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The equation of linear motion in a given coordinate axis (say $x$) in my text book:

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And throughout my mechanics course I never encountered derivative with power higher than 2 in EOM?

Why is it so?

Does additional derivatives put on any restriction in a physical sense which although mathematically constructable but not physically possible? (just like imaginary roots in quadratic equation in certain problems had imaginary or negative roots of time)

EDIT: It turns out that there exist two anser which adresses the problem in terms of lagrangian mechanics

Is there any equivalent explanation from the point of view of Newtonian Dynamics?

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marked as duplicate by ACuriousMind Aug 5 '18 at 17:09

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ @ACuriousMind : I went through the links given by the mods as the duplicates .Can somebody post an answer digestible by a high-school student? $\endgroup$ – user203163 Aug 5 '18 at 17:15
  • $\begingroup$ If you are dissatisfied with the answers already present, you have two options: You can ask a question about something specific you didn't understand in those answers, or once you have a bit of reputation you can offer a bounty to solicit answers more suited to you. $\endgroup$ – ACuriousMind Aug 5 '18 at 17:22
  • $\begingroup$ @ACuriousMind: I went through different stuff online to atleast be able to make a gist of what the given answers are saying.I found that Lagrangian formulation is equivalent to Newton's Formulation. So my questions is what would be the equivalent explanation of answers there in newtonian dynamics. Lagrangian might be beautiful to physicist eyes but Force is more intuitive and digestible for an high schooler? $\endgroup$ – user203163 Aug 5 '18 at 17:55
  • $\begingroup$ The principle of relativity of inertial motion and Newton's second law (F=ma) are intimately related. The force law relates amount of change of position to amount of force, and only the second time derivative of position is present in the force law. But in a universe where acceleration would also depend on current velocity there would be no relativity of inertial motion. We assume that in order to exist a universe must be self-consistent. Maybe a universe without relativity of inertial motion cannot be self-consistent. Force law with third derivative: maybe also no self-consistent Universe. $\endgroup$ – Cleonis Aug 6 '18 at 13:15
  • $\begingroup$ This form of your question has been marked as a duplicate. I think you can re-submit this question, asking specifically for an answer in terms of newtonian dynamics. In my opinion that should not be marked as a duplicate. The lagrangian formulation and the newtonian formulation are mathematically equivalent. If a meaningfull answer in terms of lagrangian formulation exists it must be possible to give an answer in terms of newtonian formulation. $\endgroup$ – Cleonis Aug 6 '18 at 13:35
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As a rule of thumb anything higher than a second derivative will correspond to something nonlocal.

Think about it this way, I expect that I should be able to solve my equations of motion by only knowing whats going on locally. But If I had to know all the derivatives of my function, then that's enough to construct the function everywhere using the Taylor series which means that I have to know whats going on everywhere.

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  • $\begingroup$ Can't we evaluate the higher order derivtive using the first and second one? Can't there be feasible complex physical system where this must be case that you need to know whats going on everywhere or at least withing some boundary of the system? $\endgroup$ – user203163 Aug 5 '18 at 17:17

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