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Scientific Technical Review,Vol.LVII,No.3-4,2007

UDK: 512.514. 53:929

COSATI: 12-02

Leonhard Euler (1707-1783)

and Rigid Body Dynamics

Edited by Katica (Stevanovi

) Hedrih, PhD (Eng)

The introductory test of this volume is dedicated to the 300th anniversary of the birth of Leonhard Euler, one of the
most distinguished scientists of the 18th century in the fields of mathematics and mechanics. Born on 15 April 1707 in
Basel, Switzerland, he devoted his life to mathematics instead to theology, much against his father wish and owing to
famous Bernoulli. Although the majority of his work concerned elementary mathematics, he also contributed
significantly to astronomy and physics.
Partially blind in his late twenties and totally blind in his late years, Euler was nonetheless exceptionally productive in
his studies – 500 titles appeared during his lifetime and 400 more were published after his death. Laplace used to tell
his students: “Read Euler, read Euler! He is the teacher of us all!”

Faculty of Mechanical Engineering University of Niš, Mathematical Institute SANU, Aleksandra Medvedeva 14, 18000 Nis, SERBIA

Introduction

statement attributed to Pierre-Simon Laplace

expresses Euler's influence on mathematics: "

Read

Euler, read Euler, he is the teacher (master) of us all

The Euler Committee of the Swiss Academy of Sciences

was founded in 1907 with the task to publish all scientific

books, papers and the correspondence of Leonhard Euler
(1707-1783) in a scientific edition.

Euler was one of the leading and most famous scien-
tists in the mathematics as well as in the mechanics of
the 18th century.

Euler's books and papers are edited in the Series I-III, the

correspondence in the Series IV. In the last 90 years 71
volumes of the Series I-III have been published. The last

three volumes are in preparation and should appear shortly.
The Series IV with Euler's scientific correspondence will

contain 10 volumes. Four volumes have been published and
three volumes are in preparation.

Fugure 1.

Aleksandr Mikhailovich Lyapunov (1857 – 1918) Editor of two

volumes of

Euler’s collected works

This huge endeavor can only succeed with the aid of

internationally acclaimed scientists as coworkers and with
the financial support of the Swiss National Science

Foundation, the Swiss Academy of Sciences and many
long-term substantial contributions from Swiss industrial

corporations.

A. M. Lyapunov opened a new page in the history of

global science. He also contributed as an editor of two

volumes of

Euler's collected works

. He took part in the

publication of Euler's Selected Works and was the editor of

the 18th and 19th part of this miscellany.

The “Euler phenomenon”

Three factors go a long way towards explaining the

“Euler phenomenon”: First of all, his - perhaps unique -
gifted memory. He seemed to have remembered whatever

he had heard, seen, thought, or written in his whole life, as
countless contemporaries confirmed. For example, in his
advanced age he was able to delight his relatives, friends,

and acquaintances with a literal (Latin) recitation of any
song from Virgil’s Aeneis, and he could reproduce by heart

the minutes of the academy meetings decades later, not
mentioning his memory concerning mathematics.

Furthermore, Euler’s prodigious memory went hand in
hand with a rare ability to concentrate. Noise and bustle in
his immediate environment hardly disturbed his thinking:

“A child on his knees, a cat on his back - this is how he
wrote his immortal works,” reported Thiébault, his

colleague from the Berlin academy. The third factor in the
“Euler mystery” is, quite simply, constant, meticulous

work.

Reputation

Leonhard Euler’s influence and reputation were already

impressive during his lifetime. For almost two decades he
was (according to Andreas Speiser) the intellectual leader
of the Protestant part of Germany, and (according to Eduard

Winter) he performed inestimable services as the “golden
bridge between two academies”. The 10 volumes of his

correspondence testify to this role, as does the fact that,
during his Berlin years, Euler published 109 papers in the

K.S.HEDRIH: LEONHARD EULER (1707-1783) AND RIGID BODY DYNAMICS

Petersburger Kommentare and 119 papers in the Mémoires
of the Berlin academy. And although Euler’s energy was

sufficient for him to keep up his activities at both
institutions, the institutions themselves could not easily

cope with the almost inexhaustible tide of Euler’s
productivity. To judge simply from the extent of his work,

Euler is in the company of the most prolific members of the
human race, e.g, Voltaire, Leibniz, Telemann or Goethe.
The directory of Euler’s writings published by G. Eneström

(1910-1913) takes up an entire volume and contains almost
900 titles, some 40 books among them.

Productivity

The following table summarizes the extent of Euler’s

writings specified by him as ready for publication, arranged

according to decades (not included are a few dozen works
that have not yet been dated):

Year Works

1725–1734

1735–1744

1745–1754

150

1755–1764

110

1765–1774

145

1775–1783

270

Specific disciplines

With respect to specific disciplines, the writings are

classified approximately as follows:

−

Algebra, number theory, analysis

40%

−

Mechanics and other physics

−

Geometry, including trigonometry

18%

−

Astronomy

11%

−

Ship theory, artillery, architecture

−

Philosophy, music theory, theology,

and anything else not included above

The classification of Euler’s purely mathematical works is
approximately as follows:

−

Algebra, combination and probability theory 10%

−

Number

theory

13%

−

Fundamental analysis and differential calculus 7%

−

Infinite

series

13%

−

Integral

calculus

20%

−

Differential

equations

13%

−

Calculus

variations

−

Geometry, including differential geometry

17%

Awards

Leonhard Euler won 12 international academy prizes,

not counting the prizes of his sons Johann Albrecht (7) and

Karl (1), which can essentially also be credited to Euler’s
account. The French King Louis XVI awarded Euler 1000

rubles for his “second ship theory”, and the Russian
empress Catherine II gave him double that amount so that

the blind doyen of Petersburg could receive a
supplementary salary in 1773.

Influence

As far as Euler is concerned, the opinions of the most

important mathematicians are unanimous. Laplace used to

say to his students: “Read Euler, read Euler! He is the
master of us all!” and Gauss explained emphatically: “The

study of Euler’s works remains the best instruction in the

various areas of mathematics and can be replaced by no
other.” Indeed, through his books, which are consistently

characterized by the highest striving for clarity and
simplicity and which represent the first actual textbooks in

a modern sense, Euler became the premier teacher of
Europe not only of his time but well into the 19

century.

Formation and Training

1707 Born on 15 April in Basel, the son of the Protestant

minister Paul Euler and Margaretha Brucker.

Leonhard Euler’s father was Paul Euler who studied

theology at the University of Basel and attended Jacob
Bernoulli’s lectures there. In fact Paul Euler and Johann

Bernoulli both lived in Jacob Bernoulli’s house while they
were undergraduates at Basel. Leonhard Euler was born in

Basel, but the family moved to Riehen when he was one
year old and it was in Riehen, not far from Basel, that

Leonard was brought up. Paul Euler had some
mathematical training and he was able to teach his son
elementary mathematics along with other subjects.

Euler’s father wanted his son to follow him into the

church and sent him to the University of Basel to prepare

for the ministry. He entered the University in 1720, at the
age of 14, first to obtain a general education before going

on to more advanced studies. Johann Bernoulli soon
discovered Euler’s great potential for mathematics in
private tuition that Euler himself engineered.

In 1720 Leonhard entered Basel University, which was

founded in 1460. Initially he studied theology, Oriental

languages and history, but soon switched to mathematics under
Johann Bernoulli (1667-1748), who became the world's most

cited mathematician following the death of Isaac Newton
(1643-1727). Quick to recognize Euler's mathematical genius,
Bernoulli challenged him by having him read the works of the

masters, and especially by instructing him personally in
contemporary mathematical research.

In 1723 Euler completed his Master’s degree in

philosophy having compared and contrasted the

philosophical ideas of Descartes and Newton. He began his
study of theology in the autumn of 1723, following his
father’s wishes, but, although he was to be a devout

Christian all his life, he could not find the enthusiasm for
the study of theology, Greek and Hebrew that he found in

mathematics. Euler obtained his father’s consent to change
to mathematics after Johann Bernoulli had used his

persuasion. The fact that Euler’s father had been a friend of
Johann Bernoulli’s in their undergraduate days undoubtedly
made the task of persuasion much easier.

Nicolaus Bernoulli

(1623-1708)

Jakob Bernoulli

(1654–1705)

Nicolaus I Bernoulli

(1687-1759)

Nicolaus Bernoulli

(1662–1716)

Johann Bernoulli

(1667–1748)

Nicolaus II Bernoulli

(1695–1726)

Johann III Bernoulli

(1744–1807)

Daniel Bernoulli (1700–

1782)

Johann II Bernoulli

(1710–1790)

Nicolaus Bernoulli

(1623-1708)

Daniel II Bernoulli

(1751–1834)

Figure 2.

Family tree of the Bernoulli family

Euler completed his studies at the University of Basel in

1726. He studied many mathematical works during his time
in Basel, and Calinger reconstructed many of the works that
Euler read following the advice of Johann Bernoulli. They

include works by Varignon, Descartes, Newton, Galileo,
van Schooten, Jacob Bernoulli, Hermann, Taylor and

Wallis. By 1726 Euler had already a paper in print, a short
article on isochronous curves in a resisting medium. In

K.S.HEDRIH: LEONHARD EULER (1707-1783) AND RIGID BODY DYNAMICS

1727 he published another article on reciprocal trajectories
and submitted an entry for the 1727 Grand Prize of the

Paris Academy on the best arrangement of masts on a ship.

In 1727, Catherine I of Russia invited Euler to join the

faculty of the Academy of Sciences in St. Petersburg. He
became chairman of mathematics there in 1733, replacing

Daniel Bernoulli. In 1735, he lost sight in one eye while
working around the clock for three days to solve a
mathematic problem that took other mathematicians months

to solve. While in Russia, he prepared some 90 papers for
publication and wrote the two-volume book Mechanics. He

collaborated with Daniel Bernoulli in the field of fluid
mechanics and derived the equation that related velocity

and pressure, which became known as Bernoulli's equation.
He also conceived of pressure as something that could
change from point to point throughout a fluid.

Figure 3.

Methodus inveniendi - Leonhard Euler

The cover page of Euler's Methodus inveniendi lineas curvas.

In 1741, at the urging of Frederick the Great, Euler

moved to Berlin and became professor of mathematics at
the Berlin Academy of Sciences, which he turned into a
major academy. Over the next 25 years, Euler prepared at

least 380 papers for publication. After his relationship with
Frederick deteriorated, he accepted the invitation from

Catherine the Great to return to St. Petersburg in 1766
where he became director of the Academy of Sciences.

Soon after his return, he became almost totally blind.
Nevertheless, he excelled at solving complex calculations
in his head. While in St. Petersburg, he worked on

developing a better theory of lunar motion that involved the
interactions of the Sun, Moon, and Earth.

Euler contributed to the subjects of geometry, calculus,

trigonometry, and number theory. He standardized

mathematical notation using Greek symbols that continue to
be used today. He also contributed to the fields of
astronomy, mechanics, optics, and acoustics, and made a

major contribution to theoretical aerodynamics. He derived
the continuity equation and the equations for the motion of

an in viscid, incompressible fluid.

Euler suffered a stroke and died on September 18, 1783

in St. Petersburg.

First Petersburg period

1727 Euler's thesis entitled De Sono (On Acoustics)

formed the basis for his application for a post as professor
of physics in Basel, but he was passed over on account of

his youth. Through the help of the Bernoullis, he was
offered a position in St. Petersburg at the Academy of

Science, founded by Peter the Great in 1725. There he
worked first as an assistant professor, then from 1730 as a
professor and member of the academy (he had no teaching

commitments, though he did write a textbook on
elementary mathematics). The principal contributions of

this early Petersburg period include a two-volume work on
mechanics, a book on music theory and Scientia navalis

(about hydrodynamics, shipbuilding and navigation), which
was eventually published in 1749.

1734 At the beginning of January, Euler married

Katharina Gsell, a daughter of a Swiss painter George
Gsell, who was working in St. Petersburg. Euler's son

Johann Albrecht was born at the end of November, the only
one of his offspring to follow in his footsteps as a

mathematician and member of the Academy. Only three of
Euler's thirteen children would survive him. He had twenty-
one grandchildren.

1738 As a result of a severe abscess, Euler lost the sight

of his right eye.

Berlin years

1741 Conscious of the political turmoil in the Russian

empire, Euler accepted Frederick II's offer of a

professorship at a newly established Prussian Academy
("Berlin Academy") and settled with his family in Berlin.

There he held a position as director of the mathematics
department. Maupertuis, who in 1736 made a name for
himself in a famous expedition to Lapland (the purpose of

which was to determine whether the Earth was indeed an
oblate spheroid) became president of the Academy, though

as a scientist, he ranked far below Euler.

In addition to hundreds of treatises written during the

Berlin period, Euler produced major works on the calculus
of variations, the theory of special functions, differential
equations, astronomy as well as a second masterpiece on

mechanics and a popular work on physics and philosophy
titled Lettres à une princesse d'Allemagne. The basic

outline of his celebrated work on algebra also dates from
the Berlin period. During this time, Euler maintained active

connections with the Petersburg Academy, and he helped to
promote interactions between the two internationally
renowned academies. Euler was a member of all the

important academies of his time and received many awards.

Second Petersburg period

1766 Frederick II's bumbling was influential in Euler's

accepting an offer from Catherine the Great to return to St.
Petersburg, where he remained until his death.

1771 In the aftermath of a failed cataract operation,

Euler lost the sight of his remaining good eye and soon
became nearly completely blind. During the great St.

Petersburg fire, he was saved from his burning house at the
last moment by the Basler artisan Peter Grimm. Yet,

amazingly, his productivity increased: approximately half
of his prodigious output occurred during this second

Petersburg period, including three-volume works on

K.S.HEDRIH: LEONHARD EULER (1707-1783) AND RIGID BODY DYNAMICS

integral calculus and optics (Institutiones calculi integralis
and Dioptrica) as well as the authoritative version of his

work on algebra.

1773 Following the death of his wife Katherina, in 1776

Euler married her half-sister Abigail Gsell.

1783 On 18 September Euler suffered a stroke and died

quickly and painlessly.

Euler's contributions to mathematics cover a wide range,

including analysis and the theory of numbers. He also

investigated many topics in geometry.

Rigid body Kinematics

Euler's contributions to mathematics cover a wide range,

including analysis and the theory of numbers. He also
investigated many topics in geometry with application to the

.kinematics. Euler equations in the area of rigid body
kinematics - rotation about a fixed point - are very important

and applicable in future development in dynamics.

Figure 4.

Sphere geometry of rotation

Figure 5.

Kinematics of the rigid body rotation around a fixed point- Euler’s

angles:

angle of precession,

angle of nutation

angle of 8 self rotation)

New Vector to Rigid body rotation around a fixed point and
instantaneous axis through a fixed point

Let us consider the special case of rotation around the

fixed point

and around the moving axis oriented by the

unit vector

with the rotation around the fixed point

using Euler’s angles: angles

of precession , angle

nutation an angle

of self rotation as well as the mass

moment vectors conected by the fixed point

and the

rotating axis, also around the fixed point.

Using Euler’s angles

(

)

, ,

ψ ϑ ϕ

, the Euler angular

velocities are defined as follows:

- anglular velocity of

precession, is defined by

in the direction of

- angular velocity of nutation is defined by

in the

direction of

(knot axis) and

-angular velocity of self

rotation is defined by

ϕ ′

in the direction of

′

(axis

of body self rotation). Overall instantaneous angular
velocity of the body rotation around the fixed point is:

( )

[ ]

[

]

[ ]

[

]

( )

[ ]

[

]

( )

[ ]

( )

Figure 6.

Rigid body dynamics around the fixed point. Mass moment vectors

coupled for the fixed point and the instantaneous axis and Linear momentum
and angular momentum for the fixed point and the rigid body dynamics

ω ϑ

′

with the components in the fixed coordinate system

Oxyz

(see Fig.7):

≡

−

∗

Figure 7.

∗

and b

∗

Dynamics of the rigid body rotation around the fixed

point- Euler’s angles:

- roll,

- pitch and

- yaw.

(

angle of precession ,

angle of nutation an

angle of self rotation)

K.S.HEDRIH: LEONHARD EULER (1707-1783) AND RIGID BODY DYNAMICS

(

)

cos

sin sin

ψ ϕ

(

)

sin

cos sin

ψ ϕ

−

(

)

cos

ψ ϕ

or in the scalar form

cos

sin sin

ψ ϕ

cos

sin cos

ψ ϕ

−

cos

ψ ϕ

= +

The components of the instantaneous angular velocity in

the moving coordinate system

ξηζ

fixed with the rigid

body are (see Fig.7):

(

)

cos

sin sin

ϕ ψ

′

−

(

)

sin

cos sin

ϕ ψ

(

)

cos

ϕ ψ

′

or in the scalar form

cos

sin sin

ϕ ψ

−

sin

cos sin

ϕ ψ

cos

ϕ ψ

= +

The linear momentum (impuls of the motion) and the

angular momentum (kinetic moment of the motion) of the
body rotation around the fixed point

expressed by the

body mass moment vectors are in the following forms:

−

∗

The linear momentum (impulse of the body mass mo-

tion)

( )

p t

of motion of a material system, or a rigid

body rotating around a fixed axis

with the angular ve-

locity

ω ω

is given in the following form:

( )

[ ]

[

]

( )

p t

n r dm

n r

∫∫∫

r r

where

( )

[ ]

n r dm

∫∫∫

r r

( )

is the mass linear moment of the body with respect to

the pole

and for the axis oriented by the unit vector

passing through the point

(see Refs. Hedrih 1991, 1992,

1993a,b, 1998a,b,c, 2001,2007).

−

∗

Angular momentum

for the rigid body rotating

around the fixed axis oriented by the unit vector

through the pole

, with the angular velocity

ω ω

in the following form:

[ ]

[

]

( )

, ,

r n r

∫∫∫

r r r

where we introduce the following notation

( )

[ ]

[

]

, ,

def

r n r

∫∫∫

r r r

for the mass inertia moment vector

( )

for the pole

and

the fixed axis oriented by the unit vector

The mass inertia moment vector

( )

has two

components: one component

is in the rotation axis

direction and corresponds to the axial body mass inertia
moment and the second component

( )

is orthogonal to

the rotation axis and is the deviational component in the
deviational plane.

( )

can be expressed in the following

form (see Refs. Hedrih 1998a,b,c, 2001,2007):

( )

(

)

( )

n J

J n

⎡

⎤

⎡

⎤

⎣

⎦

⎣

⎦

Kinetic energy for that case of the model motion is:

(

)

[ ]

[

]

(

)

( )

(

)

[ ]

( )

, , ,

, J

n r n r

n r

∫∫∫

r r r r

r r

For obtaining the necessary mass moment vector

( )

for the rotation axis oriented by the unit vector

through

the pole

, or the mass moment vector

( )

for the parallel

axis oriented by same unit vector

through the rigid body

mass center

, we can use the following vector

expressions:

( )

[ ]

[

]

( )

, ,

cos

r n r

∫∫∫

r r r

( )

[

]

[

]

( )

, ,

cos

∫∫∫

( )

[

]

[

]

, ,

n r

r r r

where cos

cos

and

cos

are cosines of the direction of

the unit vector

rotating axis orientation with respect to the

corresponding system coordinates,

is the rigid body mass,

and

is the vector position of the rigid body mass center.

For this considered case of the material system model

rotation around the fixed point, it is necessary to point out that
both vectors of mass moments are changeable with respect to
the fixed point and instantaneous axis of rotation. This happens

because the instantaneous axis and the rigid body change their
relative positions and the relation to each other during the body

rotation motion around the fixed point.

Euler equations in the vector form, using the mass

moment vectors coupled to the fixed point and the
instantaneous axis are:

( )

dp t

ω ω

⎡

⎤

+ +

⎣

⎦

∑

( )

, J

r F

r G

ω ω

⎡

⎤

⎡

⎤

⎡

⎤

⎣

⎦ ⎣

⎦

⎣

⎦

∑

, (A)

as well as

, L

r F

r G

⎡

⎤

⎡

⎤

⎡

⎤

= +

⎣

⎦

⎣

⎦ ⎣

⎦

∑

, (B)

This previous vector equation (B) is equation of the

K.S.HEDRIH: LEONHARD EULER (1707-1783) AND RIGID BODY DYNAMICS

motion and it is possible to express it in the scalar form by
three differential equations of Euler‘s type:

(

)

(

)

(

)

r F

r G

−

⎡

⎤

⎡

⎤ ′

′

⎣

⎦

⎣

⎦

∑

(

)

(

)

(

)

M ,

r F

r G

−

⎡

⎤

⎡

⎤ ′

′

⎣

⎦

⎣

⎦

∑

(

)

(

)

(

)

r F

r G

−

⎡

⎤

⎡

⎤ ′

′

⎣

⎦

⎣

⎦

∑

or: in the following shorter form:

−

For the case in which the coordinate axes of the moving

coordinate system fixed to the rigid body are body principal
mass inertia moment axes, from the previous Euler system

of differential equations we have:

(

)

1 01

2 3

ω ω

−

(

)

2 02

1 3

ω ω

−

(

)

3 03

1 2

ω ω

−

The first vector equation from the system (A) of the

vector equations is possible to obtain kinetic pressure to the

fixed point (bearing, or to kinetic reaction or necessary
action by the body).
Citations
* The most cited mathematician of all times

* Over 20 formulae of elementary mathematics bear Euler’s
name
* An even greater number of formulas and notions of

advanced mathematics bears his name.
* The most fruitful mathematician of all times

(800 pages per year)

Some elementary Euler formulae and theorems

Figure 16.

Euler’s line

Figure 8.

Euler’s line

cos

sin

Figure 9.

Euler’s formula

Graph theory

Figure 10.

The problem of seven Koenigsberg bridges

If all nodes are even, the graph can be drawn by starting

from each of them (1736).

A graph with more than two odd nodes cannot be drawn

in a single stroke.

Figure 11.

Examples of graphs.

K.S.HEDRIH: LEONHARD EULER (1707-1783) AND RIGID BODY DYNAMICS

Concluding Remarks

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 18th century. In contrast to Newton's

geometry-related procedure in the Principia, Euler
formulated mechanical laws preferentially in terms of the

differential calculus. Euler claimed that ''those laws of
motion which a body observes when left to itself in

continuing either rest or motion pertain properly to
infinitely small bodies''. Geometrically, these bodies can be
considered as points, but mechanically they are less than

any extended body, but different from mathematical points
due to their finite mass. Analytically, motion is described in

terms of infinitesimal time intervals whereas,
geometrically, it is related to straight lines and planes as

basic elements.

In fluid dynamics, the

Euler equations

govern the

compressible, in viscid flow. They correspond to the

Navier-Stokes equations with zero viscosity and heat
conduction terms. They are usually written in the

conservation form shown below to emphasize that they
directly represent conservation of mass, momentum, and

energy.

Momentum Equation for Frictionless Flow: Euler’s

Equation

Euler’s Equation

− ∇

In the Cartesian coordinate system

⎛

⎞

∂

−

⎜

⎟

∂

⎝

⎠

∂

⎛

⎞

∂

−

⎜

⎟

∂

⎝

⎠

∂

⎛

⎞

∂

−

⎜

⎟

∂

⎝

⎠

In the cylindrical coordinate system

⎛

⎞

∂

−

⎜

⎟

∂

⎝

⎠

V V

⎛

⎞

∂

−

⎜

⎟

∂

⎝

⎠

∂

−

∂

⎛

⎞

∂

−

⎜

⎟

∂

⎝

⎠

Continuity

∇ ⋅ =

Euler's contributions to Rational or Mathematical Fluid

Mechanics are important contributions as well as his

contribution to the General Theory of the Motion of Fluids.
Euler's generalization of the stream function concept to a pair

of stream functions or stream surfaces is also the result of his
research. Euler's Potential had been formulated about 100

years before a similar detailed mathematical exposure was
formulated by Jacobi and Clebsch around 1844. Furthermore,
Euler's work initiated the establishment of naval science in

Russia and influenced the art of building naval ships in
Russia in the 18th century in particular. An overview about

Euler's accomplishments in Naval Architecture and Ship
Hydrodynamics at Russia's St. Petersburg Academy of

Sciences is also an important contribution to world science
and to mechanical engineering.

The theory of magnetism for which he was awarded the

Paris Academy Prize in 1744 is also very well known. Euler
began with Descartes' idea that all magnetic phenomena are

elicited from the circulation of imperceptible conduits
throughout the corpuscular magnetic body. Euler imagined

that the magnetic body possessed pores which formed
continuous piping, parallel and bristling, similar to veins or

valves and so narrow as to only allow passage for the most
subtle parts of the ether, the elasticity of which pushes the
relaxed parts into the magnet pores. Then the force causes it

to bend onto itself at the exit only to return again and form
a type of vortex. Through this ingenious idea which was

developed after much thought, Euler was able to explain
magnetic phenomenon. The hypothesis was proved out by

experiments and these conformed to natural laws which in
turn ensured its ultimate probability.

The Opera Omnia

"Read Euler, read Euler, he is the teacher (master) of us all".

Pierre-Simon, marquis de Laplace (March 23, 1749 - March 5, 1827)

Published by Birkhauser and the

Euler Commission

Switzerland, the

Opera Omnia

is the definitive printed

source for Euler's works. The publication began in 1911,
and to date 76 volumes have been published, comprising

almost all of Euler's works.

Principle:

“

Mathematicians do not have biography, they have

bibliography

”

References

The URL of this page is:

http://www-history.mcs.st-andrews.ac.uk/Biographies/Euler.html

http://www.leonhard-euler.ch/

For more information about the Euler Commission please contact Herr

Prof.Dr.Hanspeter Kraft President of the Euler Commission
Hanspeter.kraft@unibas.ch

K.S.HEDRIH: LEONHARD EULER (1707-1783) AND RIGID BODY DYNAMICS

[1]

FINKEL,B.F. (1897).:

Biography - Leonard Euler

, The American

Mathematical Monthly 4 (12): 300.

[2]

IOAN,J.(2002):

Remarkable Mathematicians: From Euler to von

Neumann

, Cambridge, 2. ISBN 0-521-52094-0.

[3]

CALINGER,R. (1996):

Leonhard Euler: The First St. Petersburg

Years (1727–1741)

, Historia Mathematica 23 (2): 156.

[4]

CALINGER,R. (1996):

Leonhard Euler: The First St. Petersburg

Years (1727–1741)

, Historia Mathematica 23 (2): 125.

[5]

CALINGER,R. (1996):

Leonhard Euler: The First St. Petersburg

Years (1727–1741)

, Historia Mathematica 23 (2): 127.

[6]

CALINGER,R. (1996):

Leonhard Euler: The First St. Petersburg

Years (1727–1741)

, Historia Mathematica 23 (2): 128–129.

[7]

HOME,R.W. (1988):

Leonhard Euler's 'Anti-Newtonian' Theory of

Light

, Annals of Science 45 (5): 521–533.

[8]

O'CONNOR,J., EDMUND,J., ROBERTSON,F.: "Leonhard Euler".

MacTutor History of Mathematics archive

[9]

Leonhard Euler at the Mathematics Genealogy Project

[10]

WEISSTEIN,E.W.: Euler, Leonhard (1707–1783) at ScienceWorld.

How Euler did it Website containing columns explaining how Euler solved

various problems.
[11]

GLADYSHEV,G.,P. (2007): “Leonhard Euler’s methods and ideas
live on in the thermodynamic hierarchical theory of biological

evolution,”

International Journal of Applied Mathematics &

Statistics

(IJAMAS), Special Issue on Leonhard Paul Euler’s:

Mathematical Topics and Applications (M. T. A.), Vol. 11, Nu. N07,

November, 2007.

[12]

CONDORCET,M.J.:

Eulogy for Euler

(See this link).

[13]

G DU PASQUIER:

Leonhard Euler et ses amis

(Paris, 1927).

[14]

SPEISS,O.:

Leonhard Euler

(1929).

[15]

THIELE,R.:

Leonhard Euler

(Leipzig,1982).

[16]

YUSHKEVICH,A.P., WINTER,E. (eds.): Leonhard Euler and

Christian Goldbach: Briefwechsel 1729-1764 (Berlin, 1965).

[17]

ANDREWS,G.E.:

Euler's pentagonal number theorem

, Math. Mag.

56 (5) (1983), 279-284.

[18]

ANTROPOV,A.A.:

On Euler's partition of forms into genera

Historia Math. 22 (2) (1995), 188-193.

[19]

Chobanov,G., Chobanov,I.: Lagrange or Euler? I. The past,

Annuaire

Univ. Sofia Fac. Math. Méc.

73 (1979), 13-51.

[20]

CHOBANOV,G., CHOBANOV,I.: Lagrange or Euler? II. The
present,

Annuaire Univ.

Sofia Fac

Math. Inform

82 (2) (1988), 5-62.

[21]

CHOBANOV,G., CHOBANOV,I.: Lagrange or Euler? III. The
future,

Annuaire Univ. Sofia Fac. Math. Inform.

82 (2) (1988), 63-

109.

[22]

DAVIS,P.J.: Leonhard Euler's integral: A historical profile of the
gamma function,

Amer. Math. Monthly

(1959), 849-869.

[23]

FELLMANN,E.A.: Leonhard Euler 1707-1783 : Schlaglichter auf

sein Leben und Werk,

Helv. Phys. Acta

(6) (1983), 1099-1131.

[24]

FINKEL,B.F.: Biography. Leonard Euler,

Amer. Math. Monthly

(1897), 297-302.

[25]

FRASER,C.G.: The origins of Euler's variational calculus,

Arch.

Hist. Exact Sci.

47 (2) (1994), 103-141.

[26]

GRAU,C.: Leonhard Euler und die Berliner Akademie der
Wissenschaften, in

Ceremony and scientific conference on the

occasion of the 200th anniversary of the death of Leonhard Euler

(Berlin, 1985), 139-149.

[27]

GRAY,J.: Leonhard Euler 1707-1783,

Janus: archives

internationales pour l'histoire de la medecine et pour la geographie
medicale

72 (1985), 171-192.

[28]

GRAY,J.: Leonhard Euler: 1707-1783, Janus 72 (1-3) (1985), 171-192.

[29]

HOLLINGDALE,S.H.: Leonhard Euler (1707-1783):

a bicentennial

tribute

, Bull. Inst. Math. Appl. 19 (5-6) (1983), 98-105.

[30]

HOME,R.W.:

Leonhard Euler's 'anti-Newtonian' theory of light

Ann. of Sci. 45 (5) (1988), 521-533.

[31]

KNOBLOCH,E.:

Leibniz and Euler: problems and solutions

concerning infinitesimal geometry and calculus

, Conference on the

History of Mathematics (Rende, 1991), 293-313.

[32]

Loeffel,H.: Leonhard Euler (1707-1783),

Mitt.

Verein. Schweiz

Versicherungsmath. (1) (1984), 19-32.

[33]

MIKHAILOV,G.K.:

Leonhard Euler and his contribution to the

development of rational mechanics

(Russian), Adv. in Mech. 8 (1)

(1985), 3-58.

[34]

SAMELSON,H.:

In defense of Euler

, Enseign. Math

(2) 42 (3-4)

(1996), 377-382.

[35]

TREDER,H-J.:

Euler und die Gravitationstheorie

, in Ceremony and

scientific conference on the occasion of the 200th anniversary of the
death of Leonhard Euler (Berlin, 1985), 112-119.

[36]

TRUESDELL,C.:

Euler's contribution to the theory of ships and

mechanics

, Centaurus 26 (4) (1982/83), 323-335.

[37]

TRUESDELL,C.: Prefaces to volumes of Euler's Opera Omnia.

[38]

TYULINA,I.A.:

Euler's hydraulic studies

(Russian), in Investigations

in the history of mechanics 'Nauka' (Moscow, 1983), 167-177.

[39]

MAANEN,J.A.: Leonhard Euler (1707-1783):

man, worker, migrant

genius (Dutch), Nieuw Tijdschr. Wisk. 71 (1) (1983/84) 1-11.

[40]

VELDKAMP,G.R.: Leonhard Euler (Dutch), Nieuw Tijdschr. Wisk.

71 (2) (1983), 47-54.

[41]

YUSHKEVICH,A.P.:

Leonhard Euler and mathematical education

in Russia

(Russian), Mat. v Shkole (5) (1983), 71-74.

[42]

YUSHKEVICH,A.P.:

Life and mathematical achievement of Leonard

Euler

(Russian), Uspekhi Mat. Nauk (N.S.) 12 4(76) (1957), 3-28.

[43]

YUSHKEVICH,A.P.: Leonhard Euler :

Life and work

(Russian), in

Development of the ideas of Leonhard Euler and modern science
“Nauka” (Moscow, 1988), 15-46.

[44]

RAŠKOVI

,D.,

Dinamika

, Nau

na knjiga, Beograd, 1956.

[45]

HEDRIH,S.K. (2001): V

ector Method of the Heavy Rotor Kinetic

Parameter Analysis and Nonlinear Dynamics

, University of Niš

2001,Monograph, pp.252. (in English), YU ISBN 86-7181-046-1.

[46]

KECHRIS,K.: Leinhar Euler

[47]

http://en.wikipedia.org/wiki/Template:Bernoulli_family

[48]

www.mathematik.ch

[49]

http://en.wikipedia.org/wiki/Leonhard_Euler

[50]

http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Euler.html

[51]

HEDRIH,S.K. (1991): A

nalogy between models of stress state, strain

state and state of the body mass inertia moments

, Facta Universitatis,

Series Mechanics, Automatic Control and Robotics, Niš, vol.1, N 1,
1991, pp.105-120.

[52]

HEDRIH,S.K. (1992):

On some interpretations of the rigid bodies

kinetic parameters

, XVIIIth ICTAM HAIFA, Abstracts, pp.73-74.

[53]

HEDRIH,S.K. (1993a):

The mass moment vectors at n-dimensional

coordinate system

, Tensor, Japan, Vol.54 (1993), pp.83-87.

[54]

HEDRIH,S.K. (1993b):

Same vectorial inter-pretations of the kinetic

parameters of solid material lines

, ZAMM. Angew. Math. Mech.

73(1993) 4-5, T153-T156.

[55]

HEDRIH,S.K. (1998a):

Vectors of the Body Mass Moments

Monograph paper, Topics from Mathematics and Mechanics,

Mathematical institute SANU, Belgrade, Zbornik radova 8(16), 1998,
pp.45-104.

[56]

HEDRIH,S.K. (1998b):

Vector method of the kinetic parameters

analysis of the rotor with many axes and nonlinear dynamics

, Parallel

General Lecture, Third International Conference on Nonlinear

Mechanics, August 17-20, 1998, Shanghai, China, pp.42-47.

[57]

HEDRIH,S.K. (1998c):

Derivatives of the Mass Moments Vectors

with Applications

, Invited Lecture, Proceedings, 5th National

Congress on Mechanics, Ioannina, 1998, pp.694-705

[58]

HEDRIH,S.K. (2001): Vector Method of the Heavy Rotor Kinetic
Parameter Analysis and Nonlinear Dynamics, University of Niš

2001, Monograph, pp.252. (in English), YU ISBN 86-7181-046-1.

[59]

HEDRIH,S.K. (2001): Derivatives of the Mass Moment Vectors at
the Dimensional Coordinate System N, dedicated to memory of

Professor D. Mitrinovi

, Facta Universitatis Series Mathematics and

Informatics, 13 (1998), pp. 139-150. (1998, published in 2001. Edited
by Milovanovi

,G.).

[60]

HEDRIH,S.K. (2007):

For optimal time of study: vector and tensor

methods in classical mechanics

, Plenary Lecture, Proceedings

Nonlinear Dynamics, Dedicated to the 150th Anniversary of A.M.

Lyapunov, Polytechnic Kharkov, 2008. pp.98-107.

Received: 15.05.2007.

K.S.HEDRIH: LEONHARD EULER (1707-1783) AND RIGID BODY DYNAMICS

Leonardo Ojler (1707-1783) i mehanika krutog tela

Uvodni tekst ovog broja

asopisa posve

en 300. godišnjici od ro

enja Leonarda Ojlera, jednog od najpoznatijih

nau

nika XVIII veka iz oblasti matematike i mehanike. Ro

en je 15. aprila 1707 godine u Bazelu, Švajcarska.

Zahvaljuju

i znamenitom Bernuliju, koji je uticao na njegovog oca da promeni svoju odluku, posvetio se matematici

umesto teologiji. Iako je ve

i deo njegovog rada bio iz oblasti elementarne matematike dao je zna

ajan nau

doprinos i u oblasti astronomije i fizike.
U svojim radnim dvadesetim godinama Ojler je izgubio oko. Kasnije je ostao i bez drugog oka, ali je uprkos tome bio
vrlo produktivan. Tokom života objavio je oko 500 knjiga i radova, a još oko 400 radova objavljeno je posthumno.
Laplas je govorio svojim studentima: "

itajte Ojlera,

itajte Ojlera! on je gospodar sviju nas"

Leonhard Euler (1707-1783) et la mécanique du corps solide

L’éditorial de ce numéro est dédié au 300

ième

anniversaire de la naissance de Leonhard Euler, savant célèbre du 18

ième

siècle, très connu pour ces travaux en mathématique et mécanique. Il est né le 15 avril 1707 à Bâle, en Suisse. Il devait,
selon le désir de son père, étudier la théologie, mais grâce à célèbre savant Bernoulli qui a influencé le père de
Leonhard, celui-ci a pu se consacrer aux mathématiques. Bien que la majeure partie de son travail concerne les
mathématiques élémentaires, sa contribution est considérable en astronomie et en physique. A l’âge de vingt ans
Euler a perdu un œil; plus tard il a perdu son autre œil, mais malgré cela il était très productif. Au cours de sa vie il a
publié environ 500 livres et travaux; après sa mort, on a publié encore 400 travaux. Laplace disait souvent à ses
étudiants: "Lisez Euler, lisez Euler! Il est maître de nous tous".

K.S.HEDRIH: LEONHARD EULER (1707-1783) AND RIGID BODY DYNAMICS

Leonhard Euler