# Galilean Relativity is already included in Newton's Laws?

Usually I see an inertial reference frame being defined as a reference frame in which Newton's first and second laws holds. That means that if a particle is at rest, it stays at rest unless some external force acts upon it and that if it is in uniform motion on a straight line it stays that way unless some external force changes that.

Most books I've seem until today after defining one inertial frame that way says that Newton's laws are to valid on inertial frames. That is, they are laws, they are supposed to hold. Is not something like if they hold, it is an affirmation: "they do hold when we work on inertial reference frames".

Now, when reading about Special Relativity, some books says that prior to Einstein there was one "principle of relativity" that could be stated as follows:

The laws of Mechanics are invariant in every inertial reference frame

and that this is the result of Galileo's discussion about Salviatti's ship. I'm having a hard time with this because of the following line of thought:

If Newton's laws do hold on inertial reference frames by the definition of inertial reference frames and by the statement of the laws themselves, why this principle is hanging arround anyway? I mean, it seems to me like something automatic from Newton's laws already.

My question is then: "where this principle of relativity from Galileo enters Classical Mechanics? Is it something that already follows trivially from the laws like I'm supposing or it is something that must be added as another axiom of the theory?"

• "Usually I see an inertial reference frame being defined as a reference frame in which Newton's first and second laws holds." No, only the first law must hold, not necessarily the second law. By the principle of relativity the second law must hold, not by definition. – user70720 Mar 23 '15 at 2:37
• Newton's 2nd law clearly doesn't hold in a non-inertial reference frame as the acceleration of the frame mucks up the $a$ in $F = ma$. – Dai Mar 23 '15 at 13:51
• @Andy You can't possibly deduce the second law from relativity since F=ma/2 or F=-ma are obviously just as consistent with relativity and the first law. And I think you could go further and have laws such as $F=md^3x/dt^3$, which can be as consistent with relativity and the first law. – Timaeus Mar 31 '15 at 3:43

Galilean relativity does not automatically hold in Newtonian mechanics. You are correct that Newton's first two laws appear at first sight to be invariant under galilean transformations, but a galilean transformation only transforms spatial coordinates,they don't actually tell you how forces transform.

If your force laws just has forces that depend only on relative position and relative velocity, then your force law is galilean relativistic in combination with the first and second laws. Your force law could also depend on galilean invariant scalars such as mass or time, which is where the whole thing gets rather problematic.

To be a relativistic theory, all the laws (including the laws that spell out the forces, such as newton's law of universal gravitation) need to have the exact same form in every single inertial frame. Basically, if you can write down your laws without first asking which of the many equivalent frames you are in, then you are fine. If you can't, then you are not a relativistic theory.

Now, when reading about Special Relativity, some books says that prior to Einstein there was one "principle of relativity" that could be stated as follows:

The laws of Mechanics are invariant in every inertial reference frame

That is not the best formulation, since constancy of laws is already assumed implicitly. What is meant by the principle is this:

The processes in a mechanical system occur the same way whether the system is at rest or moving rectilinearly with respect to absolute space (or any inertial reference frame).

My question is then: "where this principle of relativity from Galileo enters Classical Mechanics? Is it something that already follows trivially from the laws like I'm supposing or it is something that must be added as another axiom of the theory?"

The principle is valid for some models and invalid for others. For models of many-particle system where total force on $i$-th particle $\mathbf F_{-i}(t)$ is a function of differences of coordinates and differences of velocities, it is valid. It follows directly from the equation

$$\mathbf F_{-i}( \text{\{}\mathbf r_j-\mathbf r_k\text{\}}, \text{\{}\mathbf v_j - \mathbf v_k\text{\}})=m\ddot{\mathbf r_i}$$ and the prescription for transformation of coordinates and time

$$\mathbf r' = \mathbf r - \mathbf V t$$ $$t'=t,$$ known as the Galileo transformation.

There are other models, where the Newton's laws and the Galileo transformation are valid, but the behaviour of the model itself does not obey the principle of relativity. For example, one model of two particles connected by a spring and experiencing friction force due to surrounding fluid can be formulated by the equations $$-k(\mathbf r_1 - \mathbf r_2) - m\gamma (\dot{\mathbf{r}}_1 - {\mathbf v_f}) = m\ddot{\mathbf r}_1$$ $$-k(\mathbf r_2 - \mathbf r_1) - m\gamma (\dot{\mathbf{r}}_2 - {\mathbf v_f}) = m\ddot{\mathbf r}_2$$ where $\mathbf v_f$ is velocity of the surrounding fluid (constant in any inertial frame). The equations are the same in any inertial frame but the model does not obey the principle of relativity - the particle system behaves one way while in motion with respect to the fluid and differently if at rest.

• This post got rewritten quite a bit and I am still not sure it is in the best shape. I welcome any ideas and suggestions to improve it. – Ján Lalinský May 12 '15 at 21:41