“Almost all of fluid dynamics follows from a differential equation called the Navier-Stokes equation. But this general equation has not, in practice, led to solutions of real problems of any complexity. In this sense, the curve of a baseball is not understood; the Navier-Stokes equation applied to a base ball has not been solved.”

— Robert Adair (physicist)

Source: The Physics Of Baseball (Second Edition - Revised), Chapter 2, The Flight Of The baseball, p. 22

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Robert Adair (physicist) 11

Physicist and author 1924

Paul Ormerod book The Death of Economics

Part I, Chapter 5, Mechanistic Modelling, p. 108
The Death of Economics (1994)

“Since the 1950s, the key equation of quantum gravity has been called the Wheeler-DeWitt equation. Bryce DeWitt and John Wheeler wrote it down, but in all the time since then, no one had been able to solve it. We found we could solve it exactly, and in fact we found an infinite number of exact solutions.”

Lee Smolin (1955) American cosmologist

Describing work with Ted Jacobson
"Loop Quantum Gravity," The New Humanists: Science at the Edge (2003)

“The method of successive approximations is often applied to proving existence of solutions to various classes of functional equations; moreover, the proof of convergence of these approximations leans on the fact that the equation under study may be majorised by another equation of a simple kind. Similar proofs may be encountered in the theory of infinitely many simultaneous linear equations and in the theory of integral and differential equations. Consideration ofjkbni semiordered spaces and operations between them enables us to easily develop a complete theory of such functional equations in abstract form.”

Leonid Kantorovich (1912–1986) Russian mathematician

"On one class of functional equations" (1936), as cited in: O'Connor, John J.; Robertson, Edmund F., " Leonid Kantorovich http://www-history.mcs.st-andrews.ac.uk/Biographies/Kantorovich.html", MacTutor History of Mathematics archive, University of St Andrews

“A nonlinear differential equation of the Riccati type is derived for the covariance matrix of the optimal filtering error. The solution of this 'variance equation' completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or non-stationary statistics.
The variance equation is closely related to the Hamiltonian (canonical) differential equations of the calculus of variations. Analytic solutions are available in some cases. The significance of the variance equation is illustrated by examples which duplicate, simplify, or extend earlier results in this field.
The duality principle relating stochastic estimation and deterministic control problems plays an important role in the proof of theoretical results. In several examples, the estimation problem and its dual are discussed side-by-side.
Properties of the variance equation are of great interest in the theory of adaptive systems. Some aspects of this are considered briefly.”

Rudolf E. Kálmán (1930–2016) Hungarian-born American electrical engineer

Source: New results in linear filtering and prediction theory (1961), p. 95 Article summary; cited in: " Rudolf E. Kálmán http://www-history.mcs.st-andrews.ac.uk/Biographies/Kalman.html", MacTutor History of Mathematics archive, 2010

“Fabricating astronomical data going back thousands of years calls for knowledge of Newton’s Law of Gravitation and the ability to solve differential equations.”

N. S. Rajaram (1943–2019) Indian mathematician

N.S. Rajaram: The Politics of History, p.47.

“When an equation…clearly leads to two negative or imaginary roots, [Diophantus] retraces his steps and shows by how by altering the equation, he can get a new one that has rational roots. …Diophantus is a pure algebraist; and since algebra in his time did not recognize irrational, negative, and complex numbers, he rejected equations with such solutions.”

Morris Kline (1908–1992) American mathematician

Source: Mathematical Thought from Ancient to Modern Times (1972), p. 143.

“Ideas and techniques known in quantum electrodynamics have been applied to the Bardeen-Cooper-Schrieffer theory of superconductivity. In an approximation which corresponds to a generalization of the Hartree-Fock fields, one can write down an integral equation defining the self-energy of an electron in an electron gas with phonon and Coulomb interaction. The form of the equation implies the existence of a particular solution which does not follow from perturbation theory, and which leads to the energy gap equation and the quasi-particle picture analogous to Bogolyubov's.”

Yoichiro Nambu (1921–2015) American physicist

[Quasi-particles and gauge invariance in the theory of superconductivity, Physical Review, 117, 3, February 1960, 648–663, 10.1103/PhysRev.117.648]

“Even if without the Scott's proverbial thrift, the difficulty of solving differential equations is an incentive to using them parsimoniously.”

George Pólya (1887–1985) Hungarian mathematician

Mathematical Methods in Science (1977)
Context: Even if without the Scott's proverbial thrift, the difficulty of solving differential equations is an incentive to using them parsimoniously. Happily here is a commodity of which a little may be made to go a long way.... the equation of small oscillations of a pendulum also holds for other vibrational phenomena. In investigating swinging pendulums we were, albeit unwittingly, also investigating vibrating tuning forks.<!--p.224

“Science is a differential equation. Religion is a boundary condition.”

Alan Turing (1912–1954) British mathematician, logician, cryptanalyst, and computer scientist

Epigram to Robin Gandy (1954); reprinted in Andrew Hodges, Alan Turing: the Enigma (Vintage edition 1992), p. 513.

“The physicist needs a facility in looking at problems from several points of view. The exact analysis of real physical problems is usually quite complicated, and any particular physical situation may be too complicated to analyze directly by solving the differential equation. But one can still get a very good idea of the behavior of a system if one has some feel for the character of the solution in different circumstances. Ideas such as the field lines, capacitance, resistance, and inductance are, for such purposes, very useful. … On the other hand, none of the heuristic models, such as field lines, is really adequate and accurate for all situations. There is only one precise way of presenting the laws, and that is by means of differential equations. They have the advantage of being fundamental and, so far as we know, precise. If you have learned the differential equations you can always go back to them. There is nothing to unlearn.”

Richard Feynman (1918–1988) American theoretical physicist

volume II; lecture 2, "Differential Calculus of Vector Fields"; section 2-1, "Understanding physics"; p. 2-1
The Feynman Lectures on Physics (1964)

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Robert Adair (physicist) 11

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