Central Forcesbased on FW-3

Motion of a particle with mass m subject to a central force:

)

conservative∇ x F = 0

no torque, r x F = 0 l = r x p = const.

It is convenient to work with polar coordinates:

(motion is planar, perpendicular to l)

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Conservation of energy:

Conservation of angular momentum:

first integral (involves only 1st time derivatives)

l = r x p = const.

(homework, problem 1.6)

motion in a small interval of time dt:

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Kepler’s second law:An imaginary line drawn from each planet to the Sun sweeps out equal areas in equal times.

(a simple consequence of the conservation of the angular

momentum for central forces)

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.

Effective potential:

Analyzing the motion:the angular momentum piece acts as

repulsive potential, centrifugal barrier

one dimensional potential:

E � Veff =1

2mr2 � 0

E � Veff

turning points

E = Veff (r0).

minimum energy, circular orbit

..

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Solution to the trajectory:

invert to find r(t)

plug in r(t) to find ϕ(t)

In practice this is often too complicated, but we can always find the geometric orbit r(ϕ) (at least numerically):

invert to find r(ϕ)

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Kepler’s problem (planetary motion):

Special case when the central force is gravitational force:

from the general solution:

we find:

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can be set to 0

straightforward manipulations:

Orbit solution:

For negative energy, orbits are bound and represent an ellipse.

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Conic sections:

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Ellipse:constant total distance from two foci

located at x=+f and x=-f:

eccentricity

equivalent definition:

in polar coordinates with the origin at the left focus:

follows from the law of cosines:

r

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Orbit solution: ellipse in polar coordinates with the origin at the left focus:

The orbit of a particle with negative energy in a gravitational field is an ellipse with the center at one focus.

Kepler’s first law:

energy of an orbit does not depend on angular momentum

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The square of an orbital period is proportional to the cube of a. Kepler’s third law:

Kepler’s 2nd law=

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Harmonic oscillator:

Comment on general central potential:

(homework, problem 1.10)

Orbit is an ellipse with the origin at the center!

orbits typically not closed:

if rational multiple of π then orbit is closed, otherwise it is open

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