Download PHYS4330 Theoretical Mechanics HW #8 Due 25 Oct 2011

PHYS4330 Theoretical Mechanics
HW #8
Due 25 Oct 2011
(1) (See Taylor 7.45.) In class, we found the kinetic energy of system of N particles to be
1�
Ajk q̇j q̇k
T =
2 α
where
Ajk =
�
α
mα
�
∂rα
∂qj
� �
∂rα
·
∂qk
�
Prove that A is symmetric, that is, Akj = Ajk . (This is simple.) Now show that
�
d �
Ajk vj vk = 2
Aij vj
dvi j,k
j
for any n variables vi , i = 1, 2, . . . , n. Hint: Think carefully how to take the derivative d/dvi .
(2) (See Taylor 7.46.) Show that rotational symmetry implies the conservation of angular momentum explicitly in spherical polar coordinates. Consider the transformation
(rα , θα , φα ) → (rα , θα , φα + �), α = 1, 2, . . . , N , where � is the same for each in a system
of N particles. (Note that θ is the polar angle. That is, we are discussing rotations about
the z-axis. If you want to add a potential energy, it must respect this rotational symmetry.
�
For example, U = α mα gzα .) If the Lagrangian is unchanged under this transformation,
�
�
show that α (∂L/∂φα ) = 0. Then, show that �z = α (mα rα2 sin2 θα φ̇α ) is constant.
(3) (See Taylor 7.49.) A mass m with charge q moves in a uniform constant magnetic field
B = Bẑ. Prove that B = ∇ × A where A = 12 B × r. (You can do this in a coordinateindependent way, only assuming that B is a constant field, and using some vector identities.)
Show that A = 12 Bρφ̂ in cylindrical polar coordinates. Write the Lagrangian in cylindrical
polar coordinates and find the three Lagrange equations of motion. Describe the solutions of
these equations for which ρ is a constant, and confirm that they agree with Newton’s Second
Law and the Lorentz force law.
(4) (See Taylor 11.2.) A massless spring k1 is suspended from the ceiling, with a mass m1
hanging from its lower end. A second spring k2 is suspended from m1 , and a second mass
m2 is suspended from the lower end of this second spring. The masses move only vertically.
The positions of m1 and m2 are measured by y1 (t) and y2 (t), respectively, each equal to zero
at the equilibrium position of the mass. If y is a column matrix of y1 and y2 , show that the
equations of motion can be written as Mÿ = −Ky where M and K are 2 × 2 matrices, and
find these matrices in terms of m1 , m2 , k1 , and k2 .
(5) For the following horizontal system of two masses and three springs (Taylor Figure 11.1),
study the motion for k1 = k2 = k3 ≡ k, with m1 = m2 ≡ m. Plot the motions x1 (t) and
x2 (t) for initial conditions ẋ1 (0) = ẋ2 (0) = 0 and (a) x1 (0) = 1 = x2 (0), (b) x1 (0) = 1 and
x2 (0) = −1, and (a) x1 (0) = 1 and x2 (0) = 0. (You may want to explore other combinations
as well, including nonzero initial velocities.) Use k/m = 4π 2 for the plots, and identify the
periods corresponding to the normal mode frequencies.