The potential energy of the system is the potential energy associated with the tension in
We assume that the displacements from the equilibrium positions are small.
We ignore the gravitational forces acting on the mass
A ( B C ) = B (C A ) = C ( A B )
Figure 3. Properties of the vector product between the vectors A and B.
Differentiation and Integration
Two important operations on both scalars and vectors are differentiation and integration.
Example: Problem 9.1
Find the center of mass of a hemispherical shell of constant density and inner radius r1 and
outer radius r2.
Put the shell in the z > 0 region, with the base in the x-y plane. By symmetry,
xcm = y
This differential equation is a non-linear equation due to the sin term. This equation has the
following general form:
= cx sin x + F cos ( t )
Figure 10. A damped pendulum, driven about its pivot point.
The solution to this equat
Hamilton's Principle - Lagrangian and Hamiltonian Dynamics
Many interesting physics systems describe systems of particles on which many forces are
acting. Some of these forces are immediately obvious to the person studying
x (t ) =
( 0 2 n 2 ) + 4 n 2 2
sin ( n t n )
n = tan 1 2 n 2
The solution to the third equation is
2 0 2
The solution of the following differential equation
+ 2 x + 0 2 x =
a0 + ( an cos n t + bn sin n
The phase paths will be executed in a clock-wise direction. For example, in the upper right
corner of the phase diagram, the velocity is positive. This implies that x must be increasing.
The x coordinate will continue to increase unt
This equation can be rewritten as
x ' cfw_ x ' x '+ y ' y '
( x ') + ( y ') + 1 ( x ') + ( y ') + 1
y ' cfw_ x ' x '+ y ' y '
( x ' ) + ( y ' ) + 1 ( x ' )2 + ( y ' )2 + 1
The initial co nditio ns are
% L x, 0 ! x ! 3
q ( x , 0) = %
(L " x) , L ! x ! L
q ( x , 0) = 0
Beca use q ( x , 0 ) = 0 , all of the ! r va nish. T he r are give n by
r = 2
Fm = m =
Figure 6. Geometry used to determine the forces on a volume of water of mass m, located
on the surface of the earth.
The force exerted by the moon on the center of the earth is equal to
FE = M E =
We can use this relation to calculate db/d and get the following differential cross section:
( ) =
sin 4 ( / 2 )
We conclude that the intensity of scattered projectile nuclei will decrease when the scatter
Let us now consider a system with n coupled oscillators. We can describe the state of this
system in terms of n generalized coordinates qi. The configuration of the system will be
described with respect to the equilibrium state of t