DPG 2007 AKPhil 8.2 Fr 9:30 KIP SR 3.401

**On the post-Newtonian period in the development of mechanics**

Leonhard Euler is famous as the leading mathematician of the 18^{th} century whereas Emilie du Châtelet is known for the translation of Newton’s Principia

into French. Châtelet’s Institutions de physique had been published in 1740, after Eulers Mechanica (1736) but before d’Alembert’s Traité (1743). A German translation, entitled*Naturlehre*, was rapidly published in 1743 while Euler’s comprehensive treatise *Anleitung zur Naturlehre *had been issued only posthumously in 1862. In this contribution it will be demonstrated that the followers of Newton and Leibniz merged and modified basic principles of their predecessors by introducing new principles, exemplified for Euler’s procedure to invent physical notions being completely commensurable with the Leibnizian representation of the calculus. Châtelet based the Institutions on Descartes’ concept of extension, Newton’s *Principia*and Leibniz’s principles of sufficient reason and conservation of living forces. Projected onto Euler’s program, Châtelet’s progress is inherently hampered by the restricted use of the language of calculus whereas the translation suffers losses from the lack of an adequate German physical terminology. Euler elaborated thoroughly both components in the *Anleitung*such that one can make use of Euler’s consistently formulated conceptual frame even for the analysis of contemporary problems.

DPG 2007 AKPhil 8.3 Fr 10:00 KIP SR 3.401

**Euler’s mechanics as a unified theory of matter and motion**

Leonhard Euler (1707 - 1783) is famous as the leading mathematicianof the 18th century. Though his pioneering work on mechanics hadan essential influence in 18th century, its impact on the 19th centuryhas been obscured by the overwhelming success of his mathematicalwritings.The following features make the difference to the theories of Euler’spredecessors Descartes, Newton and Leibniz: (i) a unified approach tomechanics based upon a universal model of the body and the introductionof algorithms for the modelling and solution of mechanical problems,called

Auflösungskunst, (ii) the rigorous statement on the priorityof relative motion, based upon the introduction of observers, calledZuschauer (*Mechanica*, 1736). This is comprehensively elaborated inthe *Anleitung zur Naturlehre*(published 1862, but not mentioned byMach) and maintained in the *Theoria*(1765), completed with the relativemotion of two observers who are comparing their observations.The results confirm (iii) the invariance of the equation of motion ininertial systems (maintained by Einstein) and, (iv) the explanation ofthe origin of forces. Finally, (v) the reliability of mechanics is basedboth upon experience and mathematical foundation of the algorithmswhich are turned out to be in harmony with the physical foundation ofmeasuring procedures. Einstein added the invariance of light velocitypreserving all basic essentials of Euler’s theory.

MathFest 2007, San Jose

**Geometry and Calculus in Euler’s Mechanics**

It will be demonstrated that Euler’s program for mechanics presented in the treatise*Mechanics or the analytical representation of the science of motion*paved the way for a successful development of mechanics in the 18^{th}century. In contrast to Newton’s geometry-related procedure in the

*Principia*Euler formulated mechanicallaws preferentially in terms of the differential calculus. Euler claimed that \those laws of motion whicha body observes when left to itself in continuing either rest or motion pertain properly to infinitely smallbodies". Geometrically, these bodies can be considered as points, but mechanically they are less than anyextended body, but different from mathematical points due to their finite mass. As a consequence, mechanicalquantities are subdivided into two classes, (i) in those which are only represented by

*finite*quantities, likemass and force, and (ii) those which are represented both by *finite*and *infinitesimal *quantities, like translation,time and velocity. Necessarily, the general and fundamental equation of motion *mdv **=* *Kdt*is formulatedin terms of quantities of both types. Completing Newton’s procedure, Euler made a significant progress byadjoining geometry to the calculus since, as it was demonstrated later in

*Institutiones calculi differentialis*, thefoundation of the calculus becomes independent of geometrical models and is based on the transfer of rules forfinite differences to infinitesimal differences. Analytically, motion is described in terms of infinitesimal timeintervals whereas, geometrically, it is related to straight lines and planes as basic elements. Later, Ehrenfest

made use of the same procedure to demonstrate that trajectories are only stable in 3D space.

**Euler Reconsidered** (ed. Roger Baker) Kendrick Press (2007)

http://www.kendrickpress.com/Euler.html

Leonhard Euler is famous as the leading mathematician of the 18

into French. Châtelet’s Institutions de physique had been published in 1740, after Eulers Mechanica (1736) but before d’Alembert’s Traité (1743). A German translation, entitled

DPG 2007 AKPhil 8.3 Fr 10:00 KIP SR 3.401

Leonhard Euler (1707 - 1783) is famous as the leading mathematicianof the 18th century. Though his pioneering work on mechanics hadan essential influence in 18th century, its impact on the 19th centuryhas been obscured by the overwhelming success of his mathematicalwritings.The following features make the difference to the theories of Euler’spredecessors Descartes, Newton and Leibniz: (i) a unified approach tomechanics based upon a universal model of the body and the introductionof algorithms for the modelling and solution of mechanical problems,called

Auflösungskunst, (ii) the rigorous statement on the priorityof relative motion, based upon the introduction of observers, calledZuschauer (

MathFest 2007, San Jose

It will be demonstrated that Euler’s program for mechanics presented in the treatise

made use of the same procedure to demonstrate that trajectories are only stable in 3D space.

http://www.kendrickpress.com/Euler.html

Introduction

1. Newton and Leibniz on time, space, place and motion

1.1. Newton and Leibniz on time and space

1.2. Order and quantification

1.3. Euler’s relational theory of motion and forces

1.4. The modification of Euler’s theory in the 19th century. Helmholtz

2. Euler on absolute and relative motion

2.1. From Ptolemy to Descartes

2.2. Euler on space, time and motion

2.3. Euler’ program for Mechanics. Mechanica 1736 [E15/16]

2.4. Thirty years later. Theoria 1765 [E289]

2.4.1. On absolute and relative motion

2.4.2. The analysis of motion using thought experiments

2.5. The complete analysis. Anleitung 1750, published 1862 [E842]

2.5.1. The introduction of observers or onlookers, Zuschauer

2.5.2. The priority of relative motion

2.5.3. The states of rest and motion. Internal principles

2.5.4. The unified concept of forces. External principles

2.5.5. The invariance of equation of motion

3. Einstein’s theory of relativity

3.1. Mechanics and electrodynamics

3.2. Galileo and Lorentz transformations

Summary and conclusions

References

1. Newton and Leibniz on time, space, place and motion

1.1. Newton and Leibniz on time and space

1.2. Order and quantification

1.3. Euler’s relational theory of motion and forces

1.4. The modification of Euler’s theory in the 19th century. Helmholtz

2. Euler on absolute and relative motion

2.1. From Ptolemy to Descartes

2.2. Euler on space, time and motion

2.3. Euler’ program for Mechanics. Mechanica 1736 [E15/16]

2.4. Thirty years later. Theoria 1765 [E289]

2.4.1. On absolute and relative motion

2.4.2. The analysis of motion using thought experiments

2.5. The complete analysis. Anleitung 1750, published 1862 [E842]

2.5.1. The introduction of observers or onlookers, Zuschauer

2.5.2. The priority of relative motion

2.5.3. The states of rest and motion. Internal principles

2.5.4. The unified concept of forces. External principles

2.5.5. The invariance of equation of motion

3. Einstein’s theory of relativity

3.1. Mechanics and electrodynamics

3.2. Galileo and Lorentz transformations

Summary and conclusions

References