Difficulty with Mathematical Methods of Classical Mechanics

In summary, the conversation revolves around a discussion of proving a mechanical system's acceleration in relation to Newton's First Law. The conversation also mentions the use of proof by contradiction and resources for learning how to do math-based proofs in the context of physics. The attempt at solving the problem involves using invariance with respect to translations and rotations, but the process is not mathematically rigorous. A recommended resource for learning more about mathematical proofs in physics is mentioned.
  • #1
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 [itex]\textbf{x}[/itex], [itex]\textbf{v}[/itex], or [itex]t[/itex], and by Example 3 the vector [itex]\textbf{F}[/itex] 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.
 
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  • #3
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 [tex]\mathbf{a}_i=f_i(\mathbf{x}_j-\mathbf{x}_k, \mathbf{v}_j-\mathbf{v}_k)[/tex], 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 [itex]f_i[/itex] 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?
 

Related to Difficulty with Mathematical Methods of Classical Mechanics

1. What is classical mechanics?

Classical mechanics is a branch of physics that deals with the motion of macroscopic objects. It is based on Newton's laws of motion and the law of universal gravitation.

2. What are the mathematical methods used in classical mechanics?

The mathematical methods used in classical mechanics include calculus, differential equations, and vector algebra. These tools are used to describe and analyze the motion of objects in a mathematical way.

3. Why do some people find classical mechanics difficult?

Some people may find classical mechanics difficult because it requires a strong understanding of mathematical concepts and equations. It also involves abstract thinking and the ability to visualize and analyze complex systems.

4. How can I improve my understanding of classical mechanics?

There are several ways to improve your understanding of classical mechanics. One way is to practice solving problems and working through examples. You can also read textbooks or watch video lectures to gain a better understanding of the concepts.

5. What are some real-life applications of classical mechanics?

Classical mechanics has many real-life applications, including predicting the motion of planets and satellites, designing machines and structures, and understanding the behavior of fluids and gases. It also forms the basis of many other branches of physics, such as thermodynamics and electromagnetism.

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