Is motion in infinitesimal interval is linear? As a kind of thought experiment I tried to think if a motion (including circular motion), when divided into infinitesimal time intervals is always linear motion (whether each interval of the motion is at the same direction). Intuitively the answer seems to be yes, but if it is true, I want to hear more comprehensive explanation.
 A: It would be easy to say that the mathematics of regular curves trivially answers your question. And I am afraid that your intuitive answer is also biased by mathematics.
From the perspective of Physics, things are less trivial.
The starting point to analyze your thought experiment is to realize that the macroscopic motions we can observe directly with optical devices like microscopes are limited to object sizes and distances not much smaller than one micron. Below those distances, we need other measuring devices. We know regular curves well model that motion in the 3D space from the macroscopic scale down to a micron-scale. At least for a finite range of time intervals between two measurements.
All this experimental scenario justifies our belief that regular curves can describe motion in a subset of $ \mathbb{R}^3$, parametrized with a continuous parameter $t$ in a closed interval of real numbers. In the model, math tells us that a regular curve looks like a straight line segment in the sufficiently small neighborhood of every point.
However, this is just a fact valid for the model. It is an extremely accurate model if we have to calculate the trajectory of a macroscopic body. Still, the model's validity cannot be extrapolated beyond the scale it has been experimentally tested. If we extrapolate the model towards a much smaller scale of distances and times, we must be aware of the uncertainty connected with any extrapolation.
We know that by increasing our spatial resolution, we arrive at the atomic or sub-atomic scale of distances and times sooner or later. We know for sure (from other experiments) that the concept of regular trajectories breaks down at such a scale. Therefore, the regular curve model should be seen as a completely satisfactory model only within its regime of validity.
A: The infinitesimal is always linear by definition. This comes from the definition in calculus which takes the infinitesimal curve to be its linearisation. In higher dimensions, it is the linearisation of a surface and so on.
Physically speaking, this model breaks down at very small distances. Quantum mechanics suggests that we need a very different idea of the continuum. This is still in the process of being invented ...
A: Imagine a particle following a track. And this track has a 90° turn in it. At the exact moment it reaches the turn, call it $t=0$, what is the direction of velocity? Velocity gives us the infinitesimal displacement of the particle with ${\rm d}\vec{x} = \vec{v}\, {\rm d}t$. These are the little line segments you are talking about.
So to answer your question, we have to decide what to do in this special case. Is there a velocity defined at the sharp corner? If velocity is always defined, then motion can be decomposed into infinitesimal segments.
There are two ways the above example makes physical sense, kind of:

*

*At the exact moment the particle is on the turn $t=0$, the velocity is tangent to the path in the future (positive time). But if you were to rewind time (since laws of physics are time sense invariant) the velocity would have to be tangent to the path in the past (negative).


*At the exact moment the particle is on the turn $t=0$, the velocity is zero and therefore has undefined direction. But how does it move to the next location as time progresses (or reveres) if it has zero velocity? A particle can have zero instantaneous velocity and move if it has acceleration. For example throw a ball in the air, and at the top the ball has zero velocity.
The first approach is the differential geometry approach, dealing with the kinematics of motion, and the second approach is the mechanical approach considering the dynamics of motion.
