Commented translation of Erwin Schrödinger’s paper ‘On the dynamics of elastically coupled point systems’ (Zur Dynamik elastisch gekoppelter Punktsysteme) [original]

Review

Commented translation of Erwin

Schrödinger’s paper ‘On the

dynamics of elastically coupled point

systems’ (Zur Dynamik elastisch

gekoppelter Punktsysteme)

Mathematics and Mechanics of Solids

2021, Vol. 26(1) 133–147

Article reuse guidelines:

sagepub.com/journals-permissions

DOI: 10.1177/1081286520942955

journals.sagepub.com/home/mms

Uwe Mühlich

Universidad Austral de Chile, Chile

Bilen Emek Abali

Technische Universität Berlin, Germany

Francesco dell’Isola

Sapienza Universidad de Roma, Italy

Received 29 May 2020; accepted 28 June 2020

Abstract

The paper ‘Zur Dynamik elastisch gekoppelter Punktsysteme’ by Schrödinger (1914) does not seem to have attracted the

attention that it deserves. We translate it into English here and we discuss its results in detail, with a view to its possible

influence in the modern theories of generalised continua. The clever solution found, in terms of Bessel functions, by

Schrödinger of the problem of the vibrations of a linear infinite chain of molecules gives some important methodological

hints for contemporary research.

Keywords

Generalised mechanics, metamaterials, nonlocal theories

1. Introduction

In his paper ‘Zur Dynamik elastisch gekoppelter Punktsysteme’, Erwin Schrödinger considers the vibration of

an infinite mono-atomic chain with nearest-neighbour interactions. Denoting the deviation of the nth atom or

mass point from its initial position by ξn, the well-known mathematical formulation of this problem reads

md2ξ

dt2=f[ξn+1−ξn]−f[ξn−ξn−1], (1)

with n=0, ±1, ±2, ...,±∞, where mindicates the mass of each atom and fis the elastic stiffness of the

interaction linear force between two contiguous atoms.

Corresponding author:

Uwe Mühlich, Universidad Austral de Chile, Independencia 631, Valdivia, Los Ríos, Chile.

Email: [email protected]

134 Mathematics and Mechanics of Solids 26(1)

As the reader might expect, this classical model of discrete physics was already well known and established

at that time together with Fourier’s method as the standard tool for solving this kind of problem. However, using

the substitutions

x2n=pfdξn

dt,x2n+1=pf[ξn−ξn−1],

Schrödinger ingeniously derives the general solution for arbitrary initial conditions in the form

xn=∞

k=−∞

kJn−k(νt)

with Jlbeing the Bessel function of the first kind and order l, initial values x0

kand ν=2qf

m. The result for the

specific case of a single initial perturbation of the 0th atom/mass point

ξn=J2n(νt)

can be inferred easily from the general solution. It turns out that this solution shows a number of surprisingly

peculiar properties. In particular, a kind of ‘dissipation’ effect in the chain of atoms is observed.

In fact, if one considers the natural homogenised continuum equation derived from (1), i.e. the D’Alembert

wave equation, it is easily proven that for the same initial condition, the corresponding solutions for the discrete

system and the D’Alembert equation differ remarkably.

The paper was published in 1914 in German in the journal Annalen der Physik. In the meantime, English has

become the Lingua Franca in science and Schrödinger’s contribution seems to have been neglected completely

in the following literature (see e.g. [1] and in particular the section ‘The Vibrating String as a Limiting Case of

a Chain of Oscillators’).

Here, we provide an English translation of Schrödinger’s paper. One reason for which said translation is

deemed necessary concerns the indisputable value of the scientific contribution that it contains.

It is undoubted that Erwin Schrödinger’s contribution to physical sciences are so outstanding that the paper

in which we are interested may be considered, with some good reason, to be secondary. On the other hand,

the analysis that it contains seems to us still topical and may initiate an important debate among the scholars

interested in continuum mechanics. In particular it seems that it can help to motivate the development of so-

called generalised continuum models, in particular when evolutionary problems are considered, in which inertial

parts of the action functional cannot be neglected. Indeed, we believe that some questions and information

provided by Schrödinger’s paper are still valuable and interesting enough to be communicated to a present-day

audience.

To this aim let us recall that in 1913 a seminal paper by Ernst Hellinger (see [2–5]) clearly set up the

applicability limits of the so-called first gradient continuum theory, as developed based on the fundamental

contributions by Cauchy. Hellinger clearly underlined that more general models were possible and desirable,

as already pointed out by Gabrio Piola, whose works Hellinger seemed to know and appreciate. This point has

already been discussed extensively, and therefore we refer the interested reader to [6–9]

We do not know whether Schrödinger was fully aware of the paper by Hellinger. In fact, in 19141generalised

continuum models had been already explored by only a few scholars. The pioneering work by Piola (1848) was

described by Hellinger in his work of 1913 (see [2]) and the celebrated work by Cosserat brothers was published

in 1909. Therefore, the possibility to better capture the discrete nature of a chain of molecules by such theories

was not clear in the literature prior to Schrödinger’s considered paper. It is not easy to understand if he, at the

end, was interested at all in this specific problem. Many of the statements by Hellinger imply, on the other hand,

that Hellinger himself was aware of the fact that with a higher-order theory, e.g. including non-local, strain

gradient, micromorphic continua or other effects, one might be able to produce results that reflect the ‘discrete’

behaviour effectively shown by the results in [10].

In any case, it is certain that Schrödinger’s analysis supports the conclusions that lead Hellinger to demand

the development of generalised continuum theories, as envisaged by Piola. In the works by Piola and/or in the

paper by Hellinger, the following are clearly stated.

(i) Every continuum model is an idealisation of an underlying discrete/discontinuous nature. Hence, ‘discrete’

mechanics has the right to exist individually alongside continuum mechanics. It may appear astonishing

Mühlich et al. 135

that at the time of Piola, and then for Hellinger and Schrödinger, discrete mechanics was considered the

most ‘reliable’ description of nature, and that continuum mechanics was regarded by all of them as a kind

of ‘approximation’, valid macroscopically for systems made up of a large number of particles, whereas

Mach and Boltzmann had a violent controversy on this point at the beginning of the 20th century (see e.g.

[11–13])

(ii) Simple continua do not include enough information about the underlying mechanical structure to account

for many macroscopic phenomena, among which, for instance, we must surely include wave dispersion.

(iii) Generalised continua are designed to account for those macroscopic phenomena, which are related to an

underlying microscopic structure.

The aim of the translation provided within the next section is to stay as close as possible to the original text

rather than to use an elegant and elaborated English.2

2. The translated text of Schrödinger’s paper

The translation is organized according to the individual pages and corresponding page numbers of the original

paper. Footnotes are placed at the end of individual pages using the same numbering which has been used in the

original paper.

Page 916

4. On the Dynamics of Elastically Coupled Point Systems

Often it has been claimed1and this is a sort of profession of faith for an atomist, that all partial differential

equations of mathematical physics relating spatial and temporal variations of any physical variables (tempera-

ture, deformation, field strength, etc.) are in a rigorous sense incorrect. This is because the mathematical symbol

of the differential quotient demands the limit towards arbitrarily small volumes, whereas we are convinced that

we have to stop generating such ‘physical’ differential quotients at ‘physically infinitesimal small’ volumes, i.e.

volumes still containing many molecules; if we were continuing the limit process further, then the correspond-

ing quotients, which so far indeed came closer and closer to a certain limit, practically reaching it very closely,

would start to vary again significantly in order to approach perhaps eventually a true limit. For the latter, despite

the fact that we would not be able to measure it, these simple laws which are reflected by the partial differential

equations, and which are always valid for the aforementioned ‘pseudo limits’, would not have validity.

Whenever we want to take the aforementioned point of view seriously, we have to fulfill two tasks: Firstly,

one has to derive all differential equations, obtained in a strict sense by considering a continuous medium, now

also in the sense indicated above in terms of difference equations which are deduced from a model postulating

the existence of molecules.

1L. Boltzmann, Ann. d. Phys. 60. p. 231; 61. p. 790. 1897.

Page 917

This task is usually not the most difficult one, in many cases it has been accomplished already1. It is, however,

not a rewarding task either. It only provides a proof that the atomistic vision of a phenomenon is possible, i.e.

compatible with our experiences (observations) but it never proves them, the observations demand an atomistic

view point. And we are at a permanent disadvantage compared with the phenomenologist who reaches the goal

usually quicker by means of a simple and plausible approach, whereas we are achieving it only after a sequence

of tedious and complicated thoughts which may even be perceived partly as sloppy.

Hence, atomistic theory faces a second problem, and only its solution can prove eventually that the physical

atomism has the right to exist alongside with phenomenological theories. It is about finding and predicting

exactly those conditions under which the differential equations, which are valid in the continuum hypothesis,

yield indeed measurably incorrect results due to the atomistic nature of matter. The most important and probably

only success in this direction can be found in the context of the kinetic gas theory: Temperature jump in heat

conduction as well as the finite sliding in dilute gases at the channel walls are facts predictable by a molecular

theory, but not by a phenomenological theory of heat transport or inner friction.

136 Mathematics and Mechanics of Solids 26(1)

Obviously, the only way to find such cases is to compare step by step the predictions of a continuum approach

with those made by a presumably adequate molecular model for many different conditions.3This is more diffi-

cult for the molecular models as these mechanical models possess an enormous number of degrees of freedom

and the task consists of integrating the equations of motion for arbitrary initial conditions. On the other hand,

as is well known, certain information about such systems can be derived without performing the integration;

1Historically, it is usually the other way around: the derivation based on atomistic ideas is in general older.

Page 918

we owe this circumstance to the mechanistic foundation for thermodynamics, the ‘Physics of Equilibrium States’

by Gibbs and Ludwig Boltzmann. But, whenever it is not dealing with equilibrium states, in particular when

computing spatial and temporal variations of certain coordinate functions, the general methods of statistical

mechanics fail, and integration of the equations of motion seems necessary.

The in the following laid out method for integrating a well-known problem of motion, which recently gained

new importance for computing specific heat capacities of solid bodies1and effects of temperature on interfer-

ence patterns of X-rays2, seems to me from this general point of view to have sufficient interest to be discussed

in its own right. The problem refers to a system of mass points which in the limit converges, in the aforemen-

tioned sense, to the partial differential equation of the vibrating string, i.e. the one-dimensional wave equation.

It seems to me that the solution laid out here reveals relationships between the point system’s laws of motion

and the propagation of waves in a one-dimensional elastic medium better than the conventional method.

The mechanical system to be studied is as follows. Finitely many point masses, all with same mass m, are

aligned and we enumerate them by 0, ±1, ±2, ±3, .... Two mass points with subsequent numbers are adjacent;

namely, the greater number is the right neighbour of the smaller numbered mass point. Let us count from the

coordinate of a mass point being at zero along the line in one of the directions called the positive direction

(which we refer to as towards the right-hand side in the sequel). Each of two neighbours shall exert a force

on each other, more specifically, the force between the points nand n+1 vanishes if they are separated by a

distance +a;

1Born u. v. Karman, Phys. Zeitschr. 13. p. 297. 1912; see also 14. p. 15 and 65. 1913; Hans Thirring, Phys. Zeitschr. 14. p. 867. 1913; 15. p. 127 nd 180.

1914.

2P. Debye, Ann. d. Phys. (4) 43. p. 49. 1914; E. Schrödinger, Physik. Zeitschr. 15. p. 79. 1914.

Page 919

in case of a distance a+ξ, however, the n+1th experiences a force −fξfrom and, exerts a force fξon;

the nth point, where fis constant. A stable equilibrium configuration can be defined by placing the 0th point

mass at the origin and ±1st, ±2nd, ±3rd ... being at ±a,±2a,±3a.... We aim to analyze the motion of this

now fully defined mechanical system, but starting only from initial configurations for which the distances of

all mass points from their aforementioned equilibrium configuration are below a specifiable and finite bound.

This ensures simultaneously that there are never any infinite forces applied to a point mass.1We introduce as

coordinates the deviation of mass points from these equilibrium positions as follows:

. . . ξ−n,ξ−n+1,...,ξ−2,ξ−1,ξ0,ξ1,...,ξn−1,ξn...

All values are finite at the initial time as well as after each finite time step. The equations of motion read

md2ξn

dt2=f(ξn+1−ξn)−f(ξn−ξn−1), n=0, ±1, ±2, ... (1)

Our interest in the infinite system of equations (1) is due to the fact that it converges in the limit (a=0;

m/a,fa finite) to the differential equation for the infinite and stretched string or the infinite elastic truss and,

moreover, a three-dimensional generalisation of this system of equations yields in the same way the equations

of the linear theory of elasticity for an infinite anisotropic medium (see the works by Born and von Karman

cited previously). The principle objective of this research is to inspect especially these limit processes and the

question for which cases the motion of the point system is indeed similar to the motion of the continuum.

Mühlich et al. 137

As is widely known, the system (1) can be integrated by using the ansatz

ξn=anexp i(ν0t+nφ), (2)

1Demanding that forces in the initial configuration are bounded is not sufficient. As a counter example, consider the points 0, ±1, ±2, ±3, ... brought

to the initial positions 0, ∓a,∓2a,∓3a,..., respectively. Such initial conditions are excluded here. In the case of physical applications, we are anyway

solely interested in small deviations from the equilibrium configuration.

Page 920

where φis arbitrary, ν0and the anare some functions of φ. The most general solution of (1) can be written as

a continuous sequence of solutions of type (2) (for continuously varying φ). This method is similar to Fourier’s

method for the string. It has the disadvantage that in the same way as the latter, a Fourier analysis of an arbitrary

initial configuration is required to assess the resulting motion, and this motion is given as a continuous sequence

of normal vibrations so that the final vibration can hardly be comprehended.

The structure of (1) allows for another solution method which provides a better overview in some cases. In

order to show this, we introduce instead of ξnand ˙

ξnnew coordinates; we denote the entities

...pf(ξ−2−ξ−1), √m˙

ξ−1,pf(ξ−1−ξ0), √m˙

ξ0,

pf(ξ0−ξ1), √m˙

ξ1,pf(ξ1−ξ1), ...

by using the symbols

...x−3,x−2,x−1,x0,x1,x2,x3...,

respectively, such that

x2n=√m˙

ξn,x2n+1=pf(ξn−ξn+1). (3)

Quantities xnare actually not sufficient to describe the state of motion. They determine only the velocity

and the relative positions of point masses. Since the position of the system as a whole (including a possible

rigid translation which can be superimposed to any motion) is of no importance to us, we may ignore the final

coordinate (we can choose an arbitrary value for some ξ).4By rewriting (1) with the aid of the variables xn, we

obtain using the short-hand notation

ν=2rf

m,(4)

the following equation:

dx2n

dt= −ν

2(x2n+1−x2n−1). (5)

Page 921

Owing to (3) and (4), the identity

dx2n+1

dt= −ν

2(x2n+2−x2n)(6)

holds for all nfrom −∞ to +∞. Inspection of (5) and (6) reveals that for all nfrom −∞ to +∞,

dxn

dt= −ν

2(xn+1−xn−1)(7)

applies.

The latter is, however, nothing else than one of two fundamental relations (by ignoring the factor ν) between

three Bessel functions, which follow each other with a distance of 1 regarding the order parameter. Thus, we

realise that we obtain a solution by identifying one of the xnwith an arbitrary Bessel function with argument νt;

any other xkcan be obtained by changing the Bessel function’s parameter as much as the difference of the order

parameter.

Loading more pages...