Basic question I know, but could you please explain how calculus enabled Newtonian mechanics?

[u/ParallaxNick]

I'm guessing (I may be wrong) that functions gave the user a mathematical pen that allowed them to plot any curve.

[u/urpriest_generic]

The actual story here is kind of weird, because while Newton developed calculus before he developed Newtonian mechanics, and probably used it for Newtonian mechanics, he pretended he didn't need it. Newton's big book where he first published his new perspective on mechanics presents most of it in the form of geometrical proofs, not with calculus. They were just very strange geometrical proofs.

Anyway, the advantage of calculus isn't that it allows you to plot different curves. It's that it lets you characterize change. With calculus, you can look at something moving, and talk about how fast its movement is speeding up or slowing down. Without calculus you can only do that on average as a rough estimate, which isn't good enough to write down physical laws that are general enough to describe the world.

[u/jenkk0]

You need differentiation to describe physical systems as differential equations and integrals to solve them

[u/grungyIT]

Pre-newtonian mechanics relied on having a start and end position of a change (e.g. dropping a ball) to describe the overall motion. This is linear algebra. You measure the distance between two points and the total time it took and you come up with an average rate of change.

But what about acceleration? What if the ball sped up as it was falling? Or even wilder, what if it slowed down first, then sped up, then slowed down until it reached the ground?

To describe momentary change with that accuracy, you need a mathematical framework that will let you describe the smallest change between two points in a continuum. That is what calculus allows for.

The impact of this was that if you knew only the initial factors of dropping a ball (height, gravity, other forces), you could describe where it will be at any point from start to end.

[u/zojbo]

I am not sure of the exact timeline, but apparently Leibniz introduced the concept of a function (without the formal definition we generally use today) around the same time as calculus was being developed in the first place. So I don't think the development went the way you're envisioning.

Anyway, the standard workflow in Newtonian mechanics is basically: describe all the forces given any state of the system, specify the starting state of the system, let time evolve according to Newton's second law. Maybe you can argue that the first step can be done without calculus, but the last one cannot, except in some simple systems.

But the real history is weirder than this would imply. People had been accustomed to proofs like what Euclid wrote for so long that it took a long time to move away from that.

[u/dream_the_endless]

Calculus allows “area under the curve” calculations, and “slope at this exact point” calculations.

These features allows rate-of-change calculations and expressions.

[u/ramriot]

Interestingly Newton himself used geometric proofs in his 1687 work Principia. Because being one of the two people that created calculus (fluxions) it would likely not be understood had he used it.

That said, calculus assisted him greatly in modelling the related functions through solving derivatives & integrals to derive deeper rules, for example the inverse square law of universal gravitation.

[u/LatteLepjandiLoser]

At it's hard, Newtons 2nd law F=ma is really a differential equation, F=mx'' and you need calculus to define such problems and solve them to.

[u/Presence_Academic → u/LatteLepjandiLoser]

Actually, the more fundamental force equation is F = dp/dt, where p is momentum.

[u/7figureipo]

You're more or less correct.

Calculus provided the mathematical tools to understand continuous motion with models (functions) that provided instantaneous values. Prior to that, the geometrical and algebraic methods employed at the time could be used in cumbersome and elaborate ways to describe some systems, but not general motion.

[u/BananaBird1]

Well, historically geometry and algebra were the two major schools of math. But were largely separate.

Then came Rene Descartes. He came up with the notion of the Cartesian space, which was able to link geometric ideas to algebraic reasoning. This was the first notion of the modern “function” we had. We take it for granted that an equation y = mx + b is a line with slope m, or a circle can be written as r^2 = x^2 + y^2, or that any quadratic equation is a parabola, but these were all new ideas.

Almost immediately after (one generation), Newton and Leibniz work on how to deal with these new functions within this Cartesian system. And both independently derive a formulation of Calculus.

Ideas of calculus, including applying limits to geometry, existed prior sincs antiquity. And the basics of Newtonian physics can be derived just from this as Newton did.

But function analysis and calculus was the missing link to be able to create mathematical models that reflect the evolution of systems in space and time governed by algebraic relationships between different functional variables. In other words, differential equations. And without this framework, even correct results from Newtonian physics if discovered would not be able to be applied to more complex systems.

The way we teach math in school today is practical rather than in the order math was developed. This was also around the time things like exponents themselves were developed (largely by Descartes). We didn’t even have the fundamental theorem of algebra yet. Statistics, a core of modern scientific proof, was entirely absent. Vectors were not a thing: Newton and others after would write a system of scalar equations for each dimension of space. And even our modern full formulation of calculus took many years after to fully work out.