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Courses Differential equations theory course, is it useful?

  1. Sep 21, 2016 #1
    Hello,


    I have a department elective course called "Differential equations theory" but I have no idea if it is going to be useful for me as a physicist (I'm interested in the theory/ minor math).

    The description of the course is as follows : The fundamental theorem of existence and autism, linear differential equations, systems of differential equations, self-system, and change parameters. Solutions volatile, functions characteristics, stability theory, the theory of Abanov.

    I tried to search the web, but I didn't find that much of info about the included description.

    Can you help me on the level of the course and if its useful ?
     
  2. jcsd
  3. Sep 21, 2016 #2

    micromass

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    You should ask the instructor.
     
  4. Sep 28, 2016 #3
    Math theory courses are generally useless for physicists, since they are concerned with questions that are irrelevant to physics. Being able to write and understand proofs has nothing to do with deducing the correct equations from empirical observations and then numerically solving them, which is what you would be doing as a physicist.

    More useful would be a course which covers the numerical solution of differential equations, which is something practically all physicists must learn about and be skilled in.
     
  5. Sep 28, 2016 #4

    micromass

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    That sounds more like statistics than physics....
     
  6. Sep 28, 2016 #5
    Fitting a model from the data is where the statistics comes in, constructing the model involves no statistics, although machine learning is trying to blur this line.
     
  7. Sep 28, 2016 #6

    micromass

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    You can most definitely fit models to data using statistics. It happens all the time. How else would you construct a model other than by using theoretical mathematics? I agree that numerical algorithms are very important, but it is most definitely not the case that this is ALL of physics. For many physics you need theoretical mathematics.
     
  8. Sep 28, 2016 #7
    I don't understand what you're saying. I said, essentially, that statistics only plays a role when I need to compare my model to data, i.e. the physicist is still needed to build the model in the first place, which is not statistics, and the majority of physicists are model builders, not geniuses like Einstein or Newton.

    Theory in mathematics occasionally produces useful proofs regarding convergence of an algorithm or whatnot, and it's nice when that happens, but the practitioner of physics does not need or really benefit from any skill in writing proofs himself or even reading them. They are simply a convenience that appears on occasion; there are many useful algorithms and mathematical methods which do not have these luxuries (simulated annealing, particle swarm, path integrals, deep learning, the renormalization group, the KPZ equation up until recently etc etc).

    Another problem is that mathematicians will attempt to answer questions that most physicists would find uninteresting I would think, such as whether or not an arbitrary differential equation has solutions which are unique. It's sort of like telling an electrical engineer to take quantum physics in the physics department rather than the engineering department; she might care about quantum mechanics insofar as it tells her how transistors work, but she might have zero interest in relativistic quantum mechanics since it's not relevant to her.
     
  9. Sep 28, 2016 #8

    micromass

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    How are Einstein and Newton not model builders?

    OK, so I take it you don't see the solvability of the three body problem and its sensitivity of initial conditions as true physics?

    I understand you're an applied solid state physicist and what you say is true for YOUR field. Please don't generalize it though.
     
  10. Sep 28, 2016 #9
    Well now this is becoming a philosophical debate I suppose, but the first thing to point out is that I distinguish something like Newtonian mechanics from a model of a baseball which employs Newtonian mechanics. Guessing the form of the potential function is to me modeling; constructing a theory of mechanics which is relativistically invariant is different. But the distinction is not wholly clear.

    My reaction is based on the fact that I distinguish theoretical physics from mathematical physics. So far as I know there is no historical precedent for knowledge of theorems and writing proofs making a contribution to theoretical physics, although it has made contributions to mathematical physics. Secondly my reaction is based upon the fact that theoretical physics is much larger than simply quantum gravity or topological materials, which are subfields driven by a new philosophical approach to physics, which I am not judging as right or wrong.
     
  11. Sep 28, 2016 #10

    micromass

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    But proofs are used all the time in physics. In order to study rotation in classical mechanics you need the perpendicular axis theorem. The proof of this result is an actual proof. You seem to be thinking only of difficult proofs in algebraic topology, and I'm not sure why.

    Even in real analysis, the epsilon-delta criterion for continuity is a very physical thing. It describes certain physical motions in a deep way.

    To me, there is no distinction between physics and math. Mathematics is a subset of physics.
     
  12. Sep 28, 2016 #11
    I guess I'm just saying that I think the typical physicist can get away with the physicist's "proof" in Goldstein or what have you and needn't concern herself with a course that covers such things in a much higher level of abstraction and rigor, and moreover that such a course is unlikely to be helpful to a theoretical physicist, but I don't need this to be the hill I die on :p
     
  13. Sep 28, 2016 #12

    micromass

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    And I agree with that. No problem. But that doesn't mean that physics that uses advanced mathematics isn't physics.
     
  14. Sep 28, 2016 #13
    Sorry, what?
    Well anyway, I think it really depends on what you want. First of all, are you actually interested in the course? Is it at the right level for you? (Ask the instructor directly, as micromass said).
     
    Last edited: Sep 28, 2016
  15. Sep 28, 2016 #14
    Ah, that's not what I meant: the usage of say, Lie group theory or differential geometry in HEP or GR is not "not physics" by my view. Elaborating on this, however, would not really be appropriate for this thread.
     
  16. Sep 28, 2016 #15

    micromass

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    GR is not physics?? Or do you mean that GR shouldn't be done by differential geometry?? Both are insane statements.
     
  17. Sep 28, 2016 #16
    There is a double negative in my post :oldwink:

    EDIT: i.e. I didn't say that GR isn't physics or shouldn't involve differential geometry, I said that I consider it to be physics.
     
    Last edited: Sep 28, 2016
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