How did Newton establish the conservation laws in the Principia?

Question

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.

Answer 1

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.

Answer 2

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.