Phase Space
- Approximate solutions to N-body problem can be arrived at through performing successive iterations of calculus
- However, this does not always work
- Some infinite series converge, others do not
- Williams Hamilton - developed idea of phase space
- Used position & momentum in description of particles (instead of force)
- Six dimensional space which describes position (3 dimensions) & momentum (3 dimentions)
- 2 particles need 12-dimensional phase space
- Single point in phase space represents entire state of a system
- Position & momentum of every point in the box
- Poincare Recurrence Time - when a particle passes over the same point in phase space
- Poincare - analyzed only cross section (small part) of phase space (Poincare Section)
- Found that orbits of 3 bodies typically diverge (instability is normal)
- If a particle trajectory cut Poincare Section even a tiny distance away from the original point, the system can follow completely different patterns
- Many systems are very sensitive to initial conditions, and move away non-linearly
- Can't predict future (Laplace wrong)
- Lewis Fry Richardson - predictions of weather (forecasting)
- Based on work of Vilhelm Bjerknes
- Measured important properties such as temperature & pressure
- Apply physics to determine how conditions will affect each other
- Weather Prediction by Numerical Process
- Contrasted to 'synoptic' forecast (chart reading)
- Ever heard of people trying to read stock charts (technicals) to predict the future? That's basically the same concept as 'synoptic' forecasting. Thanks to Richardson, weather forecasters realized long ago that the weather from prior periods does not have much bearing on what the weather for future periods will be. Instead, meteorologists now use various readings such as temperature and pressure. This would be akin to market forecasters who rely on indicators such as consumer confidence, inflation, factory output, etc. (although these indicators are often forecasts themselves, hence perhaps explaining the unreliability of most 'market forecasts' - forecasts built on forecasts)
- Edward Lorenz - Butterfly Effect (real atmosphere is extremely sensitive to the starting position)
- Does the Flap of a Butterfly's Wings in Brazil Setoff Tornadoes in Texas
- Sometimes weather is more/less chaotic (predictable)
- The New Scientists Guide to Chaos - Franco Vivaldi
- Daniel Kirkwood - influence of Jupiter & Saturn sometimes causes asteroids to swing out of normal orbit (chaos)
- Theoretically works in reverse, but in practice asteroids collide w/ inner planets
- Calculations can be used to approximate historical & future orbits (but not exactly due to rounding)
- Irrational numbers are the reason for chaos
- Small deviations subvert Laplace's attempts to "predict" universe
- Only universe (God) itself can predict the universe (freewill?)
- Also shows that universe cannot be reversed
- Must show particles (all) move back to the present state, which cannot be stated (irrational numbers have inifinite decimals)
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