BEZOUT IDENTITIES WITH INEQUALITY CONSTRAINTS

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MA4248 Weeks 4-5.
Topics Motion in a Central Force Field,
Keplers Laws of Planetary Motion, Couloumb
Scattering
Mechanics developed to model the universe
- 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
!
http//www-groups.dcs.st-andrews.ac.uk/history/Ma
thematicians/Ptolemy.html
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PTOLEMAIC THEORY
Earth
Planet
Earth is fixed, each planet moves in a circular
epicycle whose center moves in a circle with
center near the Earth.
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REVOLUTION
Nicolaus Copernicus (1473-1543) produced a
heliocentric (versus geocentric) theory of
cosmology
Galileo Galilei (1564-1642) questioned
authority - refuted Aristotles 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
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GEOMETRY OF ELLIPSES
2a-r
r
2ea
a semi-major axis (half of horizontal diameter)
angle
e eccentricity
foci
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r
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ALGEBRA OF ELLIPSES
2a-r
r
2ea
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r
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KEPLERS LAWS
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
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ANGULAR MOMENTUM AND TORQUE
The angular momentum , torque (about any
fixed point) of a body with movement, force is
(vector cross-products is orthogonal to both
vectors and its magnitude equals the area of the
parallelogram)
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CENTRAL FORCE
A force acting on a body is central if its
direction is along the line connecting the body
to a fixed point.
body
central force
fixed point
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CENTRAL FORCE
For a central force
therefore
Therefore the angular momentum
is constant.
(why?), the body
Therefore, since
moves in a plane. When does it move in a line?
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POLAR COORDINATES
Construct an orthonormal coordinate system
Define polar coordinates
and unit-vector valued functions (for r gt 0)
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POLAR COORDINATES
Therefore, the velocity of a particle is
and its acceleration is
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CENTRAL FORCE
Therefore, if a body moves in a central force,
then
hence
and
Remark
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KEPLERS SECOND LAW
Remark 1 Since L is constant and
equals the rate at which the radius vector sweeps
out Area, Keplers Second Law holds for any
central force
Remark 2 If F is conservative
and
where
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INTEGRATING THE EQUATION
Remark 3We can obtain a differential equation
for r
In general, the integral on the right will not
be an elementary function
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GRAVITATIONAL COULOMB FORCES
Remark 4 If
then the effective potential
has this graph --gt for k gt 0 (what if k lt 0 ?)
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INTEGRATING THE EQUATION
Remark 5We can also substitute the identity
to obtain
let u 1/r
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KEPLERS FIRST LAW
Remark 6 This has the form of a conic section
with semi-latus-rectum
and eccentricity
for
Circle
Ellipse
Parabola
Hyperbola
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KEPLERS THIRD LAW
Remark 8 The semi-major axis is determined by
the energy
Remark 9 The rate of area swept out
Remark 10 The area of the ellipse (b semi-major
axis)
for
Remark 11 The period
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KEPLERS EQUATION
Remark 12 The true anomaly is the angle
from pericenter and the eccentric anomaly is
Remark 13 Integrating the equations of
motion for r yields Keplers (transcendental)
equation
for
Remark 14 For Earths orbit
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SUPERINTEGRABILITY
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 gt 0) force where the
Laplace-Runge-Lenz vector defined by
for
is a constant of motion (Problem 12, page 25)
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