Lagrangian Mechanics
Problem:
Consider the problem of a charged particle moving in an electric field superimposed on a
magnetic field. The force on the particle is given by F=-ÑV+(e/c)(v´H).
Assume a constant magnetic field in the z-direction, so that H = H0k.
(a) What are the rectangular equations of motion?
(b) Assume the Lagrangian to be of the form
.
Determine 
.
(c) Determine the Hamiltonian and the canonical equations of motion.
(d) Are there terms beyond the total energy in the Hamiltonian? If yes, what does
their existence imply?
 | Solution:
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Problem:
A particle constrained to move on a spherical surface of
radius R is projected horizontally from a point at the level of the center so that
its angular velocity relative to the axis is w.
If w2R>>g, show that its depth z below the
level of the center is given approximately by
.
Problem:
Suppose that the magnetic field in a circular particle accelerator is symmetrical about a
vertical z-axis, so that in cylindrical polar coordinates B(r,q,z)=Bz(r,z)k+Br(r,z)h, where k and h are unit vectors in
the z and r directions, respectively.
In this case B=Ñ´A, where A = Af(r,z)m, and m
is a unit vector in the f direction. The Lagrangian for a
particle of charge e and mass m in the field B is
, where c is
the speed of light. Find two constants of the non-relativistic motion. Next
investigate the stability of the orbital motion in the following way. Assume the
particle is still moving non-relativistically and is confined to the z=0 plane.
Let
r=a+a , where a is a
constant radius, and a is a very small change from a.
Find the equation of motion for a, making suitable
approximations along the way. Under what conditions will the motion be stable,
though possibly oscillatory, in r?
Problem:
(a) Show that two Lagrangians L1 and
L2, which differ only by the total derivative of a function of q
and t, i.e.
, describe
the same motion for q.
(b) Find the Lagrangian and Hamiltonian of a pendulum consisting of a mass m
attached to a massless rigid rod AB of length l free to move in a vertical
plane. The end A of the rod is forced to move vertically, so that its displacement
from the fixed point O is a given function of time g(t).
Gravity acts vertically downward.
(c) Show that the vertical acceleration of the point A,
, has the same effect on the equation
of motion as a time varying gravitational field.
| Solution:
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Problem:
(a) Derive the (Euler)-Lagrange equation of motion for a one-dimensional
system from a variational requirement (Hamilton’s Principle) on the time
integral of the Lagrangian
, where
q is a generalized coordinate and
.
(b) Consider the pendulum illustrated in the figure with l the length
of the string, m the mass of the ball, Fg the
gravitational force, and f the angular displacement.
(You may assume the string to be of fixed length and negligible mass). Use
Lagrange’s equation to derive an equation of motion that neglects terms of order
f3 and higher.
(c) The preceding example corresponds to harmonic motion if the amount of
displacement is suitably restricted. Suppose that instead the periodic motion
occurs in a one-dimensional potential that is not necessarily harmonic. Derive
a general expression for the period of the motion in this case.
