Is galilean relativity principle derived from the Newton's laws or other statements or not?

It's equivalent to Newton's laws, though, historically, this wasn't understood but much later.

Vice versa. Newton's laws were inspired by Galilean relativity principle, which was stated 50 years earlier (based on ship sailing experience). But Galilean relativity can be trivially derived from Newton laws, if you wish.

Dear Sirs ,

How can the principle be derived from the Newton's laws? Could you write?

If in the 2nd law the force acting on a particle equals zero (far enough from other bodies), the particle can move with arbitrary constant velocity - that is Galilean invariance. As I understand it.

Dear Miccola,

You wrote the 1 st Newton's law which is also called as the law of inertia. The law was discovered by Galilea before Newton. See my first message to know what galilean relativity principle is. It is a classical version of Einstein relativity principle.

Galilean relativity principle is the statement that Newton's laws are invariant under the transformations of the Galilean group. And this can be checked. Conversely, the equations of motion that are invariant under the Galilean group, describe Newton's laws. The simplest way to show this is by using the principle of least action, that provides an equivalent description and by showing that the classical action is invariant under the Galilean group; conversely, writing the most general action, invariant under the Galilean group, that leads to equations of at most second order in time. It's this last property that is required, also-this was noticed, in fact, by Ostrogradskii in the 1850s.

How should you derive the galilean relativity principle from the Newton's laws without the least action principle? Please, write your variants.

The simplest textbook answer I know is as follows. I think it is wrong in general case. Let us have two inertial frames moving at constant speed relativelty each other. There is an accelerating body which we observe in these inertial frames. It is clear that the body acceleration will be the same in the two frames. Then the textbook author assume by definition that masses of the body will also be the same in the two inertial frames. Finally as the force is defined from the 2nd Newton's law one concludes that the forces will be the same in the two frames. So it is proven that the 2nd Newton's law is invariant under the galilean group transformation in this simplest case.

Why do I think the above proof is wrong? Let us to think about the proof from the view of axiomatic physics approach. Consider we have only the three Newton's laws, nothing else. For example any forces laws (e.g. Newton's law of universal gravitation, Hooke's law, etc.) are not discovered so far. The Newton's laws tell only that there are certain quantitative relationships between previously unknown physical quantities: "inertial frame", "mass", "force". The Newton's laws can not tell us about which parameters the above physical quantities depend on. The experiment only tells about it. Therefore in general case a mass can depend on inertal frame parameters (for example parameters of the reference body of the inertial frame, space length change and time intervals change in respect to other inertial frame, etc.). So in the above proof the masses of the accelerating body measured in the two inertial frames can be different in general case.

That is why I think the galilean relativity principle is an independent principle of classical mechanics. The principle can not be derived from the Newton's laws. Please express your view

Dear Anatoly,

If you are satisfied by the 1st Newton's law, which is basically equivalent to Galilean relativity, there is nothing to prove.

But if you mean to axiomatically construct a logically self-consistent mechanics without Galilean relativity, like non-Euclidean geometries proposed by mathematicians in 19th century, that should be possible, of course.

There are plenty of mechanical systems without translation invariance - a pendulum, a bent railway, a body in an "irremovable potential field", etc. But they are used to be well handled by existing formalism: systems with nonlinear constraints - by Lagrangian, potential motions - by Hamiltonian.

So, it is unobvious whether there is need for something new physically. And mathematically, it must be just part of non-Euclidean geometry, already well developed.

Dear Miccola,

I did not understand you. The 1st Newton law is not equivalent to the galilean relativity principle. These statements are different.

According to the principle of Galilean relativity, if Newton's laws are true in any reference frame, they are also true in any other frame moving at constant velocity with respect to the first one.

Newton lived approximately 100 years earlier than Newton. but now we know that Galileo principle is linked with Newtonian low of gravitation, It follows from the Newtonian theory of gravity that all material bodies possess the acceleration g = GM/r*2. Thus principle of Galileo is true at the same distance from the center of the Earth r. Thus Newton theory of gravitation includes principle of Galileo as the partial case.

IMHO, all "derivations" of the Galilean transformations based on Newton's second law are flawed, because they contain an additional assumption: invariance of force and/or acceleration (and of mass, btw). A well known counter-example is electromagnetism, which, ipso facto, is inconsistent with Galilean transformations. We have to make assumptions about dynamics so as not to have internal inconsistensies in the theory. The minimal correct answer therefore is that of Stam Nicolis

Can we write purely mechanical axioms, principles that produce Lorentz transformations and other results of relativistic mechanics?

The below article shows that Lorentz transformations are deduced from Pauli principle. So we should start from quantum mechanics.

“Is galilean relativity principle derived from the Newton's laws or other statements or not?”

- Galileo formulated the experimentally observed by everybody, but understood as the fundamental scientific principle only by him, relativity principle in 1632, whereas Newton, basing on this principle – and on few other Galileo findings, formulated basic laws in mechanics in 1686.

Cheers

The derivation of the Galilean principle could be viewed from a simple perspective. Apply the triangle law of vector addition (vertices: one event and two observers in two inertial frames of reference). As the relative velocity between the frames is uniform, a double derivative with respect to time would give you the required results (a=a' => F=F').