1. MA4248 Weeks 4-5. Topics Motion in a Central Force Field, Kepler’s Laws of Planetary Motion, Couloumb Scattering Mechanics developed to model the universe http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Ptolemy.html 1 - follow the seasons, predict eclipses and comets, compute the position of the moon and planets Babylonians (2000-300BC) – arithmetical models Claudius Ptolemy (85-165AD) – geometric models based on epicycles that prevailed for 1400 years !

2. PTOLEMAIC THEORY 2 Earth is fixed, each planet moves in a circular epicycle whose center moves in a circle with center near the Earth. Earth Planet

3. REVOLUTION 3 Nicolaus Copernicus (1473-1543) – produced a heliocentric (versus geocentric) theory of cosmology Galileo Galilei (1564-1642) – questioned authority - refuted Aristotle’s claim that heavy bodies fall faster - championed Copernican theory over Ptolemaic - sentenced to house arrest by the dreaded Inquisition Tycho Brahe (1546-1601) – observational astronomer - Danish King helped him build 37-foot quadrant - compiled, over 20 years, most accurate records - Emperor Rudolph II sponsored his move to Prague and collaboration with Kepler

4. GEOMETRY OF ELLIPSES 4 2a-r r a = semi-major axis (half of horizontal diameter) r 2ea e = eccentricity foci angle

5. ALGEBRA OF ELLIPSES 5 2a-r r r 2ea

6. KEPLER’S LAWS 6 Johann Kepler (1571-1630) – mathematician who believed in “the simplicity and harmonious unity of the universe” (quote page 323 David Burton) I.Each planet moves around the sun in an ellipse, with the sun at one focus. II.The radius vector from the sun to the planet sweeps out equal areas in equal intervals of time. III. The squares of the periods of any two planets are proportional to the cubes of the semimajor axes of their respective orbits:

7. ANGULAR MOMENTUM AND TORQUE The angular momentum, torque (about any fixed point) of a body with movement, force is 7 (vector cross-products is orthogonal to both vectors and its magnitude equals the area of the parallelogram)

8. CENTRAL FORCE A force acting on a body is central if its direction is along the line connecting the body to a fixed point. 8 fixed point body central force

9. CENTRAL FORCE For a central force 9 therefore Therefore the angular momentumis constant. Therefore, since (why?), the body moves in a plane. When does it move in a line?

10. POLAR COORDINATES Construct an orthonormal coordinate system 10 Define polar coordinates and unit-vector valued functions (for r > 0)

11. POLAR COORDINATES Therefore, the velocity of a particle is 11 and its acceleration is

12. CENTRAL FORCE Therefore, if a body moves in a central force, then 12 hence and Remark

13. KEPLER’S SECOND LAW 13 Remark 1: Since L is constant and equals the rate at which the radius vector sweeps out Area, Kepler’s Second Law holds for any central force Remark 2: If F is conservative and where

14. INTEGRATING THE EQUATION 14 Remark 3:We can obtain a differential equation for r In general, the integral on the right will not be an elementary function

15. GRAVITATIONAL & COULOMB FORCES 15 Remark 4: If then the effective potential has this graph --> for k > 0 (what if k < 0 ?)

16. INTEGRATING THE EQUATION 16 Remark 5:We can also substitute the identity to obtain let u = 1/r

17. KEPLER’S FIRST LAW 17 Remark 6: This has the form of a conic section with semi-latus-rectum and eccentricity forfor Circle Ellipse Parabola Hyperbola

18. KEPLER’S THIRD LAW 18 Remark 8: The semi-major axis is determined by the energy Remark 9: The rate of area swept out forfor Remark 10: The area of the ellipse (b semi-major axis) Remark 11: The period

19. KEPLER’S EQUATION 19 Remark 12: The true anomaly is the angle from pericenter and the eccentric anomaly is forfor Remark 13: Integrating the equations of motion for r yields Kepler’s (transcendental) equation Remark 14: For Earth’s orbit

20. SUPERINTEGRABILITY 20 Remark 15: A mechanical system is integrable if you can express the state as a function of time (even an non-elementary function). This requires constants of motion (such as angular momentum) and is a very special condition. In very special cases additional constants of motion exist that ensure closed orbits. These superintegrable systems include motion in a central –k/r (k > 0) force where the Laplace-Runge-Lenz vector defined by forfor is a constant of motion (Problem 12, page 25)