The document discusses apses and apsidal distance in orbital mechanics. It defines apses as the two points in an orbit that are nearest to and farthest from the center of motion, forming the major axis of the orbit. The length of this major axis is called the apsidal distance. At an apse, the velocity of the orbiting body is perpendicular to the radius vector from the center of motion. The document then discusses using the (u,θ) equation to solve differential equations to find orbital paths given the central acceleration.

1. Ms. M. Mohanamalar, M.Sc., M.Phil.,
Ms. N. Malathi, M.Sc.,
DYNAMICS (18UMTC41)
II B.Sc. Mathematics (SF)

2. Apse and Apsidal Distance

3. In mechanics, either of the two points of an orbit which are
nearest to and farthest from the centre of motion. They are called
the lower or nearer, and the higher or more distant apsides
respectively. The "line of apsides" is that which joins them, forming
the major axis of the orbit. The length of the line of apsides is
called the apsidal distance

4. If there is a point A on a central orbit at which the velocity
of the particle is perpendicular to the radius OA, then the point A is
called an Apse and the length OA is the corresponding apsidal
distance.
Hence at an apse, the particle is moving at right angles to the radius
vector.

5. We know that where u = , r being the
distance from the centre of force and p is the perpendicular from
the centre of force upon the tangent.
At an apse, p = r = . Hence from the above relation we
get = 0 at an apse.

6. Here we consider the problems namely given the value of the
central acceleration P, we will find the path.
We use the (u,) equation
To solve this differential equation, we multiply both sides by 2 ,
we get

7. Hence we get
Integrating both sides with respect to ,