Two-Body Central-Force Problems
Hasbun Ch 8
Taylor Ch 8
Marion & Thornton Ch 8
Central Force Examples
Gravitational Force
Coulomb Force
Conservative force, so the
force can be derived from a
potential energy, U(r1, r2)
Also a conservative force,
described by the potential
energy function:
r = r1 - r2
r1
r2
O
White Boards
Write the Lagrangian for a two-body central-force
system.
The Problem: find the possible motions of these two
bodies
Center of Mass and
Relative Coordinates
What generalized coordinates should we use to solve
the two-body central force problem?
Relative position, r
Position of center of mass, R, where
The total momentum of the system
is:
CM
R
r2
O
r1
B/c total momentum is constant,
CM frame is an inertial reference
frame.
White Boards
Write r1 and r2 in terms of R and r.
Write the kinetic energy T in terms of R and r.
Write the Lagrangian L in terms of R and r.
White Boards
Write the equations of motion from the Lagrangian
(one equation for R, one equation for r)
Center of Mass Frame
If we choose the center-of-mass frame as our
reference frame (we can do this b/c it’s an intertial
frame), then
The center of mass part of L is zero, so the Lagrangian
becomes:
Note this is the Lagrangian for a single particle moving in a central potential!
Conservation of Angular Momentum
Angular momentum of the two particles is conserved.
Rewrite this in the COM frame
Because the direction of angular momentum is constant, the motion
remains in a fixed plane, which we can take to be the x-y plane.
Always a 2D problem in COM frame
White Boards
The easiest coordinates for this 2-D problem are polar
coordinates.
Remember that the velocity in polar coordinates is
given by
Write the Lagrangian in polar coordinates, and find
the equations of motion.
Equations of Motion
Angular Equation
Radial Equation
Centrifugal
Force
White Board
Use the phi equation to eliminate phi-dot from the
radial equation
White Board
Use the phi equation to eliminate phi-dot from the
radial equation
Bound & Unbound Orbits
At rmin, the energy
is equal to Ueff
Matlab Problem
Write down the actual and effective potential
energies for a comet (or planet) moving in the
gravitational field of the sun. Plot the 3 potential
energies involved (U, Ucf, Ueff) and use the graph of
Ueff vs r to describe the motion as r goes from large
to small values.
Use
G=1
m1=m2=l=1
Matlab Problem