Newton's 2nd law of motion is most often written in the differential form

$\sum F = {dp \over dt} $

but can also be expressed in an integral form

$ p = \int\sum F dt $

Each form of expressing the connection between force and momentum is mathematically correct, but if we adhere to causality we are limited to the second expression (Forces lead to momentum (movement) - not vice versa).

This limitation is further exemplified when one attempts to simulate the equations in a digital computer. If feedback is involved the first expression can lead to algebraic loops, while the second gracefully evolves simulated motion. Algebraic loops do not represent physical reality (even the fastest of all physical systems are limited by the speed of light!)

So other than what is defined to be the independent/dependent variables, and/or the operation of either differentiation vs integration, there is nothing in the mathematics that tells us which is the 'correct' formulation of the laws of motion. The presence of differentiation in the mathematics implies prediction - knowing the future and therefore expresses the physical system in a non-causal manner. Rewriting the system in an integral form expresses the system in a causal way. The integral form in a sense reveals the roles of each factor in cause and effect.

Are there perhaps other ways (missing equation(s), other types of mathematics) that can help to resolve the ambiguity that calculus can cause? Is $F = ma$ incomplete? Perhaps some expression involving entropy is required for completeness. Physical reality favors memory and shuns prediction.

BTW Newton's law just served as a simple example towards my question. The ambiguity can be realized in any other mathematics involving calculus that model physical systems (including Maxwell's equations).

  • $\begingroup$ Causality is definitely encoded in Maxwell's equations. Are you implying Newton's second law isn't causal somehow? $\dot{p}$ depends on the local in time and space force, whatever it is. $\endgroup$ – webb May 28 '14 at 21:32
  • $\begingroup$ They are both correct formulation of the laws of motion. But one formulation is useful when you know $p(t)$ and the other is useful when you know $F(t)$ which may depend on what you are modeling. $\endgroup$ – tpg2114 May 28 '14 at 21:48
  • $\begingroup$ Taking the derivative of any quantity implies prediction. Reality never involves predicting the future. Therefore Newton's 2nd law is only causal when expressed in mathematics without derivatives. Writing the expression in an integral form reveals the causality of the physics. This is also true for Maxwell's equations. Is there another way to redeem any reality in expressing physics using derivatives? $\endgroup$ – docscience May 29 '14 at 14:42

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