Ian Bruce

The first 6 chapters is a sort of update
of the *Mechanica*, which Euler
acknowledges as being lacking in a number of ways, although a great improvement
on what was available previously. The dynamics of particles is presented
without reference to the originators of the theory, whom we might regard mainly
as Galileo, Newton, and Huygens, mostly in an analytical manner, though of course
dynamics has never escaped from the clutches of geometry completely. One sees
here situations where vectors would now be used, and we have to remember that
Euler's exposition set the stage as it were for future developments; his
methods lend themselves to the use of different kinds of forces, knowledge of
which was almost completely lacking at the time. The editor's preface of the
Opera Omnia edition for the whole book has been translated and can be accessed
here. It is a 'must read' in understanding Euler's treatise, and you should
refer to it as needed. As indeed also you may wish to refer to the resume,
where the main results used subsequently are set down again for your
convenience. Note that all text and lines in red on the figures have been added
by the translator to the original diagrams found in the St. Petersburg edition
of this work in the Euler Archive; I am grateful for this resource supplied by
Dartmouth College. I have of course read the *O. O*. edition of this work, and occasionally I note corrections
made by the original editor of this work, Charles Blanc, for which I am also
grateful.

Click here *for the translated O O
preface* :

Click here for the 1st Chapter : *Introduction*
: In this chapter on kinematics, Euler takes a long hard look at the
philosphical ideas then current regarding motion. Having convinced himself that
he is right, he moves on to considering motion along a straight line, along a
plane curve, and finally in three dimensions. Along the way he shows the great
benefits to be made by resolving the speed of a point in either of these
situations into components. One is struck by the apparent modern viewpoint
adopted in this work, although it dates from 1765.

Click here for the 2nd Chapter :*
Concerning the Principles of Internal Motion *: In this chapter basically on
what we now call Newton's First Law, Euler wonders about the situation of
particles within a body, and introduces the idea of external and internal
actions. After considerable reflection he introduces the idea of absolute rest
and motion, in practice rel. to the fixed stars, and introduces the idea of
inertia to account for the state of bodies not affected by external actions.

Click here for the 3rd Chapter : *Concerning
the External Causes of Motion or Forces *: In this chapter basically on what
we now call Newton's Second Law, Euler starts by considering the origin of
force, from which he is led to the idea of impenetrability. He introduces the
concept of inertia, which is identified with mass, and he finally produces what
is Newton's Second Law analytically, as we know it. The need for unit
quantities to be used in defining force is introduced. Euler considers the
resolution of forces along orthogonal axis as a means of shortening
calculations in general. We become aware of how far Euler has moved on in his
thinking since he wrote the Mechanica, and these initial 6 chapters are a sort
of rewriting of this great work, dealing with point masses or corpuscles. We
have done Euler the honour of perpetuating his methods more or less, to the
present day, which is why one has a certain sense of 'deja vu' when viewing
this chapter : but this is where it all began, so to speak. A definite read for
physics students, both young and old!

Click here for the 4th Chapter :*
Concerning Absolute Measures Sought from the Fall of Weights* : In this
chapter Euler replaces his former ratio method of comparing speeds and
distances fallen by weights. There is a general move towards the use of absolute
quantities rather than ratios, and we can see the beginnings of modern
kinematics.

Click here for the 5th Chapter : *Concerning
the Absolute Motion of Bodies Acted on by Forces in General *: Euler finally
presents his new theory of dynamics, in which an analytical form of Newton II
is put in place, forces are resolved along normal axes in 2 and 3 dimensions,
and a number of problems are used to demonstrate dynamics more or less as we
know it today. The absolute nature of physics is presented for the first time,
and there is a move away from the relative ideas prevalent in the Mechanicae;
Euler here tackles the theory of lunar motion from this new base with more
success.

Click here for the 6th Chapter : *Concerning
the Relative Motion of Bodies Acted on by Forces in General* : Euler gives a
detailed discussion of relative motion in two and three dimensions.

Click here *for a summary of the first
10 chapters : *

Click here for the 1st Chapter : *Concerning
the Progressive or Rectilinear Motion of Rigid Bodies. *

The
real work now begins, where the emphasis changes from point masses to extended
bodies......At present chapters 1 - 10 are available in translation. Euler
discusses the definitions of the terms rigid body, progressive motion, centre
of mass, and introduces the idea of internal forces, along with various
theorems and problems for extended bodies.

Click here for the 2nd Chapter : *Concerning
Rotational Motion about a Fixed Axis with no Disturbing Forces. *

Euler
carefully discusses the definition of angular motion of a body about a fixed
axis between two arbitrary points on a laminar body. He goes on to find the
forces exerted on such an axis due to the uniform rotation of the body about
this axis, and shows how such forces change on changing the reference points of
the axis.

Click here for the 3rd Chapter : *Concerning
the Generation of Rotational Motion. *

Euler
carefully discusses the starting conditions for the rotary motion of a body
about a fixed axis between two arbitrary points on a body. The response of the
individual particles of the body are here presented as if the particles were
connected to the axis by threads, and the moment of inertia and related moments
are defined for the first time. The forces and torques exerted on the
supporting points on the axis are examined in depth : eventually these reactive
forces are put to zero when the rotation of free bodies is considered, hence
they are considered to be equal while still finite. This is a long and rather
difficult chapter, containing much material. One notes of course, the initial
idea that a body can have an acceleration, and yet be at rest initially, from
which the dependence of the initial angular increment depends on the square of
the time increment.

Click here for the 4th Chapter : *Concerning
the Disturbance of Rotational Motion Arising from Forces of any Kind. *

Euler
carefully discusses the equation governing the accelerated motion of a body
acted on by moments about a fixed axis between two arbitrary points on a body.
This chapter is useful in explaining some of the lingering mystery of the
preceding chapter, which Euler seems to acknowledge here as being rather vague
at times. He has fixed on the need to develop theorems treating moments of
inertia.

Click here for the 5th Chapter : *Concerning
the Moment of Inertia. *

Euler
sets out the main theorems governing the calculation of moments of inertia,
including the finding of the principal axes. This is a most useful chapter, and
seems to surpass that available in modern texts on dynamics. One can see that
the transition to vector and other notations follows in a simple manner from
Euler's excellent presentation.

Click here for the 6th Chapter : *The
investigation of the Moment of Inertia for Homogeneous Bodies.*

Euler
presents the moments of inertia of number of standard shapes, including a thin
rod and a ring, various laminar shapes incl. a general triangle, and solid
shapes incl. cylinder, sphere, solid of revolution, etc. Chapters 5 and 6
should be examined together.

Click here for the 7th Chapter : *Concerning
the Oscillatory Motion of Heavy Bodies.*

This
is a most interesting chapter, in which the results concerning moments of
inertia are applies to oscillating rigid bodies, incl. physical pendulums with
one or two spherical bobs attached, and an analysis of the forces acting at the
supports is presented for a body oscillating about one or two supports either
horizontally or inclined at some angle. The use of pendulums in determining the
acceleration of gravity is presented in detail.

Click here for the 8th Chapter : *Concerning
Free Axes, and the Motion of Rigid Bodies about such Axes. *

In
this chapter Euler introduces free axes of rotation, which are soon established
to act along the principal axes of the body, but which are not fixed in space.
He goes on to investigate some simple kinds of motion about such axes, making
use of the independence of linear motion and angular motion and shows that any
force acting can be replaced by a couple and a force through the centre of
mass; the couple handles the rotational motion, while the force takes care of
the linear motion.

Click here for the 9th Chapter : *Concerning
the First Generation of Motion in Rigid Bodies. *

In
this chapter Euler continues his treatment of the initial motion of a body, now
free from fixed axes. There are a number of important results here, incl. the
introduction of Euler Angles, the change from arbitrary axes to principal axes,
and the minimal vis viva (read as kinetic energy doubled ) associated with
principal axes. This is a challenging chapter, and may require considerable
thought on the part of the reader to understand, as results are derived from
first principles that nowadays are usually brushed off in text books using
vector products, etc. In this chapter, use is made of spherical trigonometry to
lessen the pain of the increasingly complex three dimensional diagrams used to
resolve forces and establish moments. The internal forces acting on an
arbitrary axis are evaluated, and the initial change in motion found about an
axis .

Click here for the 10th Chapter : *Concerning
the Momentary Change in the Axis of Rotation Produced by Forces. *

You
need to be familiar with Ch. 9 to make progress here, esp. sect. 639. Someone
should perhaps set out Euler's analysis in modern terms, although it is not the
task of this translator, as it would take me too long. It is useful to have a
printed copy of this chapter and the previous one on hand to make headway, as
there is much to-ing and fro-ing amongst long formulas, although the basic
ideas seem simple enough. A good book to have at hand is Rene Dugas's 'History
of Mechanics', Ch. VI, which like all good books is out of print, if you are
thinking of getting one; however, by trying to cover all of Mechanics,
attention to individual authors is rather limited, and Euler has suffered
considerably in this regard.

Click here for the 11th Chapter : *Concerning
the Free Motion of the said Bodies with Equal Principal Axes, Acted on by no
Forces. *

This
chapter is concerned with demonstrating the mixed or combined translational and
rotational motion of a sphere or regular solid. The more complicated bodies to
be considered in the next two chapters can of course be reduced to this simple
case by making the moments of inertia about the principal axes equal.

Click here for the 12th Chapter : *Concerning
the Free Motion of the said Bodies with Two Equal Principal Axes, Acted on by
no Forces. *

This
chapter sets out the case for bodies with two equal axes. Here it is shown how
general rotational motions can be compounded from motions about different axes.
I will comment further on this chapter later, which it is a simpler version of
the one to follow.

Click here for the 13th Chapter : *Concerning
the Free Motion of the said Bodies with Three Unequal Principal Axes, Acted on
by no Forces. *

This
chapter sets out the case for bodies with three unequal axes. Unfortunately the
general equation of motion cannot be integrated using known functions and
lapses into elliptic integrals, and this is presumably where Euler first came
across these things. A compromise is found for some particular motions, which
are presented, relating to the motion of the instantaneous axis of rotation on
a sphere, using spherical triangles. The connection to absolute space via an
extra transformation is also presented. The next chapter deals with external
forces acting on spinning tops.

Click here for the 14th Chapter : *Concerning
the Motion of Spinning Tops on a horizontal plane with all Three Principal Axes
equal to each other. *

The
chapter deals with the external force of gravity acting on a spinning top on a
smooth horizontal plane, with the three moments of inertia equal to each other
for any choice of axes through the centre of mass. The serious reader should
read again perhaps, problems 62 and 68 of Ch. 10. Most of the mathematics is
taken up with referring the axes to a reference frame at rest in an absolute
space, as well as the local frame fixed in the top. Again, the motion of the
centre of mass is abstracted out of the problem. The work is really one of
absolute genius in my humble opinion : the derivation of results more fitting to
vector analysis are produced by Euler referring to the motions of the axes on a
unit sphere, and all from elementary geometrical considerations. A working
knowledge of the sine and cosine rules for the arcs and angles associated with
spherical triangles are needed. I have set out some notes dealing with these in
the first problems, and revised problem 68, which in addition has some more
notes written into the text. Euler tells us that due to the extremely
cumbersome nature of his solution, he is to show another approach in the next
chapter, more suited to such calculations, in which the moments of inertia are
not all equal to each other.

Click here for the 15th Chapter : *Concerning
the Free Motion of Rigid Bodies acted on by any Forces. *

This
chapter sets out Euler's new method for dealing with the problems handled to
date, which he admits to being long and difficult; in a few pages he resets the
stage by realizing that the angular velocity can be resolved along the
principal axes and that the velocity of a point in a body rotating about any
axis can be resolved by a new kind of product, which is now known to us as the
vector cross product. This is derived via spherical triangles, and can be
contrasted with what we know as the dot product, which was introduced in
Problem 68. The previous problems regarding rates of change of axes, etc are
reworked and the same results obtained by much shorter expositions : Euler's
Equations for the motion of a rigid body are the confirmed result of this
exercise, and so in this chapter we see the beginnings of vector analysis.

Click here for the 16th Chapter : *Concerning
the Rotational or Turning Motions of Celestial Bodies . *

This
chapter continues to set out Euler's new method for dealing with the problems
handled to date, here he turns his interest to astronomy. He establishes the
torques acting on a celestial body produced by a distant source of
gravitational forces obeying the inverse square law. He then considers the
moon's librations as a candidate for such torques, which is not very convincing
on physical grounds. He has much more success on revisiting his earlier work on
the precession of the equinoxes treated earlier (E171). In the next chapter he
returns to spinning tops.

Click here for the 17th Chapter : *A
fuller Explanation of the motion of Tops on a Horizontal Plane without
Friction. *

In
this chapter Euler establishes his differential equations arising from the
moments of the reaction force for tops which are figures of revolution. He
looks for integrable solutions to these equations, and finds a minimum
rotational speed for stability in such cases; in the next chapter he looks at
hemispherical tops.

Click here for the 18th Chapter : *Concerning
the motion of bodies with a spherical base on a horizontal plane. *

In
this chapter Euler establishes his differential equations arising from the
moments of the reaction force for tops which in general rotate on a base which
is part of a sphere. He looks for integrable solutions to these equations, but
must be content with s.h.m. solutions for small oscillations. This was the
introduction of what is now called the tippy top, as it is able under certain
conditions to invert itself repeatedly.

Click here for the 19th Chapter : *Concerning
the motion of cylindrical bodies on a horizontal plane. *

In
this chapter the differential equations arising from the moments of the
reaction force for loaded cylinders, for which the centre of inertia does not
coincide with the geometrical centre , are established and solutions found for
the vertical motion. A special kind of cylinder is considered and solved for
s.h.m. as equivalent to a simple pendulum.
An Appendix to follow examines the effect of friction of motion on an
inclined plane.

Click
here for Ch's 1, 2, & 3 of the Appendix, *Supplement concerning the motion of rigid bodies disturbed by friction.*

Euler has put a great deal of thought into the involvement of friction with dynamical processes. It is all very well to say that he was just looking for an excuse to find some interesting phenomena on which he could exercise his amazing calculating powers, but I think that is a very unfair comment. At the time, around 1760, most but not all the facts about friction were known. The main defect was the unreliable nature of experimental results, which masked the distinction between static and kinetic friction. The first chapter sets out the laws of friction as then understood, and applies them to linear motion on a horizontal surface; the second chapter examines motion with friction down an inclined plane, including tipping or rolling of the object. The third chapter is much more extensive, and deals with the rotational motion of a regular body between end cylinders constrained by some kind of bracket, not very sophisticated, but amenable to a detailed analysis; the case of a compound pendulum of some kind held by such a construction is considered in detail. The final Scholium gives details of the remaining chapters, which appear to be quite challenging, as they cover more complex instances of friction.

inclined plane.

Click
here for Ch's 4 & 5 of the Appendix, *Supplement concerning the motion of rigid bodies disturbed by friction.*

In Ch. 4 Euler considers the introduction of friction at the point of the top, proportional to the normal force and opposing motion. He is able to write down the general equations for such a top, but as yet cannot provide a solution to this more general problem.

Ch. 5 is concerned with the motion of a ball or globe on a surface with friction. He considers the general situation where the ball or globe is proceeding in one direction on a horizontal plane with friction present, while it spins and abrades or grazes in another direction along the surface. The initial propositions deal with the kinematics of the situation, and the curve traced out by the centre of mass. The full treatment with torques of forces resolves along principal axes and forces applied the centre of inertia is then put in place. Euler did not use the laws of friction quite as we do now, but one can make allowances for this. This is a long and involved chapter. If someone could find the time to read it through very carefully and provide me with comments for improvement, I would be most grateful from them.

Ian
Bruce. August 10,
2009 latest revision. Copyright : I reserve the right to publish this
translated work in book form. However, if you are a student, teacher, or just
someone with an interest, you can copy part or all of the work for legitimate
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