CENTRAL FORCES
Definition:
(1) The potential is only dependent on the distance from source.
i.e.
(2)
From (1):
, i.e. the central force is radical in
direction and its magnitude is only dependent on the distance from the source.
Theorem 1:
If a particle is subject to a central force, the particle angular momentum is
conserved.
Proof:
Theorem 2:
If a particle is subject to a central force, its motion is on a plane.
Proof:
Consider at some time t
0
, construct the plane P containing the vectors
position vector
and the instantaneous momentum vector
.
The plane P has the
normal direction of
.
Consider the cross product of the position vector and the
momentum vectors
at some other time
t
1
.
Notice that
because
of the conservation of angular momentum.
The position vector
is always proportional
to the constant vector
and thus on a plane.
The EulerLagrange Equation
Define a physical quantity called Lagrangian which is defined by:
L =
T – V
where
T(x
1
,x
2
,…x
n
)
and
V(x
1
,x
2
,…x
n
)
are the kinetic energy and the potential energy of the
system.
Then the equation of motion is given by the n differential equations:
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i
= 1,2,…N
This is known as The EulerLagrange Equation and we are not going to give a proof on
the present course.
This equation is indeed logical equivalent to the Newton’s laws of
Motion.
Example:
Consider the springmass system:
.
Applying the EulerLagrange Equation yielded:
The Effective Potential of a Central Force
Consider a mass subject to a central force which has a potential of
V(r)
.
Using the polar
coordinates (r,
θ
) with the source placed at the origin.
Applying the EulerLagrange Equation on this case yields:
The first equation is the equation of motion for radical direction.
The angular momentum
has magnitude of
, where
is the tangential
component of the linear momentum.
But
, therefore
Therefore the second equation of the EulerLagrange Equation is the conservation of
angular momentum.
L
is determined by the initial condition.
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 Winter '07
 Harris
 Angular Momentum, Force, Kinetic Energy, Noether's theorem

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