### Truesdell: Essays in the History of Mechanics

#### by Carson Reynolds

Surprisingly, Truesdell reports that “the main kinematical properties of uniformly accelerated motions, still attributed to Galileo by the physics tests, were discovered and proved by scholars of Merton College — William Heytesbury, Richard Swineshead, and John Dumbleton — between 1328 and 1350. Their work distinguished *kinematics*, the geometry of motion, from *dynamics*, the theory of the causes of motion.”

Here Truesdell provides a concrete definition for dynamics as distinct from mechanics. As an aside, let’s consider exactly what mechanics is, and how dynamics relates to it. Mechanics deals with forces and their effect on bodies. These forces might produce equilibrium, as would be the case with statics, or produce motion as understood by dynamics. Truesdell says further “The connection of force to motion is the primary problem of Western mechanics . . .” This evokes an approach common to mechanics following Newton: namely if dynamics can be generally understood, then much of the results in statics can be addressed by its rules. “They sought a simple, *general* statement about the motion of all bodies, from which the motion of a particular body in particular circumstances would follow mathematically.” Of course, there are other programs for the development of mechanics:

“The purpose of mechanics is to describe how bodies change their position in space with “time.” I should load my conscience with grave sins against the sacred spirit of lucidity were I to formulate the aims of mechanics in this way, without serious reflection and detailed explanations. Let us proceed to disclose these sins.”

-Albert Einstein, *Relativity, the Special and General Theory*

“‘Force is the cause of motion, motion the cause of force’ (Arund. 205*r*.);

Pronouncements of this kind give rise to proceedings toward sanctification of Leonardo as a precursor of Galileo and Newton by naive enthusiasts who select a sentence in which such words as ‘gravity’ and ‘force’ can be interpreted in a more or less modern meaning and enshrine it as a prophecy of modern science.”

“For Example, at one point Leonardo writes, ‘Every action done by natures id done in the shortest way’ (Arund. 85*v*., *cf*. Q. Anat. IV 16*r*), and his enthusiasts have interpreted this pronouncement as implying Fermat’s principle of least time in optics and Maupertuis’ principle of least action in analytical dynamics.”

“At the head of Book I stand the famous *Axioms, or Laws of Motion*:

‘I. Every body continues in its state of rest, or of uniform motion straight ahead, unless it be compelled to change that state by forces impressed upon it

‘II. The change of motion is proportional to the motive force impressed, and it takes place along the right line in which that force is impressed.

‘III. To an action there is always a contrary and equal reaction; or, the mutual actions of two bodies upon each other are always equal and directed to contrary parts.'” “From a few simple axioms, all the major properties of the motions of bodies had been proved.” Newton made no distinction between planetary and terresterial movement, following Kepler.

“While *he* [the modern scientist] may regard Newton’s laws as equivalent to the differential equations called “Newton’s equations” in modern textbooks, there is no evidence that Newton himself thought of ever used his principles in general mathematical form.” “The first to go substantially beyond Newton in the three-body problem was the man who found out how to set up mechanical problems once and for all as definite mathematical problems, and this man was Euler. The year in which the ‘Newtonian equations’ for celestial mechanics were first published is not 1687 but 1749, as we shall see.”

“While Newton’s approach led ultimately to mechanics as we know it today, most of the life work of Euler was required in order to clarify and develop the Newtonian concepts, to supplement them by equally important new ideas, and to demonstrate how real problems can be solved.”

“In Newton’s *Principia* occur no equations of motion for systems of more than two free mass-points or more than one constrained mass-point; Newton’s theories of fluids are largely false; and the spinning top, the bent spring, lie altogether outside Newton’s range.”

Truesdell makes the distinction between the “principle of momentum” and the “law of momentum” employed by Cannon and Dostrovsky. This is an implicit argument about the nature of Newton’s statements on momentum and their historical development.

“His [Euler’s] paper called ‘Discovery of a new principle of mechanics’, published in 1752, presents the equations

Fx=M*ax, Fy=M*ay, Fz=M*az

where the mass M may be either finite or infinitesimal, as the axioms which ‘include all the laws of mechanics.’ Later he called them ‘the first principles of mechanics.'”

“Descartes, Leibnitz (1686, 1695), who by no means despised special problems, introduced the concepts of *live force* and *dead force*. The dead force is only the old force of position, known to the schoolman and nowadays called potential energy, but live force is mass times velocity squared, twice what is now called kinetic energy.”

“Contrary to the usual claims, neither did D’Alembert reduce dynamics to statis, nor did he, here or anywhere, propose either of the two forms of the laws of dynamics now usually called ‘D’Alembert’s principle’, these being due to Euler and to Lagrange, respectively, at a later period. D’Alembert was the first to give a general rule for obtaining equations of motion of constrained systems.” “D’Alembert . . . was the first to derive a *partial differential equation* as the statement of a law of motion . . .”

“Euler sought later an approach to mechanics as a while that would yield directly and easily the equations of motion of a rigid body as a special case. This he achieved in a memoir published in 1776, where he laid down the following laws as applicable to *every part of every body*, whether punctual or space-filling, whether rigid or deformable:

Law 1. The total force acting up the body equals the rate of change of the total momentum. [F=M’]

Law 2. The total torque acting upon the body equals the rate of change of the total moment of momentum, where both the torque and the moment are taken with respect to the same fixed point.” [L=H’]

“Three problems were critical and central for the development of a general mechanics. All three consisted in discovery of differential equations of motion for particular kinds of space-filling bodies:

1. A rigid body.

2. A perfect fluid.

3. An elastic bar.

All three of these problems were solved by Euler.”

“In this way Euler in 1771 was able to write down the general equations of mechanics for a plane deformable line. After doing this, he found it eay to state the general principle of moment of momentum . . .”

me parece muy interesante todo lo de Euler en cuanto al desarrollo de la mecánica. Sobre todo en lo que se refiere a sus fundamentos. Estoy trabajando sobre la introducción de la mecánica en los libros de texto, con énfasis en el concepto de masa, y desde una perspectiva didáctica. Les agradezco el artículo y me gustaría saber dónde hay más sobre el mismo tema.

I would like receive more information on the general equations of mechanics for a plane deformable line by euler and general principle on mechanics of machine.