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In the contemporary axiomatic description the conservation laws in mechanics are deduced from Noether's Theorem using the homogeneity and isotropy of space and time.

How did Newton, if he did, establish the conservation of linear momentum and energy in the Principia?

I'm not looking for the most general concept of these as conservation laws; but possibly them in a limited form; say elastic collision.

I've had a brief look and the first surprise is that he eschews the use of calculus and uses instead the theory of conic sections, i.e. he is relying on Euclid's Elements; I take it because this was the most favored method as calculus was in a way too new for natural philosophers then.

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  • $\begingroup$ Newton's work is heavily influenced by natural philosphers preceding him: the celebrated giants his famous quote refers to. In particular, the principle of inertia which nowadays we understand as a conservation law had a long evolution before Newton crystallised it. I believe this hsm post is key to understanding the process. $\endgroup$ – hjhjhj57 May 22 '15 at 22:54
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Newton wrote the Principia in 1684-1686. At that time, he would not have expected any reader in Europe (with the possible exception of a few friends) to have known the method of fluxions (calculus), which he had never published. In order to have his work understood, he needed to write it without using either the techniques or the notation of fluxions. (Leibniz's first publication of the techniques of calculus was not until 1684, so Newton also would not have anticipated that many of his readers could be familiar with the ideas through Leibniz, and in any case readers who knew Leibniz's notation would not have understood Newton's.)

Newton clearly understood what we would today call vectors, as you can clearly tell from the Principia. On pp. 84-86, he presents what we would today call the parallelogram method of vector addition (Cor. I), and states that the momentum vector is conserved (Cor. III). The fact that he doesn't use the term "vector" (which was invented much later) is irrelevant.

He doesn't discuss conservation of energy because conservation of energy wasn't discovered until the 19th century. In the 17th century, there was no notion of the mechanical equivalent of heat, and thermal energy was not conceived of as the kinetic energy of the random motion of molecules. In this period, the phlogiston theory was brand-new.

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In Principia Newton presents a picture based on forces rather than energy and momentum, and he did not have the concepts of vector and of mechanical energy at his disposal. Moreover, Newton opposed Leibniz's idea of putting kinetic energy ("vis viva") at the center of dynamics on philosophical grounds, because he considered forces to be more fundamental. There was no notion of potential energy at the time, and some of the forces treated in Principia, such as air resistance, are not potential, so when they operate even total mechanical energy is not conserved.

Conservation of kinetic energy and momentum was mostly considered at the time in the context of elastic collisions, and there was "vis viva controversy" as to which of them was the "true" quantity of motion. Neither could be considered a universal law at the time, because the nature of heat was not understood, and frame dependent components of a vector were not an item. In fact, Newton worked with two different notions of momenta, mass times speed, going back to Descartes, and the modern one, but only in terms of components. He explicitly notes that Cartesian "motion may be got or lost", and derives conservation of components from the third law of motion:"if the bodies meet with contrary motions there will be an equal deduction from the motions of both".

In an anticipation of modern view Boscovich in 1745 characterized vis viva as the measure of force acting through a distance, and momentum as the measure of a force acting through a time, and D'Alembert included this characterization into the second edition of his Traite de Dynamique (1758). But neither the concept of a vector, nor the general notion of energy, required to give conservation laws centrality in physics, emerged until the 19th century.

As for use of calculus, there is evidence that Newton originally worked out Principia in terms of Euclidean geometry, and connected it to calculus only after the first publication. Conversion of Principia into the calculus notation was done by Euler in Mechanica (1736). Lagrange in Mecanique Analytique (1788) introduced potential energy (without the name, proposed by Rankine in 1853), and explicitly derived conservation of mechanical energy from Euler's reformulation in essentially the modern way. Modernizing the notation, Newton's second law $m_i\frac{d^2x_i}{dt^2}=-\frac{\partial U}{\partial x_i}$, where $U$ is the potential of forces, is multiplied on both sides by $\frac{dx_i}{dt}$ and summed over the (generalized) coordinates. Using the product rule on the left, the mutivariable chain rule on the right, and moving everything to the left gives: $$\frac{d}{dt}\left(\sum_i\frac12m_i\left(\frac{dx_i}{dt}\right)^2+U\right)=0,$$ i.e. the sum of kinetic and potential energies is conserved.

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    $\begingroup$ 'he did not have the concept of a vector'; there's a diagram in the first chapter of the Principia where he explains that the sum of two forces exhibited as two sides of a parallelogram is its diagonal - this looks like a vector; even if not expressed as we might do it today. $\endgroup$ – Mozibur Ullah May 21 '15 at 9:54
  • $\begingroup$ @Mozibur Ullah Such diagrams were drawn already in antiquity, by Heron of Alexandria for example, and possibly even by Aristotle. It looks like "addition of vectors" only because we know what vectors look like. Newton does not manipulate them as items, or expresses them in coordinates, or takes dot products of them, etc. Which is what he would need to express or attempt to prove conservation of momentum. See brief history of vectors here math.mcgill.ca/labute/courses/133f03/VectorHistory.html $\endgroup$ – Conifold May 22 '15 at 1:41
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    $\begingroup$ I agree that there is a distinction; but that shouldn't obviate the fact Newton recognisably had the concept; the article quoted called Newtons use of it vectorial entities; after all just because the integers that we have now are part of the real line - a notion that the Greeks didn't have; should we say that they too didn't have a notion of integers? It shouldn't surprise me that the parallelogram law goes back to Hero, though it does - simply because it's such an intuitive law; it makes me curious as to whether Hero had a concept of force. $\endgroup$ – Mozibur Ullah May 22 '15 at 11:46
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    $\begingroup$ Newton chose to clothe his dynamics into the language of Euclidean geometry, still the standard of rigor in his time, to avoid mathematical objections similar to those later leveled by Berkeley in the Analyst. Can you provide some documentation for this claim? It seems unlikely on the face of it, since it makes him sound like he had precognition. I've never seen this claim made before today. It seems far more likely that he avoided fluxions in the Principia simply because he wanted to be understood by his audience. People in England in 1687 hadn't heard of calculus in any form. $\endgroup$ – Ben Crowell May 22 '15 at 19:43
  • $\begingroup$ The mathpages link is interesting, but it doesn't support your answer, since it says, "Newton obviously recognized the vectorial nature of momentum." I don't think your answer is believable if we simply look at the relevant part of the Principia: archive.org/stream/100878576#page/84/mode/2up If you read pp. 84-86, it's extremely clear that he's presenting what we would today call the parallelogram method of vector addition (Cor. I), and stating that the momentum vector is conserved (Cor. III). $\endgroup$ – Ben Crowell May 22 '15 at 19:55

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