Conservation of Linear Momentum Magnitude Along a Trajectory Explained

Question

I was once criticized for "taking angular momentum as momentum going
in a circle". I was loosely trying to state, in classical mechanics,
that in using conservation of momentum, one can switch between linear
and angular momentum, in a problem when one is not concerned with
rotation of the body on itself, and even treat rotational
motion with linear momentum. I think that, in a very real sense,
angular momentum can also be seen as momentum going in a circle. This is because circular motion can be seen as velocity going in a
circle, meaning of course that its direction changes to stay
tangential, even though there are other more abstract formulations or formalizations with a different dimentionality.

Actually I realize that this is hardly a physics problem, but rather
pure kinematics. Very simply because when considering only linear (not
necessarily straight) motion of a mass, one can simply factor out the
mass and deal with speed and acceleration, rather than momentum and
forces.

The idea is that the speed (momentum magnitude) of a body (mass) is
constant when all accelerations (forces) are orthogonal to the
trajectory, whatever the shape of that trajectory.

This does not seem too original.

It provides a very simple treatment of some single body (angular) momentum conservation
problem, but no one seem to ever use it.

It is particularly useful if one has to analyze strange trajectories,
for example imposed by rails.

Of course, it can be extended to the case of non-orthogonal
accelerations (forces) by projecting the acceleration (force) on the
trajectory tangent to get the speed (momentum magnitude) variation.

So I would like to know the proper mathematical formulation of this,
or a web reference where this is discussed and formulated
mathematically, especially in the case of non orthogonal forces. I
could not find it myself, but it may be a question of having the right
keywords.

I am also curious as to why this seems not much considered in
practice. I feel it gives beginners or amateurs a wrong perception of
momentum conservation laws which are much more interesting when used
to analyze interactions between parts of a system. Dynamics with a
single mass is hardly dynamics.

Answer 1

If you know about vectors then all you need is Euler's laws of motion for a rigid body.

1. For linear motion there is $$ \begin{aligned} \vec{p} &= m\, \vec{v}_{\rm cm} \\ \sum \vec{F} &= m\, \vec{a}_{\rm cm} \end{aligned}$$
2. For angular motion there is $$ \begin{aligned} \vec{L}_{\rm cm} &= I\, \vec{\omega} \\ \sum \vec{M}_{\rm cm} &= I\, \vec{\alpha} + \vec{\omega}\times I \vec{\omega} \end{aligned} $$
3. For any constraint a reaction force does no work, and hence $\vec{F} \cdot \vec{v} =0$

Now you can solve any problem in rigid body mechanics, without friction or contacts.