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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 XVII–XVIII 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. 52–57.

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. 6–16 (in Russian).

6. G.U. Stepanov. Inertia forces - fiction or reality? In: Mechanics.

Contemporary problems, Moscow, MGU, 1987 (in Russian).