In mechanics, observers are of paramount importance, since each observer observes from a given system of reference or frame of reference or simply frame.
We can ask whether the laws of motion change if we change from one frame to another. The answer is yes: in general, the equations of motion depend on the chosen frame. If there is an observer travelling in a roller coaster and such observer is paying attention to a simple motion, the laws for this motion can look daunting.
However, the purpose of theoretical mechanics is to deal with frames for which the laws of mechanics take their *most simple* form. It is an aesthetic purpose, if you like.
A property is *homogeneous* when it has the same value everywhere. For example, imagine a perfectly white wall: in it, colour would be an homogeneous property, since every point in the wall has the same colour. And, if the wall remains white forever, colour is also homogeneous in time.
A property is *isotropic* when it has the same vector value for all directions. If we locate ourselves at the centre of a perfect sphere, all directions look the same to us.
What we want for our laws of motion is that they don't depend on the particular position an object is, or on the particular time we observe. We neither want them to depend on the direction we look, or whether we observe from present to future or from present to past. Well, regarding the isotropy of time, maybe we would be happy to find mechanical laws that are different from present to future than from present to past, but it seems that it is not the case.
We can always choose a frame or reference for which space is homogeneous and isotropic and for which time is homogeneous. (We postpone the discussion of isotropic time.) We call such frame an inertial frame, or an inertial system of reference (iframe) .
In an inertial frame, a free body which is at rest will always remain at rest. This is a beautiful foundation: it finds the most extreme simplicity of a frame by choosing a free body in an homogeneous and isotropic space and in an homogeneous time. From such foundation we can build all mechanics. We could have made another choice, but good luck trying to depart from another foundation!
The homogeneity of space implies that the Lagrangian cannot depend on |r>. Otherwise, some points would be different than others, which would be not an homogeneous thing.
The homogeneity of time implies that L cannot be an explicit function of t. But wait, we have been considering L(q,q̇,t) all the time! Yes, but now we are considering a free particle (or an isolated mechanical system), so for such cases, L cannot explicitly depend on t.
If L cannot depend on t and |r> but we also know that L does not depend on second derivatives and beyond, what is left? Only velocity |v>. For now, L can only depend on |v>.
But wait again! Space is also considered isotropic, which means that our laws cannot depend on whether we look from one orientation or another. This kills the dependence on |^v^>, so we are left with just v. But beware of v, since it can have a minus or a plus sign, and this goes against isotropy as well. This means we have v² as the only explicit dependence for L. Conclusion: for a free particle, L=L(v²)=L(<v|v>), or if you prefer, L=L(|v|).
Using the E-L equations, the term ∂L/∂|r> is = |o>. This leaves us with
(d/dt)(∂L/∂|v>) = |0> => ∂L/∂|v> = |constant>.
But as L=L(<v|v>), every time the operator ∂/∂|v> hits a <v|v> inside L, ∂/∂|v> must kill all velocity dependence on <v|v>. Otherwise we would end up with a ∂L/∂|v> depending on |v>. Also, if you are confused by bra's and ket's, just write all velocities as bra's. The operator ∂/∂|v> hits |v> and <v| indistinctly. In summary, if ∂/∂|v> over <v|v> must give a constant, since ∂(<v|v>)/∂|v> = 2|v>, this can be constant only if
|v> = |constant>.
Conclusion, free particles have constant velocity when observed in inertial frames. This is the *law of inertia*.
If we choose another inertial frame that moves uniformly in a straight line wrt to our previous inertial frame, the laws of motion must be the same in that second frame, so for an observer in such frame, free particles will also move with constant velocities. Those velocities will be different wrt to the first frame, but the equations of motion will be the same.
From experiments, we learn that the laws of motion in two inertial frames are the same not only in free motion: they are entirely equivalent in all respects.
We also know that there are infinite inertial frames that we consider: all of them must be mechanically equivalent, meaning that the laws of motion are the same for all of them. The properties of space and time are also the same for all of them. This is called the Galileo's relativity principle: the paramount foundation of classical mechanics.
So inertial frames are special, and by no means the only frames we can choose. By default, we are always going to consider inertial frames, unless we specify the contrary.
Galileo's principle means that there does not exist a privileged or absolute inertial frame over another. Hence the term relativity.
Let K be an iframe and K' another iframe moving with velocity |u> wrt K. A particular point is observed in K as having the radius vector |r>, while the same point is observed by K' with the vector radius |r'>. These two radius vectors are related by
|r> = |r'> + |u>t.
What is a static point of K is a moving point for K'. There are no absolutely static points in the universe.
In classical mechanics, if K observes an instant of time t and K' observes an instant t', we can relate these observations by
t = t'.
The assumption that time is absolute (the same for all inertial observers) is one of the foundations of classical mechanics.
The two latter expressions are called a Galilean transformation.
The equations of motion of classical mechanics must be invariant under such a transformation.
The two assumptions under a Galilean transformation agree with experiments up to a certain degree of accuracy. In particular, for objects moving at slow speeds wrt the speed of light, the experimental agreement is extraordinary. However, classical mechanics breaks for speeds which start to approach the speed of light. This is not the subject of this study, though. And you should not downplay the importance of this classical approximation, since all our everyday experience and intuition is based on it. Moreover, the speed of light is a huge quantity, and unless you are considering very extreme conditions, the approximation of classical mechanics is beyond excellent. Never try to move to Einstein's relativity until you completely master Galileo's relativity and Newtonian mechanics.