PHYSICS: A BRIEF SUMMARY
MIGUEL A. LERMA
1.
Introduction
This is a brief introduction to Physics intended for “the impatient”.
Its purpose is to give a brief summary of a number of core theories in
Physics. Usually it takes several years for a Physics student to learn
these theories, but for some practical purposes all you need to know
can be told in the time it takes to read a booklet like this one.
This work is conceived as a dynamic document, that will be posted
on the web and modified periodically to expand some sections, correct
possible mistakes, and include further subjects of interest. Look for it
at
~mlerma/courses/e1199s/physics.pdf
Please, send me your suggestions—email address at the end.
2.
Mechanics
2.1.
Newton’s Laws.
Ordinary Mechanics is ruled by Newton’s laws.
The motion of a particle is described by
(2.1)
F
=
m
a
,
where
F
is the applied force,
m
is the mass of the particle, and
a
=
d
v
/dt
=
d
2
r
/dt
2
is the particle’s acceleration, with
v
being its velocity
and
r
is position vector.
In coordinates equation (2.1) looks like this:
(2.2)
F
i
=
m
d
2
x
i
dt
2
(
i
= 1
,
2
,
3)
.
2.2.
EulerLagrange equations.
Newton’s law as described above is
easy to use in Cartesian coordinates for mechanical problems without
constrains, but it can be generalized in a way that makes it easier to
apply to more general situations.
In one dimension Newton’s law is
(2.3)
m
¨
x

F
(
x, t
) = 0
,
Date
: July 10, 2013.
1
2
MIGUEL A. LERMA
where the dot denotes time derivative.
If the force derives from a
potential
V
(
x, t
), then
F
(
x, t
) =

∂V
(
x, t
)
/∂x
. On the other hand, by
using the kinetic energy
T
( ˙
x
) =
m
˙
x
2
/
2, and the momentum
p
=
m
˙
x
=
∂T/∂
˙
x
we see that
m
¨
x
=
dp/dt
, hence
(2.4)
d
dt
∂T
∂
˙
x
+
∂V
∂x
= 0
.
Now we introduce the
Lagrangian
function,
L
(
x,
˙
x
) =
T
( ˙
x
)

V
(
x
),
and the equation becomes:
(2.5)
d
dt
∂L
(
x,
˙
x
)
∂
˙
x

∂L
(
x,
˙
x
)
∂x
= 0
.
Its generalization to any number of (non necessarily Cartesian) coor
dinates
q
1
, q
2
, . . . , q
n
is the
EulerLagrange
equation:
(2.6)
d
dt
∂L
(
q
k
,
˙
q
k
, t
)
∂
˙
q
k

∂L
(
q
k
,
˙
q
k
, t
)
∂q
k
= 0
(
k
= 1
,
2
, . . . , n
)
.
It turns out that not only mechanical systems but also many other
physical systems can be described by an equation like (2.6) with a
suitable Lagrangian
L
. The choice of Lagrangian is dictated by physi
cal experience, although some authors (such as Landau) have tried to
derive it from general principles.
2.3.
Hamilton’s Principle.
The
action
of a physical system with a
given Lagrangian
L
(
q
k
,
˙
q
k
, t
) between times
t
1
and
t
2
is defined by the
integral
(2.7)
S
(
q
k
(
t
)) =
Z
t
1
t
0
L
(
q
k
(
t
)
,
˙
q
k
(
t
)
, t
)
dt.
That integral depends on the path
q
(
t
) followed by the system between
t
0
and
t
1
. Equation (2.6) turns out to be equivalent to the fact that
the action (2.7) is a critical point (usually a minimum) in the space of
paths with fixed endpoints
q
k
(
t
0
) and
q
k
(
t
1
):
(2.8)
δS
=
δ
Z
t
1
t
0
L
(
q
k
,
˙
q
k
, t
)
dt
= 0
.
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 Physics, MIGUEL A. LERMA