What is the difference between translation and rotation ?
If this were a mathematics site, the question would be at best naive.
But this is physics site, and the question must be interpreted as a
question about physical theory, that is about hypotheses that can be tested,
subjected to experiments and possibly disproved by experiments.
Restatement of the question
After about 3 days, 5 answers, 160 views and some more comments,
taking these contributions into account (hence the length).
First I thank all users who commented or tried to answer my ill
stated question, and I apologize for not doing better. Hopefully, they
did help me improve my understanding and the statement of my
question. You can look at the awkward prior formulations of the
question which better explain the existing answers.
I am trying to understand whether and how translation differs from
rotation, and whether or why it is a necessary physical concept or
possibly only a mathematical convenience.
There are two sides to the issue I am raising, one regarding
space(time) symmetries, and one regarding motion. From what (very)
little I understand of Noether's theorem, these cannot be unrelated as
laws governing motion have to conserve charges that are derived from
the space(time) symmetries. This may have been one source of my
initial confusion.
One of my point is that if there are situations when rotations is not
distinguishable from translation: infinitesimal angles of rotation as
suggested by user namehere. Then relevant phenomena can be
analyzed either as rotations or as translation, with proper
accomodation, particularly to account for the existence of a radius
when rotation is concerned, which changes dimensionality and the
mathematical apparatus.
Of course this requires "care" when considering phenomena involving the
center of rotation or phenomena indefinitely distant.
One example is torque, moment of inertia, angular momentum, vs force,
mass and momentum. The possible undistinguishability of translation and
rotation would seem to indicate that they are really two guises for
the same set of phenomena. They relate to two distinct symmetries, but
is that enough to assert that they are fundamentally different ? This
is precisely what is bothering me in the last comments of user
namehere attached to his answer and motivated my question initially.
Actually, it was someone telling me that "angular momentum is not
linear momentum going round in a circle" that started me on this
issue, as I was not convinced. It may have other manifestations, but it is also that.
I am aware that the mathematical expressions, including
dimensionality, are significantly different for translational and
rotational concepts, and somewhat more complex for rotational
concepts, as remarked by user namehere. But rotational concepts have
to account for the existence of a center and a radius which may be
directly involved in the phenomena being considered: this is typically
the case for the moment of inertia which has to account for a body
rotating on itself.
If we consider a translational phenomenon about force, mass and
momentum, occuring in a plane. We can analyse it indifferently as
translational, or as rotational with respect to a center of rotation
sufficiently distant on a line orthogonal to the plane, so that all
radiuses may be considered vectorially equal up to whatever precision
you wish. Since the radiuses may be considered equal, they can be
factored out of the rotational formulae to get the translational ones.
That is, the rotational mathematics can be approximated arbitrarily
well by its translational counterpart. This should accredit the
hypothesis that it is the same phenomena being accounted for in both
cases.
I am not trying to assert that rotating frames should be inertial
frames. I am only asking to what extent physicists can see a
difference, and, possibly (see below), whether inertial frames
actually exist. When do rotational phenomena differ in substance from
translational ones ? Is there some essential phenomenon that is
explained by one and not by the other ? And conversely ?
Mathematics is only a scaffolding for understanding problems. They are
not understanding by themselves. Mathematical differences in
expression do not necessarily imply a difference in physical essence.
Then one could also question whether translation is (or has to be) a
meaningful physical concept. This can be taken from the point of view
of space(time) symmetries or from the point of view of motion. Why
should it be needed as a physical concept, or can it (should it) be
simply viewed as a mathematical convenience ? Does it have meaning
independently of the shape (curvature) of space ? (I guess relativists
have answers to that).
Take the 2D example of the surface of a sphere. What is translation in
that space ? The usual answer is "displacement along a great
circle". This works for a point, but moderately well for a 2D solid,
as only a line in that solid will be able to move on a great circle.
I guess we can ignore that, as any solid will be "infinitesimal" with
respect to the kind of curvature radius to be considered.
However there is the other problem that, very simply,
every translation is a rotation, and in two different ways, with a
very large but finite radius. But it probably does not matter for
scale reasons.
Now, there is also the possibility that I completely missed or
misunderstood an essential point. Which would it be ?
Best Answer
I hope I am interpreting your question right: One of the defining differences between translation and rotation is that translation is commutative. If I move forward 1 and right 1, it is the same as moving right 1 and then forward 1. The same can not be said of rotation. If I rotate my phone clockwise parallel to my body then clockwise perpendicular to my body I end up with a different end position than if I do the same rotations in a different order.
We don't measure translation or rotation though. We can define a translation by measuring distance, speed, direction, etc. We can define a rotation by measuring angles, angular velocity, direction, etc. These measurements are limited by the precision of our measuring equipment.
Any real motion is a combination of many different motions. Pure translation is really how we simplify real motion by excluding the things we don't care about. It is a simplification, and as such, I'm not sure it needs to be "proven".