ArticlePDF Available

Inertia forces and methodology of mechanics

Authors:
  • Dniepropetrovsk National University, Ukraine

Abstract and Figures

It is shown that the present-day methodology of mechanics treating “inertia forces” as fictitious ones is inconsistent and leads to a contradiction in continuum mechanics. Newton's laws inevitably lead to the conclusion about the existence of real “Newtonian inertial forces” at all points of accelerated bodies. But the usual formulation of the second law excludes these forces from the class of “physical forces”. A real sense of the d'Alembert principle consists in changing the treatment of this law, by considering the term mW as a force, which allows one to bring both laws of mechanics in concordance. The translatory and Coriolis inertial forces are components of the real Newtonian inertial force. Forces acting at all points of a body are the same in all reference frames, inertial and non-inertial ones.
Content may be subject to copyright.
239
Mechanics of Systems with Internal Resonances
240
Appendix
Inertial Forces and Methodology of
Mechanics
1
It is shown that the present-day methodology of mechanics treating
“inertia forces” as fictitious ones is inconsistent and leads to a
contradiction in continuum mechanics. Newton's laws inevitably lead to
the conclusion about the existence of real “Newtonian inertial forces” at
all points of accelerated bodies. But the usual formulation of the second
law excludes these forces from the class of “physical forces”. A real
sense of the d'Alembert principle consists in changing the treatment of
this law, by considering the term mW as a force, which allows one to
bring both laws of mechanics in concordance. The translatory and
Coriolis inertial forces are components of the real Newtonian inertial
force. Forces acting at all points of a body are the same in all reference
frames, inertial and non-inertial ones.
1. Introduction
Fundamentals of mechanics and its methodology have been well
established during the XVIIXVIII centuries, in the times of Newton and
Lagrange [1,2]. But there is an important point in the foundation of
classical mechanics that until now has not received unambiguous
treatment. This is the problem of inertia forces, their reality or fiction. It
is obvious that this is not a particular question, but a principal problem of
the methodology of mechanics.
Generalizing and simplifying the situation, one can distinguish three
views on inertial forces the position of physics and those of
theoretical mechanics and applied mechanics.
In physics courses only the translatory and Coriolis forces, appearing
in description of relative motion, i.e., a motion in non-inertial reference
1
Translation of paper by A. I. Manevich published in “Reports of Ukrainian
National Academy of Science”, 2001, N 12, pp. 5257.
Appendix: Inertial Forces and Methodology of Mechanics
241
frames, are named “inertial force”. These forces depend upon the choice
of the coordinate system and so are considered a priori as a convenient
fib, as “pseudoforces” [3], which are introduced in order to present the
equations of motion in the same form as in the inertial reference frame.
In courses of theoretical mechanics the “inertial forces” are usually
treated in a wider sense. The D'Alembert principle introduces inertial
forces as some vectors having dimension of force, which are invented in
order to use the language and notions of statics in dynamics. The
simplest formulation of this principle for material points follows from
Newton's second law mW = F + R (F, R are the active force and the
reaction force respectively) after transferring all the terms of this
equation to the right hand side and denoting mW = Fi. The obtained
equation F+ R + Fi = 0 is usually regarded as a formal balance of forces
acting on the body, if to apply (conventionally) force Fi (“D’Alembertian
inertial force”) to the body. It is emphasized that no balancing is actually
reached, as these forces are applied not to the given body but to its braces
and to the bodies causing its motion (“accelerating bodies”).
In addition to the translatory, Coriolis and D’Alembertian inertial
forces, theoretical mechanics considers also “Newtonian inertial forces”
or “counteraction forces”, which are spoken about in the Newton's third
law. “Newtonian inertial forces”, in distinction from other inertial forces,
are treated as real ones (therefore the Newton's term is sometimes
considered as unsuccessful, and it is proposed to use only term
“counteraction forces”). To be distracted from the terminology
arguments, we may say that theoretical mechanics treats the inertial
forces applied to the moving (accelerated) bodies as fictitious, imaginary
ones, but recognizes at the same time the reality of inertial forces applied
to braces or accelerating bodies.
A quite opposite standpoint on inertial forces is agreed upon in
courses of applied mechanics and engineering. They usually regard the
inertial forces applied namely to moving bodies as real ones (similarly to
elastic, damping and other forces), which are responsible for many real
phenomena. But such a natural (and clear for engineers) treatment has no
sense within the framework of the “legalized” methodology of classical
mechanics and is considered as a primitive one, as a display of shortage
of common sense, and the known discussions on inertial forces were
devoted namely to the struggle against such notions.
But it is not only the different approaches that have been well
established in physics, theoretical and applied mechanics that cause
Mechanics of Systems with Internal Resonances
242
confusion. Within physics (as well as theoretical mechanics) one can
easily bring out an inconsistency in treatment of the subject. When
considering experiments and various concrete phenomena, authors of
modern courses of mechanics and physics often stand on the viewpoint
of applied mechanics and deal with inertial forces as with real ones (see,
e.g., Feynman physics course [3]). This inconsistency is not accidental
but is a consequence of serious difficulties in the methodology of
mechanics.
Let us consider, for example, such a simple question: “Do centrifugal
forces act in rotating bodies?”. Now mechanics gives the following
answer: “These forces are absent in inertial reference systems, but they
exist from the viewpoint of an observer in the non-inertial reference
system rotating with the body”.
But let us compare the forces for two cases. Let a material point and a
body rotate around fixed axes (Fig.1, (a), (b), (c)). In the case of a
material point the centrifugal force exists both in the inertial coordinate
system (as the force exerted by the point mass on its brace, according to
Newton's third law), and in the rotating system, where this force is the
translatory inertia force. In the case of a rotating body centrifugal forces
exist in the co-rotating reference frame but they are absent, according to
the traditional methodology, in the inertial coordinate system.
(a) (b) (c)
Fig. 1
These questions arise: why the forces acting in the rotating body are
different in the inertial and rotating coordinate systems, whilst the force
acting from the material point on the thread is the same in all reference
systems (as well as the force acting from the thread on the mass point)?
And if any rotating elementary mass exerts on its surroundings by an
elementary centrifugal force, why these forces are absent in the rotating
Appendix: Inertial Forces and Methodology of Mechanics
243
body for the observer in the inertial coordinate system? (One could put
more general question: how can force represent a physical quantity if it is
not an invariant or a tensor?).
It is only one simple example among the many questions which the
traditional methodology of mechanics cannot give an answer. The
traditional viewpoint of physics and theoretical mechanics on inertial
forces (which denies, in particular, the existence of centrifugal forces)
leads to deadlock. All attempts of consistent treatment of the
foundations of mechanics with separation of inertial forces (“fictitious”)
from real (“physical”) forces, which were made repeatedly (e.g., see
monograph by Ishlinskiy [4]), were unconvincing and led only to
additional confusion. Note that Newton in “Mathematical Principles of
Natural Philosophy” considered the centrifugal force as real one (regard-
less of the choice of coordinate system [1]).
In this connection we would like to draw attention to papers by L. I.
Sedov [5] and G. Ju. Stepanov [6], which assert the reality of inertial
forces. But these papers give no answer on the main question: how to
concord this statement with the foundations of classical mechanics? How
must we change its methodology?
It is shown in the paper that the cause of these difficulties lies in the
very foundations of mechanics, in certain discordance of the notion
“force” in its principal laws. These difficulties find natural resolution in
the framework of the proposed interpretation of the fundamentals of
mechanics.
2. Existence of Spatial Inertial Forces
We have to begin from the very beginning. Let us consider a body 1 on
which a force F acts (from a body 2) imparting to this body an
accelerated motion with respect to an inertial reference system (Fig. 2,
(a)). The body 1 acts on the body 2 with force Fi = F, which will be
called “inertia force” (following Newton [1]).
Let us consider the body 1 not as a material point but as a solid
(for simplicity as a uniform bar). Cutting body 1 in an arbitrary cross-
section on two parts and writing equations of motion for each part, we
easily ascertain using Newton's third law that the longitudinal force in
the bar varies along the length from 0 to Fi (in this case linearly, Fig. 2,
(c)). Such a change of the force in statics would testify the existence of
spatial forces. Does this statement hold in dynamics, i.e., do spatial
Mechanics of Systems with Internal Resonances
244
inertia forces act in the body? Dividing the body on several parts and
applying Newton's laws to each part, we also easily ascertain that force
Fi equals to sum of the Newtonian inertial forces for all these parts.
Continuing such division up to infinitely small volumes, we come to the
conclusion that the force Fi is the sum (integral) of the elementary inertia
forces for all elementary parts. Thus, spatial Newtonian inertial forces
act in the accelerated body 1.
Fig. 2
The reasoning presented here is elementary, but as a matter of fact
even it is not necessary. It is enough to ask the question: what gives rise
to force Fi, the existence of which is asserted by the third law? The usual
answer this force appears according to the third law explains
nothing. The cause of force may be only a change in the state of the
body. But the change is that the body 1 begins to move with acceleration
(with respect to the inertial reference frame). Hence force Fi is generated
by accelerated motion of body 1 with respect to the inertial coordinate
system. But if a body with mass m generates a force mW then each
elementary mass dm must generate an elementary inertial force dmW
(obviously, additiveness of mass implies namely this).
The spatial Newtonian inertia forces are the same inertia forces,
which are introduced in the d'Alembert principle. Hence
“D’Alembertian inertia forces” are as real as the force Fi. (Of
course, one may name these forces “spatial counteraction forces” but it
will be shown below that our terminology is preferable).
Appendix: Inertial Forces and Methodology of Mechanics
245
The use of the solid model instead of the material point model enables
us to make sure that the standpoint of theoretical mechanics ”inertia
forces are real ones but they are applied not to given body but to its
braces or “accelerating bodies” has no sense. It is inconsistent to
declare that the spatial inertia forces are fictitious but the force Fi is real.
In this sense there is no difference between inertial forces and, e.g.,
gravity forces. It would be strange to assert that the pressure force of a
body lying on a table is real but the gravity forces at each point of the
body are fictitious. However during more than two hundred years
classical mechanics declares such a statement with respect to inertial
forces considering spatial inertial forces as imaginary ones but their sum
Fi as a real one!
Let us remember known experimental observations described in
popular books on mechanics. In the middle of the nineteenth century
English botanist Night plants bean seeds on the felloe of a rotating
wheel. It is known that plants always grow against the gravity force.
How did these plants grow? Their stems were directed inward (to the
axis of rotation), and roots outwards. This experiment unambiguously
has shown that each cell of the plant undergoes the action of the
centrifugal force, i.e., Newtonian (or D’Alembertian, that is the same)
inertia force.
Thus the following reciprocally connected conclusions can be drawn
from the Newton's laws:
each material point at accelerated motion gives rise to an inertial
force; spatial “Newtonian inertia forces” act in accelerated
bodies;
these spatial forces are identical to “D’Alembertian inertia
forces” which are, hence, real forces.
It is important to make the following remark. Accepting the third law
we thereby acknowledge that not only a cause of accelerated motion is a
force but, inversely, any accelerated motion generates a force. The
inertia forces are not a cause of the motion of the body; they are
themselves caused by the motion. Newton wrote: “This force is
manifested by body only when another force applied to it causes a
change in its state”[1]. Hence for each body we should discern forces
causing its motion, and forces caused by the motion, inertia forces.
Distinction between forces causing motion and those caused by
motion is clearly seen on the example of rotating bodies. A torque
causes rotation, the rotation causes accelerations of all points and
Mechanics of Systems with Internal Resonances
246
therefore centrifugal forces are induced which are balanced by
centripetal forces (due to deformations of the body).
Of course, the division of forces on “causing motion” and “caused by
motion” one has sense only with respect to a given body, and there is no
actual difference between inertial forces and those forces which are
acknowledged by physics as real ones. If we consider two interacting
bodies, these forces commute the force caused by the motion of body
1 (its inertial force) causes the motion of body 2, and inversely.
Generally speaking, main forces acting at collisions of bodies are
inertial forces; other forces are caused by the inertial forces or negligible.
3. “Paradox” of Forces Balance. Discordance of Notion
“Force” in Two Main Laws of Mechanics
But let us consider now the set of forces applied to body 1. This set
force Fi and the Newtonian (or D’Alembertian) spatial inertial forces
turns out to be balanced. The question arises how can the body move
with acceleration, if applied forces are balanced?
The way out of this “contradiction” is apparent, it follows naturally
from the reasoning above about distinction of forces causing motion of
body, and forces generated by the motion.
Inertial forces enter in the left hand side of the second law equation.
Term mFi is not simply a product of mass by acceleration, but also a
force, namely an inertia force (with the opposite sign). Forces causing
motion of the body are written at right hand side, and forces caused by
the motion at the left. Hence sense of the second law is: under action of
forces a body performs such an accelerated motion that inertial
forces generated by this motion counterbalance the applied forces.
Thus the above “paradox” is resolved simply by changing the
treatment of the second law, namely by considering it as equality (at
opposite directions) of two forces the force applied to body and
causing its acceleration and the inertial force caused by this acceleration.
Such an idea is an essence of the D’Alembert principle (though it was
not initial idea of the principle author).
We would like to repeat that the balance of forces really occurs due to
accelerated motion. Forces applied to body are balanced not only in
statics but also in dynamics. The distinction between dynamics and
statics is only that in dynamics a part of the forces appears as a
result of accelerated motion.
Appendix: Inertial Forces and Methodology of Mechanics
247
Physics always identified two notions rest state (uniform motion)
of the body and the balance of forces applied to the body. We should
separate these notions. If forces initially applied are balanced then the
body is in the rest state; but if these forces are not balanced then
balancing is achieved due to inertial forces.
Let us compare again two principal laws of mechanics. We should
note a principal difference between these laws. Newton's third law
implies all forces causing motion and caused by motion (only due to
Newtonian inertial forces does this law hold). But the second law, in its
usual formulation, names “force” only forces causing motion, but not all
forces acting on the body in the process of motion. Only the terms
entering in the right hand side are named “force”, and by that inertia
forces are excluded from the class of “physical forces”. Hence, in
certain sense we may speak about certain discordance of notion
“force” in two principal laws of mechanics.
Namely this difference in treatment of notion “force” by two main
laws is the cause of the contradiction in methodology of classical
mechanics. This contradiction remains implicit in the dynamics of
material points, but manifests itself in dynamics of solids. Recognition
that mFi is also a force eliminates the contradiction between these two
laws. Classical mechanics could not reject the D’Alembert principle; it
has become the foundation of analytical dynamics. But classical
mechanics has not changed its methodology, the inherent contradiction
between both laws has not been overcome, and D’Alembertian inertial
forces have been introduced in mechanics as “a convenient fib”, as “a
fiction”, not as real forces.
It should be emphasized that the very technique of mechanics is non-
contradictory as analytical dynamics is based on account of the term
mFi as a force and on balancing forces applied to body in dynamics as
well as in statics. The problem is the elimination of the contradiction
between the techniques of mechanics and its methodology.
It goes without saying that the inertial forces manifest themselves in
stresses, deformations and failures of bodies; they perform work. They
have potential, and this is the body's kinetic energy [5].
4. Non-inertial Reference Frames. Invariance of Forces in
Mechanics
Mechanics of Systems with Internal Resonances
248
Let us consider the translatory and Coriolis inertia forces in non-inertial
reference frames, i.e., namely those forces with which physics relates
exclusively notion “inertial forces”. It is evident, in the light of the
above said, that such an approach, when inertia forces are connected only
with the relative motion, is principally limited and does not give
possibility grasping the meaning of the problem.
The more general viewpoint, which recognises the reality of
D’Alembertian inertia forces, enables us easily to bring to light the sense
of the translatory and Coriolis inertia forces. Inertial forces are
determined by acceleration with respect to the inertial reference frame,
i.e., by the absolute acceleration of the material point. In non-inertial
reference frames the absolute acceleration is the sum of the translatory,
Coriolis and relative accelerations We, Wc, Wr and so the full, or absolute,
inertial force is sum of three items:
Fi = mWa = m(We + Wc + Wr) =Fei + Fci + Fri ,
where Fri = mWr is the relative inertia force (this term was introduced
for the first time, apparently, by L. I. Sedov [5]).
In non-inertial coordinate systems two components of the full inertial
force translatory and Coriolis forces are transferred to the right
hand side of the equation of motion. So the distinction of non-inertial
reference frames from inertial ones is only the division of the full inertial
force into three components, two of which being forces caused by
motion, are regarded formally as forces causing the motion.
Of course such breaking up of the full inertial force on three terms
depends on the choice of the reference frame. But the arbitrariness of the
division of the absolute inertia force on components cannot be a reason
for treating them as “pseudoforces”, similarly as projections of a force on
coordinate axes are not regarded as fictitious quantities, although they
depend upon the choice of a coordinate system.
Thus the translatory and Coriolis inertia forces are only
components of the full inertia force (Newtonian, or D’Alembertian).
Now we can give an answer the following principal question: do
forces acting on a body depend upon the choice of coordinate system?
The key moment here is the presence of the relative inertia force.
Physics until now considered only two terms the translatory and
Coriolis inertia forces and could not regard them as real forces since
these forces (or any their combination) do not constitute an invariant and
depend upon the choice of the coordinate system. Only with the addition
Appendix: Inertial Forces and Methodology of Mechanics
249
of the third component relative force of inertia these forces
constitute an invariant.
Taking into account the relative inertial force, we obtain the same
force in any reference frame (though observers in different coordinate
systems may name this force differently: e. g., for one observer it may be
the translatory inertial force, for another the relative inertial force).
The above treatment of inertial forces entirely conforms (in
distinction from the present-day dominating standpoint) with the
fundamental statement of general relativity theory about equivalence of
inertial and gravity forces.
We must recognize that in the arguments an with engineer's intuition
and common sense about reality of inertial forces, applied to bodies
accelerated, mechanics is found to be wrong. With rehabilitation of
D’Alembertian inertial forces and disappearance of the phantom of
“fictitious” forces the methodology of mechanics becomes clear and
transparent.
References
1. I. Newton. Mathematical Principles of Natural Philosophy (Russian
translation by A.N. Krylov), Moscow, Nauka, 1989.
2. J. L. Lagrange. Mecanique Analytique, Paris, 1788.
3. R. P. Feynman, R. B. Leighton, M. Sands. The Feynman Lectures
on Physics, vol. 1. Addison-Wesley, Reading MA, 1963.
4. A. Ju. Ishlinskiy. Classical Mechanics and Inertial Forces, Moscow,
Nauka, 1987 (in Russian).
5. L. I. Sedov. On principal models in mechanics, Moscow, MGU, 1992,
pp. 616 (in Russian).
6. G.U. Stepanov. Inertia forces - fiction or reality? In: Mechanics.
Contemporary problems, Moscow, MGU, 1987 (in Russian).
ResearchGate has not been able to resolve any citations for this publication.
  • R P Feynman
  • R B Leighton
  • M Sands
R. P. Feynman, R. B. Leighton, M. Sands. The Feynman Lectures on Physics, vol. 1. Addison-Wesley, Reading MA, 1963.
  • A Ju
A. Ju. Ishlinskiy. Classical Mechanics and Inertial Forces, Moscow, Nauka, 1987 (in Russian).
On principal models in mechanics
  • L I Sedov
L. I. Sedov. On principal models in mechanics, Moscow, MGU, 1992, pp. 6-16 (in Russian).
Inertia forces -fiction or reality? In: Mechanics. Contemporary problems
  • G U Stepanov
G.U. Stepanov. Inertia forces -fiction or reality? In: Mechanics. Contemporary problems, Moscow, MGU, 1987 (in Russian).