Ph610 Analytical Mechanics Fall 2009 H.W. #5 (updated) Due by smapdi62

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```									Ph610 Analytical Mechanics Fall 2009            H.W. #5 (updated)               Due: 10/2/2009

HW#5: Multi-body problem-II, Coordinate transformation due to rotation
1. (30 points) Consider a system in which the total forces acting on the particles consist of
!                          !
conservative forces Fi ' and frictional forces fi . Show that if the frictional force is proportional
to the particle’s velocity, then the frictional force does not contribute to the virial theorem , i.e.,

1 N !' !
< T >= ! < " Fi • ri > .
2 i =1

2. (40 points) Consider a two-body system.
(a) Derive the following orbital equation
!dr
d! =                                ; µ = reduced mass,!=angular momentum
2#                !2 &
µr 2      E " V (r) "
µ%\$             2 µr 2 (
'
k h
(b) Consider an inter-particle interaction, V (r) = ! + 2 , i.e. a perturbed gravitational
r r
interaction. Find the exact solution r(! ) by integrating the equation in part (a).
k     h
(c) For E< 0 orbits, show that if ! >> 2 then the orbit is a precessing ellipse with a
r    r
!
precession frequency ( ! ) given by

! 2"µh
!=             where ! is the orbital period for h=0.
# "2
h                                                 !2
(d) The term         looks very much like the centrifugal barrier         , why does this term causes a
r2                                               2 µr 2
precession of the orbit while an addition to the centrifugal barrier through a change in ! does not
cause a precession?
(3) (30 points)
(a) We have shown in class that a rotation of a set of orthogonal axes into another set of
orthogonal axes is described by an orthogonal coordinate transformation, i.e.,
!
xi ' = Aij x j , where A ia an orthognal matrix, that is AA = I .
Now consider a rotation of a set of non-orthogonal axes whose metric tensor is given gij , find the
corresponding condition on the transformation matrix A.
(b) Let’s go back to orthogonal axes. Pick a point on a rigid object and let its position vector be
!
r = xi ' xi ' with respect to the body axes. Let the rigid body undergoes rotational motion. Hence
ˆ
!
this position vector is given by r (t) = xi ' xi '(t) = xi (t) xi .
ˆ           ˆ
Let xi = Aij (t)x j (t) , show that xi '(t) = Aij (t) x j
'
ˆ                 ˆ
!
"      !       !             !
(c) Extra credit (15 points) For a pure rotation, the velocity vector r (t) = ! (t) " r (t) , where ! (t)
is the instantaneous rotational frequency of the object. Here is a brute force method to show this
!
relationship in terms of the Euler’s angles. According to part b, r (t) = xi ' xi '(t) = xi ' Aij (t) x j ,
ˆ              ˆ
!
"               "                               !
"
hence r (t) = xi ' Aij (t) x j . Now express r (t) in terms of the instantaneous body axes,
ˆ
!
"            "                  "       # ˆ'
r (t) = xi ' Aij (t) x j = xi ' Aij (t) A jk xk (t) . The transformation matrices in terms of Euler’s angles are
ˆ
given in GPS (4.46) and (4.47).
!"                                                                    !     !              !
Compute r (t) in the body axes and show that it can be written as ! (t) " r (t) . Express ! (t) in
!
the body axes, i.e., ! (t) = ! i (t) xi' (t) and ! i (t) in terms of the Euler’s angles.
ˆ
!
For your convenience, A is given below
% cos! cos " # cos\$ sin " sin ! # sin ! cos " # cos\$ sin " cos!                  sin \$ sin " (
'                                                                                            *
!
A(t) = ' cos! sin " + cos\$ cos " sin ! # sin ! sin " + cos\$ cos " cos! # sin \$ cos " *
'                 sin \$ sin !                             sin \$ cos!                cos\$     *
&                                                                                            )
is expressed in terms of the Euler’s angles which are function of time.
*** The algebra is quite involved, you may use Mathematica or similar program. ***
(If you don’t have such program, the text provided an alternative derivation of ! i (t) and the
results are given in Eq. (4.87))

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