

                                                                                                     2.6

Conservation of the angular momentum

The constant of the angular momentum for the movement of our two body system can be derived as follows:

1. Multiply equation 2.4 by r (cross product from vectorial calculus):

2. As in general,  , the second term turns to zero, thus:

3. Now, as  , we have that the above equation turns out to:

The expression r x v, which is a constant of the movement, is represented by the vector h, is known as the Specific angular momentum.

Based on the above, we have seen how the Specific angular moment of a body in motion around another remains constant along all of its orbit.

That is, the Specific angular momentum is:

                                                                                                                      2.7

Now, as h is the vectorial cross product of r and v, h will always be perpendicular to the plane that contains these two vectors. And because, as seen, h is constant then r and v must always be confined to the same plane and therefore we can conclude that the movement of out two bodies does indeed remain confined within a same plane which of course is known as the orbital plane. The magnitude of vector h can be derived through the geometry depicted in figure 2.2 for vectors r and v.

Figure 2.2

If we define that the vertical direction is towards the position vector and that the horizontal direction is perpendicular, then we can define the direction of the velocity vector v by means of the angle with the vertical. The angle formed between the velocity vector and the horizontal direction is called the trajectory angle and is specially useful in the study of the movement of artificial satellites.

From the definition of the angular momentum vector, we have that its magnitude is:

                                                                                                  2.7 A.

 The sign of is that of the product r.v.

Trajectory equation:

 By means of equation 2.3 we defined the relative movement of one body with respect of the other. Even as this equation is relatively simple, its complete solution and the description of movement as a function of time is rather complex.; we can however reach a partial solution which will give us important information about the shape and geometry of the orbit.

 Recalling that the movement equation is (equation 2.4):

 If we carry out the cross vectorial product with vector h, we find

                                                                                               2.8

 Now, as , as h is constant and seeing that (by means of a well know theorem of vectorial calculus):

 Additionally, we have that (multiplying by the derivative of the unit vector):

 Therefore, we can say that:

 Equation which after integrating at both sides, is equivalent to:

                                                                                                    2.9

 Where B is a constant of integration. If we carry out the product of this equation with r, we obtain the following scalar equation:

 As in general, , we have that:

Where is the angle between the constant vector B and vector r.

Now, solving the above equation for r, we finally obtain:

                                                                                           2.10

Equation 2.10 defines the geometry of the trajectory of a body in gravitational motion around another, expressed in polar coordinates and where angle is measured from vector B towards vector r.

From analytic geometry, the general equation of the conic curves family (described ahead) expressed in polar coordinates and with the origin in one the foci is:

                                                                                              2.11

 Now, it should be clear to the reader that equation 2.11 is mathematically equivalent to equation 2.10 and thus we can conclude that the orbit of a two body system is a conic curve which retains all of its properties and characteristics.

 In equation 2.11, p is known as the parameter of the curve (also known as the "semi-latus rectum" ) and e is called the eccentricity of the conic, which y itself determines its shape and its main characteristics.

 By comparing both equations 2.10 and 2.11 we can immediately conclude that:

                                                                                                              2.12