Difficulty with Mathematical Methods of Classical Mechanics

[curlofgradient]

Homework Statement

A friend and I are going through Vladimir Arnold's Mathematical Methods of Classical Mechanics, but I think my lack of a background in pure math / proofs is seriously hampering my ability to do any of the problems in the first chapter. For example:

PROBLEM. Show that if a mechanical system consists of only one point, then its acceleration in an inertial coordinate system is equal to zero ("Newton's First Law").

Hint. By Examples 1 and 2 the acceleration vector does not depend on \textbf{x}, \textbf{v}, or t, and by Example 3 the vector \textbf{F} is invariant with respect to rotation.​

Homework Equations

Examples 1, 2, and 3 refer to the facts that Newton's equations must be invariant with respect to galilean transformations; 1 is translation through time, 2 is translation through space and the addition of a constant velocity term, and 3 is rotations in space.

The Attempt at a Solution

Honestly, I can't see how the hint doesn't already constitute a solution! I'm not sure what more the book wants from me. I could write down the equations for the three examples, but that seems too trivial. Any help would be appreciated, especially if someone could point me towards resources for learning how to do math-based proofs specifically in the context of physics.

 

[jedishrfu]

You could lookup proof by contradiction and start with the premise that the acceleration =/= zero which would imply that the other vectors are changing over time and then what...

https://en.wikipedia.org/wiki/Proof_by_contradiction

and Newton's first law:

https://www.grc.nasa.gov/www/k-12/airplane/Newton1g.html

https://en.wikipedia.org/wiki/Newton's_laws_of_motion

 

[curlofgradient]

Thanks jedishrfu, but I am not really familiar with knowing when to apply proof by contradiction, etc. I am a graduate student in physics; I have already gone through Goldstein etc. so I am familiar with Newton's laws. What I'm not familiar with is how to prove things. If you give me a system I can find a Lagrangian and solve whatever problem, but if you ask me how to prove something is true about systems in general I am usually at a loss. Just knowing how to solve this particular problem won't help me solve the next problems unless I know the general procedure for solving this sort of thing.

My attempt at this problem so far has been as follows:

By invariance with respect to translations we have \mathbf{a}_i=f_i(\mathbf{x}_j-\mathbf{x}_k, \mathbf{v}_j-\mathbf{v}_k), and the potential can only depend on relative distances between particles. However, since there is only one particle, there are no pairs of particles and therefore f_i must be a constant. Then, I guess somehow we are supposed to use invariance wrt. rotation to prove that the only constant allowed is 0? But it seems to me that any constant vector would be invariant under a rotation...

Also, it is obvious that this sort of answer isn't mathematically rigorous. It would have been helpful if the author did some examples, but obviously the reader is supposed to already be familiar with this sort of procedure. I am wondering what book would help?

 

[jedishrfu]

There's this classic

https://www.amazon.com/dp/3540636986/?tag=pfamazon01-20

Inspired by Erdos and his Book of the most elegant of proofs.