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Lecture 3 – Inertial Reference Frames (Intro to Chapter 2 in F&W)
We have been explicitly making assumptions about the frames of reference in which
we have been defining equations and now we want to make those issues more
explicit. We start with Newton’s second equation in either vector (i.e., linear algebra)
notation
d mr
F
dt
(3.1)
or tensor (i.e., component) notation
j
j
d mr
F
dt
(3.2)
These equations implicitly define a class of reference frames, the inertial frames,
simply by the fact that these equations are true in those frames with no extra
contributions. We are also explicitly using Cartesian coordinates in the second
equation (i.e., rectilinear axes that extend to infinity). We also typically assume that
these axes are embedded in a (3-dimensional) Euclidean (vector) space with a metric
defined so that we know how to construct scalar and vector products. By assumption
we take the Cartesian coordinates axes to be orthogonal, i.e., to have vanishing scalar
products, and to be specified by a (complete) set of (3) unit vectors. By definition
any vector in this space can be represented by a linear combination of the unit vectors
with appropriate coefficients (the components). We start by considering an abstract
inertial frame in some “empty” Euclidean space with enormous symmetry, i.e.,
invariant under (possibly time dependent) translations and rotations. In practice we
have in mind a frame “at rest” with respect to the distant (low number density) stars.
In this lecture we want to consider the range of other reference frames, which we are
transported to by some transformation, such that the form of Newton’s equation is, in
some sense, unchanged.
When we say “unchanged” by some coordinate transformation, there are actually two
relevant levels of constancy. The weaker level is covariance, meaning that in the
new, primed coordinates the equation has the same form,
k k
j j
j k j k r r
d mr d mr
F r F r
dt dt
(3.3)
although the right-hand-side of the new equation is allowed to have a new functional
form in the sense that the individual components are different,
j k j k
F r F r
.
This change in detail still allows the new frame to be inertial. What we are excluding
is the appearance of new terms in the equations (and we will provide examples
below). (Actually certain types of new terms can appear in the component version of
Newton if we transform to curvilinear coordinates still in an underlying inertial
frame.) If the new frame is an inertial frame, the vector version, Eq. (3.1), of Newton
remains unchanged. On the other hand, it is also possible that in the new reference
frame even the details of the component version of the right-hand-side are
unchanged, i.e., that
j k j k
F r F r
. In this case we have an invariance of the
dynamics (a stronger statement than covariance) and, as noted in Lecture 1, there will
be an associated conserved quantity (constant of the motion) as specified by
Noether’s Theorem. Examples are translational invariance when
j k j k
F r F r
leading to momentum conservation and rotational invariance in
the central force problem leading to angular momentum conservation.
Returning to the issue of covariance we consider first the simplest inhomogeneous
transformations, translations. We simply move the origin of the reference frame by a
specific (vector) distance,
0
r
. Thus in the new frame the coordinates are given in
terms of the old components by
0,
0,
k k k
k k k
r r r
r r r
(3.4)
We can rewrite Newton in terms of the coordinates in the new frame as
0,
0,
j j j
j k j k k j k
d mr d mr mr
F r F r r F r
dt dt
(3.5)
It is clear that Newton’s equation is form covariant (and still valid in the new frame if
it was valid in the old frame) as long as there is no new term on the left-hand-side of
It is important to note that, if we are comparing experiments in 2 different
laboratories in two different reference frames moving with respect to each other, we
should use both identical apparatus and identical initial conditions in the two frames.
Thus the initial conditions are identical and not related by the transformation of Eq.
(3.4).
So it is reasonable to ask – what is the complete set of coordinate transformations (the
Galilean transformations) that preserve the inertial properties of the frame of
reference. To answer this question we want consider a general functional form for
the transformations, i.e., the transformations are allowed to depend on the coordinates
themselves. The mathematical background here includes the study of vector spaces
(our Euclidean configuration space,
x
, is such a space), linear algebra and group
theory (see Lectures 4 and 5 on the web page for my Phys. 557 course).
We can express the most general transformation of this nature, including the
possibility that it is nonlinear, as (using component notation)
1 2 3
1 2 3
k k
k k
r r t t g r t r t r t t
r r t t h r t r t r t t
(3.9)
The second equation expresses the inverse transformation, which we require to exist.
The only other required features are that g k
and h
k
are differentiable, single-valued
and invertible.
Note that in the first line we are treating
r
as a function of
r t
and t, while
r t
is
a function of only t (i.e.,
dr dt r t ). The converse is true for the inverse
transformation in the second line.
Considering the time derivatives of these expressions and using the chain rule we can
obtain the corresponding expressions for the transformations of the velocity and the
acceleration (as usual in such an expression repeated indices are summed over),
j j j j j
j k k
k k
j
jk k
dr r r g g
r r r
dt r t r t
g
R r
t
(3.10)
and
2 2 2 2
2 2
2 2 2
2
j j j j j
j k l k k
k l k k
j j j
jk k k l k
k l k
d r r r r r
r r r r r
d t r r r t r t
r r r
R r r r r
r r r t t
(3.11)
where we have introduced the 3x3 matrix
j
jk
k
r
R
r
(3.12)
The inverse transformation for the velocity, for example, has the corresponding form
1
j j j j j
j k k
k k
j
jk k
dr r r h h
r r r
dt r t r t
h
R r
t
(3.13)
with the inverse matrix
1
j
jk
k
r
R
r
(3.14)
We can verify that this is the inverse by considering
1
j j
k
jk kl jl
k l l
r r
r
R R
r r r
(3.15)
where jl
is the Kronecker delta function (the unit matrix) that is 1 for j = l and zero
otherwise.
1 2 3 1 2 3
j j j j
g r r r t a b r r r c t
(3.20)
where a j
and c
j
are constants and b
j
is a linear function of the 3 components of r. The
first and last terms are just the constant translation and transformation to a reference
frame with constant relative velocity mentioned earlier. The fact that the middle term
is a linear function ensures that the matrix R is constant, i.e., not a function of t or the
3 components of r. The final step in defining the Galilean transformation is to require
that
jk jk
T R
(3.21)
In this case both the acceleration and the force transform in the same way and we
have
j
k
j jk k
d mr
d mr
F R F
dt dt
(3.22)
From Eqs. (3.18) and (3.21) it follows that
1 1
jk kj
R R R R
(3.23)
Thus the linear transformation included in the Galilean transformation is described by
a 3x3 constant matrix for which the inverse is equal to the transpose. These matrices
are the orthogonal matrices and the associated transformations are the (global or
rigid) rotations in 3-dimensions. (The name derives from the fact that right angles are
preserved by the transformations.) The associated group is O(3), although we
typically focus on the group SO(3) where the “S” means we do not include
reflections. Examples of reflections are
x x
or
x , y , z x, y, z
.
Mathematically this means that we include only matrices with determinant = +1.
More generally the above property for orthogonal matrices requires only that
2
det RR det R det I det R 1.
(3.24)
ASIDE: The orthogonal 3x3 matrices with determinant +1 constitute the
fundamental representation of the (abstract) group SO(3), i.e., the smallest matrices
whose multiplication properties provide a faithful representation of the
(multiplication) properties of the elements of the group. These matrices and the
group elements are parameterized by three continuous parameters (the Euler
angles), which we can think of as the angles by which we rotate in the three
independent planes, (xy), (yz), and (zx) (or, in the special case of 3-dimensions, as
the angles of rotation about the three unique axes perpendicular to these planes).
This parameterization is differentiable so that SO(3) is a so-called Lie group and we
can define derivatives with respect to each of the angles infinitesimally close to the
origin in parameter space, i.e., close to the identity operator in the group space. The
derivatives serve to define operators that generate infinitesimal transformations and
are called simply the generators of the group. For SO(3) there are 3 generators,
which, with appropriate normalizations, are the angular momentum operators, L
x
,
L
y
, L
z
, familiar from quantum mechanics. The generators serve to define a vector
space, called the algebra of the group, in which the vector product is provided by
the commutator, for example,
x y z j k jkl l
L L iL L L i L
The relevant structure of the algebra, and thus the structure of the group near the
identity, is specified by the 3-index tensor on the right-hand-side of the commutator
equation, which is called the structure constant of the group (the factor of i is a
result of the choice to define the L
k
as Hermitian). The transformations studied by
Lie will play an important role in our discussion of flows in phase space.
In summary we have seen that the form of Newton’s second law is covariant under
the Galilean transformations, which include translation of the origin by a constant
vector (3 parameters), transformation to a reference frame moving with constant
relative velocity (3 parameters) and constant rigid rotations of the Cartesian
coordinate axes (3 parameters). Although we have not discussed it explicitly, it
should be clear that there is also covariance with respect to a constant translation in
time,
t t ,
(3.25)
i.e., physics is the same in different time zones. Overall the Galilean transformations
are described by 10 parameters. Note that, if the form of the force exhibits invariance
with respect to time translations (no explicit time dependence), then we expect the
energy to be conserved.
