1. 1 Two of Newton’s Law of Motions 1) In the absence of any forces applied, an object at rest will stay at rest, and a body moving at a constant velocity in straight line continues doing so indefinitely. 2) When a force is applied to an object, it accelerates. The acceleration is in the direction of the force and proportional to its strength, and is also inversely proportional to the mass (m) of the object. In suitable units:

2. 2 Newton’s Law of Gravitation Gravitational force: an attractive force that exists between all objects with mass; an object with mass attracts another object with mass; the magnitude of the force is directly proportional to the masses of the two objects and inversely proportional to the square of the distance between the two objects. Where: m1 & m2 are the masses of the two objects r is the distance between the objects G is the universal gravitation constant G = 6.67384 x 10 -20 km 3 /(kg*seconds 2 )

3. 3 Consider a single object t = time m = mass of object F = force on object (vector) a = acceleration object (vector) x = location of object (vector) v = velocity of object (vector) The following equations apply: Note: F, a, v and x are all vectors and are functions of t.

4. 4 Two Gravitating Particle Masses m1m1 m2m2 Each particle has a scalar mass quantity

5. 5 Particle Positions x1x1 x2x2 (0,0) Each particle has a vector position relative to the origin:

6. 6 Particle Velocities v1v1 v2v2 Each particle has a vector velocity

7. 7 Particle Accelerations a1a1 a2a2 Each particle has a vector acceleration

8. 8 In two dimensions, vectors can be represented by two signed floats (or doubles): Vector Operations The length or magnitude of a vector is a scalar quantity and can be computed as: A unit length vector U with the same direction as a non-unit length vector A can be computed as:

9. 9 More Vector Ops A unit vector U can be given a known magnitude (length) as follows: Where U is a unit vector and force is a scalar quantity which is to be the magnitude of F. Vector addition and subtraction consist of adding and subtracting the vectors’ x and y components:

10. 10 Two-body Newtonian Gravitation Two objects of mass m 1 and m 2 with position vectors X 1 and X 2 exert a gravitational force on each other. The magnitude of the force is given by: The vector F 21, representing the gravitational force exerted by m 2 on m 1 is then: The net resulting force vector on any mass is the vector sum of all the force vectors generated by masses other than itself.

11. 11 Newtonian Gravitation Newton’s second law (F = mA) can be applied to the two body scenario:

12. 12 Newtonian Gravitation Using velocity:

13. 13 N-Body Gravitation Generalizing to N bodies from two bodies we must sum gravitational force vectors induced by all N bodies except the one we are computing the acceleration for:

14. 14 N-Body Newtonian Gravitation Simulation Problem: Plot the position of the bodies as a function of time. We need to specify the initial velocity and positions of the objects. Next we need a numerical scheme to advance the equations in time. Can use Euler’s Method …. as a first approach.

15. 15 Numerical Approach For m=1 to FinalTime/dt For n=1 to number of objects End