Theoretical Physics: Learning to make simple mathematical statements

[WWCY]

Hi all,

I am currently involved in a theoretical physics project and would like some advice. One of the main takeaways I hope to (well,) take away is the capacity to make simple physical/mathematical statements that are valid, which seems to be an important part of theoretical physics. How do you "learn" to do this?

Whenever I try to make a simple mathematical statement, I always happen upon two problems; a) I don't know how to convert my question into mathematics and b) When I do manage to write down the statement, and go through with the proof, I never where in my proof have I made faulty assumptions/leaps of physical and mathematical logic.

While the standard answer I tend to be given is to keep trying and making mistakes as it all comes with experience, is there any better way of getting experience than to just take blind stabs in the dark? I do wish to pursue research as a career, and it'd be nice to pick up such a skill before I start postgrad studies.

Cheers

 

[Dr.D]

It might be helpful to consider your equations as sentences for which the verb (in all cases) is "is." Thus when you write
A = B
you are making the statement "the quantity A is the same thing as the quantity B." The only statements you are allowed to write are those that you know to be true.

 

[WWCY]

Thanks for your response!

It might be helpful to consider your equations as sentences for which the verb (in all cases) is "is." Thus when you write
A = B
you are making the statement "the quantity A is the same thing as the quantity B." The only statements you are allowed to write are those that you know to be true.

However, much of the time it's what comes before writing these equations that gives me grief. Sometimes before writing down the "starting" equation, I need to make some assumptions of the physics behind the problem. So for instance, I would have to say "Consider phenomenon $X$, which can be described by $A_X = B$" yet I can never know for sure if $X$ is indeed described (or most accurately described) by $A_X$. Are there any guiding questions I can ask myself to make sure I'm not making any leaps of physical/mathematical logic?

 

[Dr.D]

If you are trying to describe a mechanics problem, the place to start is probably Newton's Laws (or the Lagrange equations, etc). For that problem, it is unlikely that you will need Maxwell's equations. Similarly, if you want to deal with an optics problem, Newton's Law probably will not be needed. And so on.

My point in the previous paragraph is that we attempt to apply known, general truths, to each new situation. We choose those truths from the ones known to be applicable to the class of problems we are considering.

There is also a corollary to this. If the problem of interest falls outside all classes for which we have established general truth statements, then we are exploring what (at least for us) is new phenomena. All bets are off, and we should probably seek to move ahead via an experiment or two. Later, we may attempt to propose the general truth statements, after we have some experimental evidence for them.