Numeriska ber äkningar i Naturvetenskap och Teknik

1. Today’s topic: Numeriska beräkningar i Naturvetenskap och Teknik Some Celestial Mechanics F

2. Coordinate systems Cartesian coordinates Numeriska beräkningar i Naturvetenskap och Teknik Unit vectors are orthogonal with norm 1

3. Cylindrical coordinates Numeriska beräkningar i Naturvetenskap och Teknik

4. Vector- och scalar product in cylindrical coordinates Numeriska beräkningar i Naturvetenskap och Teknik Orthogonal Right hand system

5. Spherical coordinates Numeriska beräkningar i Naturvetenskap och Teknik

6. Introductory mechanics Force law Numeriska beräkningar i Naturvetenskap och Teknik Torque Angular momentum gives:

7. A quantity that does not change with time, i.e. our case does not change along the trajectory of a planet is called a: CONSTANT OF MOTION If we can find a quantity whose time derivative is zero that quantity is a constant of motion. Numeriska beräkningar i Naturvetenskap och Teknik The angular momentum is constant in a central force field...

8. r x p is orthogonal to r, i.e. r is orthogonal to L which is constant. Numeriska beräkningar i Naturvetenskap och Teknik Central force 1 Angular momentum is a constant of motion 2. Motion is in a plane

9. Numeriska beräkningar i Naturvetenskap och Teknik In order write down the equationsof motion weneed the acceleration in cylindricalcoordinates. Thisproblem relieson the calculusyoulearn in mathclass!

10. Velocity in cylindrical coordinates Numeriska beräkningar i Naturvetenskap och Teknik Motion in the plane due to central force Angular velocity Radial velocity

11. In the same way… acceleration in cylindrical coordinates Numeriska beräkningar i Naturvetenskap och Teknik and also using the same method we can derive

12. Look at this at home! Acceleration in cylindrical coordinates Numeriska beräkningar i Naturvetenskap och Teknik

13. Acceleration in cylindrical coordinates Numeriska beräkningar i Naturvetenskap och Teknik Ins. from above gives that we have TWO components

14. Equations of motion in the central force system Numeriska beräkningar i Naturvetenskap och Teknik with the acceleration in the plane this can also be written as:

15. Equations of motion in the plane in cylindrical coordinates Depends explicitly on the force Numeriska beräkningar i Naturvetenskap och Teknik Can be integrated without defining F Now, we note that i.e. Which gives

16. Sector velocity Numeriska beräkningar i Naturvetenskap och Teknik Kepler’s second law

17. Rho direction: Equations of motion in the plane in cylindrical coordinates We have two functions of time, rho and phi. We want ONE! Numeriska beräkningar i Naturvetenskap och Teknik The angular momentum can be used to switch between rho and phi! since We have Substitution gives:

18. Numeriska beräkningar i Naturvetenskap och Teknik The energy is a second constant of motion...

19. A second constant of motion A conservative force, i.e. a force with potential Numeriska beräkningar i Naturvetenskap och Teknik WHY of interest? Examples of such forces?

20. A second constant of motion Numeriska beräkningar i Naturvetenskap och Teknik How to get a first order time derivative out of this?

21. A second constant of motion Numeriska beräkningar i Naturvetenskap och Teknik We think of the chain rule again and multiply by These are equal

22. Look at this at home! Continue by looking at the left hand side in the eq. below Numeriska beräkningar i Naturvetenskap och Teknik l.h can be written We now have time derivatives on both sides of this equation! i.e.

23. Lets identify the terms! Numeriska beräkningar i Naturvetenskap och Teknik Kinetic energy from radial motion Potential energy From L constant we have (still) Kinetic energy from motion in phi

24. Solving the equations of motion • Onecannoweither try tointegratewithrespectto the time, t, or, onecansolvewithrespectto the angle. • Two steps for a ”straightforward” solution. • Transform equationto be distancerho as functionof • the anglephiinsteadoftime. • 2. Make a 1/rho substitution tocreate a standard linear diff. eq. withconstantcoefficients. Numeriska beräkningar i Naturvetenskap och Teknik

25. Solving the equations of motion From second order time derivative to second order derivative in phi: Numeriska beräkningar i Naturvetenskap och Teknik Apply it two times

26. Numeriska beräkningar i Naturvetenskap och Teknik At this point we have but Binet!

27. Numeriska beräkningar i Naturvetenskap och Teknik Solving the equations of motion Binet’s equation for the kepler case (1/r2 ) Second order diff equation. (solve with characteristic equation!)

28. Numeriska beräkningar i Naturvetenskap och Teknik Different orbits Reference direction when α is zero

29. Numeriska beräkningar i Naturvetenskap och Teknik Different orbits Investigate in the project!

30. Numeriska beräkningar i Naturvetenskap och Teknik Other thoughts: Which velocity, in which direction, will give circular orbit? Is there a maximum velocity for a planet to stay in a closed orbit around the Sun? If the velocity is below the escape velocity, how does different start angles influence the shape of the orbit? Can you create ellipses and circles from the same starting speed? If a small planet passes close by another planet (e.g. an elliptic orbit that passes close to a jupiter like planet) what will happen. Why? (Voyager slingshots). If we integrate what should be a closed orbit with bad precision what will happen?

31. Numeriska beräkningar i Naturvetenskap och Teknik Orbital motionρ(t)

32. Numeriska beräkningar i Naturvetenskap och Teknik Orbital motionρ(t) This integral can in principle be solved t(ρ) but its inversion ρ(t) is not possible in ”simple functions”. The same is true for the angle as a function of time.

33. Numeriska beräkningar i Naturvetenskap och Teknik Extra

34. Numeriska beräkningar i Naturvetenskap och Teknik Variable substitution... Half major axis Eccentric anomaly Mean anomaly (ohmega constant, if e=0) Actual angle = true anomaly)

35. Numeriska beräkningar i Naturvetenskap och Teknik After this substitution... Kepler’s third law (can also be found from geometrical considerations)

36. Numeriska beräkningar i Naturvetenskap och Teknik Generally at time t Kepler’s equation Only numerical solution How find ρ(t)? Gives ρ for this t!

37. Numeriska beräkningar i Naturvetenskap och Teknik Two body problem For two interacting bodies the mass above is substituted by the so-called reduced mass Three body problem... Many tried to solve it (Poincare and others) but no solution exists in simple analytical form. Power series expansions exist. The problem has a very interesting background story. As an example, find and read on your own the story behind the Mittag-Leffler prize.

38. Numeriska beräkningar i Naturvetenskap och Teknik Notera att volymelementet i cylinderkoordinater är: