Galilean invariance and kinetic energy

[Order]

I tried to look this up on the internet. I know there is a book about it but I forgot its title.

I know that you can prove that the kinetic energy should be proportional to velocity squared by saying that this is the only Galilean invariant definition of kinetic energy.

Can someone help me remember how this is defined? (Or maybe better, give the title of the book? I would like to read it!)

 

[WannabeNewton]

Are you referring to Landau and Lifgarbagez Classical Mechanics? This is the only book I know of where that argument is given, within the first few pages even. But before you go to the trouble of finding the book, is this the argument you are talking about: https://www.physicsforums.com/showpost.php?p=4380393&postcount=9?

 

[Order]

Are you referring to Landau and Lifgarbagez Classical Mechanics? This is the only book I know of where that argument is given, within the first few pages even. But before you go to the trouble of finding the book, is this the argument you are talking about: https://www.physicsforums.com/showpost.php?p=4380393&postcount=9?

That will have to do if I cannot find anything else. But from what I remember the book had a single author and the argument was easier to follow. I translated it into something that a 15-year old could understand.

I don't quite follow the Lagrangian argument myself actually and why you can't choose it to depend linearly on the speed. (I never took the Lagrangian mechanics course because of studies abroad.)

 

[WannabeNewton]

Oh then I'm not sure which book it is then. The aforementioned book is the only one I know of myself where I've seen an argument resembling what you asked for. Sorry I couldn't be of more help and good luck!

 

[PJC01]

Galilean invariance refers to the principle that the laws of physics should remain the same for all observers moving at a constant velocity. This means that the fundamental physical quantities, such as kinetic energy, should remain the same regardless of the frame of reference.

The concept of kinetic energy is closely tied to the concept of velocity. It is defined as the energy an object possesses due to its motion. In classical mechanics, the kinetic energy of an object is given by the formula 1/2 * mv^2, where m is the mass of the object and v is its velocity.

The idea that this formula is the only Galilean invariant definition of kinetic energy can be understood by considering the implications of Galilean invariance. If we were to define kinetic energy using a different formula, it would have to take into account the relative velocity between the observer and the object. This would violate Galilean invariance, as the value of kinetic energy would be different for different observers.

As for the book you are looking for, it is likely "Galilean Invariance" by Eugene C. G. Sudarshan and N. Mukunda. This book delves into the mathematical foundations of Galilean invariance and its implications in classical mechanics. It may be a helpful resource for understanding the concept in more depth.