Direct Energy, 2018a

264 11.8 Problems
11.4. Figure 11.2 illustrates three possible paths for the mass spring system
and their corresponding actions. The paths considered are:
x 1 (t) =2t − 1
x 2 (t) =2t 2 − 1
x 3 (t) =− cos(πt)
For each path, calculate the action using Eq. 11.36 to verify the
values shown in the gure. Assume a mass of m =1 kg and a spring
constant of K = π 2 m J . 2
11.5. The gure shows a torsion spring. It can store potential energy 1 2 Kθ2 ,
and it can convert potential energy to kinetic energy 1I ( )
dθ 2
2 dt . In
these expressions, θ(t) is the magnitude of the angle the spring turns
in radians, and ω = dθ is the magnitude of the angular velocity in
dt
radians per second. K is the torsion spring constant, and I is the
(constant) moment of inertia.
(a) Find the Lagrangian.
(b) Use the Euler-Lagrange equation to nd a dierential equation
describing θ(t).
(c) Show that energy is conserved in this system by showing that
dH
=0.
dt
(d) Set up Hamilton's equations.
11.6. The purpose of this problem is to derive the shortest path y(x) between
the points (x 0 ,y 0 ) and (x 1 ,y 1 ). Consider an arbitrary path
between these points as shown in the gure. We can break the path
into dierential elements d −→ l = dxâ x + dyâ y . The magnitude of each
dierential element is
|d −→ l | = √ √
( ) 2 dy
(dx) 2 +(dy) 2 = dx 1+ .
dx