I'm a mathematician learning physics from scratch, for my own curiosity and interest. Starting from the basics, I'm trying to get a deep grasp of Newton's laws of motion.
V.I. Arnold describes Galileo's principle of relativity as following (the bold font is my addition):
There exist coordinate systems (called inertial) possessing the following two properties:
- All the laws of nature at all moments of time are the same in all inertial coordinate systems.
- All coordinate systems in uniform rectilinear motion with respect to an inertial one are themselves inertial.
From Newton's laws, we can deduce the existence of inertial systems (the first law) satisfying the second property, and also that the behavior of all mechanical systems is the same in all inertial coordinate systems. If you are in a sealed train, measuring forces between bodies and their resulting motions will not differentiate a train at rest from a train at a constant velocity. But Galileo's principle uses an seemingly "stronger" formulation, stating that no physical phenomenon ("law of nature") will differentiate between the cases.
Are the two equivalent? Does every physical phenomenon stem from the behavior of mechanical systems? Or could we imagine a universe in which Newton's laws are correct but there's still an elusive, tricky experiment to refute Galileo's principle?