50
Introduction
with coordinates for this space, it is ok to think of
Q
as Euclidean space
for purposes of this section. However, the distinction turns out to be an
important general issue.
Example 1.
Here
Q
=
R
3
since a point in space determines where our sys
tem is; the coordinates are simply standard Euclidean coordinates: (
x,y,z
)=
(
q
1
,q
2
3
).
Example 2.
Here
Q
=
S
1
, the circle of radius
R
since the position of the
particle is completely determined by where it is in the hoop. Note that the
hoop’s position in space is already determined as it has a
prescribed
angular
velocity.
The Lagrangian
L
(
q,v
)
.
The Lagrangian is a function of 2
n
variables,
if
n
is the dimension of the conFguration space. These variables are the
positions and velocities of the mechanical system. We write this as follows
L
(
q
1
,...,q
n
,v
1
,...,v
n
)) =
L
(
q
1
n
,
˙
q
1
,...,
˙
q
n
)
.
(1.6.1)
At this stage, the ˙
q
i
s are not time derivatives yet (since
L
is just a function
of 2
n
variables, but as soon as we introduce time dependence so that the
q
i
s are functions of time, then we will require that the
v
i
s to be the time
derivatives of the
q
i
s.
In many (but not all) of our examples we will set
L
=
K
E

P
E
, i.e. as
the di±erence between the kinetic and potential energies.
The sign in front of the potential energy in this deFnition of
L
is very
important. ²or instance, consider a particle with constant mass, moving
in a potential Feld
V
, which generates a force
F
=
∇
V
. We will see
shortly that the equation
F
=
ma
is a particular case of the basic equations
associated to
L
, namely the Euler Lagrange equations, therefore necessarily
we need the minus sign before the
P
E
. There are other deeper reasons why
the minus sign that involve relativistic invariance.
4
Recall also that the