2.2
Calculus of variations
61
and this is a larger number.
We have found that, sure enough, tcycloid < tstraight . The particle spends less
time on the cycloid than on the straight slide, in spite of the fact that loses speed
on the way up toward (x1 , z1
Chapter 1
Newtonian mechanics
1.1
Reference frames
An important aspect of the fundamental law of Newtonian mechanics,
F = ma,
(1.1.1)
is that it is formulated in a reference frame which is either at rest or moving with
a uniform velocity (the velocity mus
146
Hamiltonian mechanics
We have learned that |J| = 1, and in practice we may always design the canonical
transformation so that its Jacobian is in fact J = +1. This gives us, then, the
statement that [f, g]Q,P = [f, g]q,p , and the Poisson bracket is in
2.9
Additional problems
109
(b) Derive the equations of motion for the particle. Show that the equation
for can be expressed in the form
1 2
+ () = ,
2
and nd an expression for the eective potential (). (The constant
is proportional to the particles tot
1.4
Mechanics of a system of bodies
25
But dr1 dr2 = d(r1 r2 ) = dr12 , so this can be expressed as
2
W12 =
1
F12 dr12 + F13 dr13 + F23 dr23 .
At this stage of the derivation we incorporate the fact that the mutual forces
are derived from a potential. As
3.3
Liouvilles theorem
133
p
R(0)
q
Figure 3.11: The initial region R(0) in phase space has an elliptical boundary. The
ellipse is centered at (, p) = (0, p0 ) and it has semiaxes and p . An angle (not shown)
parameterizes the position on the boundary.
th
2.5
Generalized momenta and conservation statements
85
On the other hand,
L
qa
qa
=
f
f
qb +
qb
t
b
2f
2f
qb +
,
qa qb
qa t
=
b
and this is equal to the previous result, because the order in which one evaluates
second partial derivatives does not matter.
3.2
Applications of Hamiltonian mechanics
121
limited to the interval 0 0 , and we have the usual situation of a pendulum
oscillating back and forth between the limits 0 . The phase trajectories representing this bounded, oscillatory motion are closed cur
2.1
Introduction: From Newton to Lagrange
49
for this problem, and we write q . (We do not need a label a in this case, as
there is only one generalized coordinate.) The relation between and the original
Cartesian coordinates is x = sin and z = cos , with
1.3
Mechanics of a single body
13
l cos
x
l
l sin
z
Figure 1.4: Motion of a pendulum. For convenience the z axis is taken to point downward. The angle between the position of the pendulum and the vertical is denoted (t).
T
T
Tcos
Tsin
mgcos
mg
mg
Figur
2.7
Motion in a rotating reference frame
97
^ = r
z ^
Q
up
r
east
so
ut
h
P
^ ^
y =
^ ^
x =
Figure 2.20: The local Cartesian frame (x, y, z) at P . The x direction points south, the
y direction points east, and the z direction points up. The position ve