Page 213 - Intro to Tensor Calculus

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Figure 2.2-7. Pulley and mass system

I 18. Let φ = ds , where s is the arc length between two points on a curve in generalized coordinates.
dt
p
m n
(a) Write the arc length in general coordinates as ds = g mn ˙x ˙x dt and show the integral I, defined by
equation (2.2.35), represents the distance between two points on a curve.
(b) Using the Euler-Lagrange equations (2.2.36) show that the shortest distance between two points in a
2
d s
i j k i dt 2
i
generalized space is the curve defined by the equations: ¨ + ˙ x ˙x =˙x
x
jk ds
dt

2 i
j
d x i dx dx k
(c) Show in the special case t = s the equations in part (b) reduce to + =0, for
ds 2 jk ds ds
i =1,... ,N. An examination of equation (1.5.51) shows that the above curves are geodesic curves.
(d) Show that the shortest distance between two points in a plane is a straight line.
2
1
(e) Consider two points on the surface of a cylinder of radius a. Let u = θ and u = z denote surface
coordinates in the two dimensional space defined by the surface of the cylinder. Show that the shortest
p
2 2
2
distance between the points where θ =0,z =0 and θ = π, z = H is L = a π + H .
1
i j
I 19. For T = mg ij v v the kinetic energy of a particle and V the potential energy of the particle show
2
that T + V = constant.
i
i
∂x i ,
Hint: mf i = Q i = − ∂V i =1, 2, 3and dx i =˙x = v ,i =1, 2, 3.
dt
I 20. Define H = T + V as the sum of the kinetic energy and potential energy of a particle. The quantity
r
H = H(x ,p r ) is called the Hamiltonian of the particle and it is expressed in terms of:
i
• the particle position x and
r
j
• the particle momentum p i = mv i = mg ij ˙x . Here x and p r are treated as independent variables.
(a) Show that the particle momentum is a covariant tensor of rank 1.
(b) Express the kinetic energy T in terms of the particle momentum.
∂T
(c) Show that p i = .
∂ ˙x i