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Mechanistic Images in Geometric FormHeinrich Hertz's 'Principles of Mechanics'$

Jesper Lützen

Print publication date: 2005

Print ISBN-13: 9780198567370

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198567370.001.0001

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Cyclic coordinates

Cyclic coordinates

Chapter:
(p.208) 18 Cyclic coordinates
Source:
Mechanistic Images in Geometric Form
Author(s):

JESPER LÜTZEN

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198567370.003.0018

Abstract and Keywords

In Heinrich Hertz's account of conservative systems, the concept of a cyclic coordinate, and, in particular, what he called an adiabatic cyclical system, enters as an important technical tool. This chapter explains the meaning of these concepts as understood in the ordinary image of mechanics. First, the historical development of the concept and its important mechanical and mathematical properties are discussed. An example of a simple mechanical system that has a cyclic coordinate is then presented. This example is only meant as a didactical device. The chapter concludes by examining Hermann von Helmholtz's use of cyclic motion in his mechanical model of thermodynamics. Helmholtz's papers on this topic are important for the present book because they were the point of departure for Hertz's ideas on cyclic coordinates. This chapter also considers Edward John Routh's modified Lagrangians and J.J. Thomson's equation of hidden cyclic motion.

Keywords:   cyclic coordinates, mechanics, adiabatic cyclical systems, Hermann von Helmholtz, cyclic motion, thermodynamics, Edward John Routh, Lagrangians, hidden cyclic motion

In Hertz’s account of conservative systems the concept of a cyclic coordinate, and, in particular, what he called an adiabatic cyclical system enters as an important technical tool. In this chapter I shall explain the meaning of these concepts as understood in the ordinary image of mechanics. I shall first give an account of the historical development of the concept and its important mechanical and mathematical properties. Then I shall give an example of a simple mechanical system that has a cyclic coordinate. This example can then serve as a simple standard example to which I shall refer in later chapters when I want to give a concrete example of some of the properties of the systems discussed by Hertz. Hertz himself did not give any examples, but I have found it suggestive to complement his entirely general theoretical considerations with a simple example. I do not suggest that Hertz himself primarily had such simple systems in mind when he wrote his Mechanics. The example is only meant as a didactical device. At the end of this chapter I shall discuss Helmholtz’s use of cyclic motion in his mechanical model of thermodynamics. Helmholtz’s papers on this topic are important for the present book because they were the point of departure for Hertz’s ideas on cyclic systems.

18.1 Routh and modified Lagrangians

A generalized coordinate of a mechanical system is called a cyclic coordinate if it does not enter explicitly into the expressions of the kinetic or potential energy of the system – or almost equivalently in its Lagrangian or Hamiltonian. For example, if a change of the coordinate does not change the mass distribution of the system but only cyclically permutes the masses (e.g. the turning of a wheel), then the coordinate is cyclic. This is the origin of the name. The importance of such coordinates was discovered in several steps. First, it was observed that if the Hamiltonian                    Cyclic coordinates                does not contain a certain generalized coordinate q ρ; Hamilton’s equations

(18.1)                    Cyclic coordinates
(p.209) imply that the conjugate generalized momentum p ρ(t) is a constant during the motion. This simple way to determine integrals of motion was used by Joseph Liouville (see (Lützen 1990, p. 707)) around 1850.

A deeper understanding was obtained by Edward John Routh (Routh 1877 b) in connection with the Lagrangian formalism: Let T and V represent the kinetic and potential energy, respectively, of an isolated conservative mechanical system considered as functions of its generalized coordinates q ρ and their time derivatives                    Cyclic coordinates                                   Cyclic coordinates                . Then, as explained in Chapter 2, Lagrange had shown (Lagrange 1788) that the equations of motion can be written (see eqn (2.13))

(18.2)                    Cyclic coordinates
where P ρ is the force in the direction of the coordinate q ρ. If we express the force as minus the gradient of the potential energy we get
(18.3)                    Cyclic coordinates
or
(18.4)                    Cyclic coordinates
where the Lagrangian L is defined by
(18.5)                    Cyclic coordinates
If the Lagrangian does not depend explicitly on some of the coordinates                    Cyclic coordinates                but only on their time derivatives, Lagrange’s equations (18.4) imply that
(18.6)                    Cyclic coordinates
This corresponds to the observation made above concerning the Hamiltonian formalism. In this case, Routh suggested (Routh 1877 b, §20) that one could use eqns (18.6) to express                    Cyclic coordinates                in terms of the constants c ρ. If we substitute these expressions into the expression for L we obtain a function                    Cyclic coordinates                whose partial derivatives with respect to                    Cyclic coordinates                and q ρ for ρ = 1, 2,…, k can be written as
(18.7)                    Cyclic coordinates
or
(18.8)                    Cyclic coordinates
(p.210) and
(18.9)                    Cyclic coordinates

Therefore, if we follow Routh and make a so-called Legendre transform, defining the modified Lagrangian L′ by

(18.10)                    Cyclic coordinates
and again use eqn (18.6) to eliminate                    Cyclic coordinates                , this function will, by eqns (18.4), (18.8) and (18.9), satisfy the equations
(18.11)                    Cyclic coordinates
i.e. equations exactly similar to Lagrange’s equations (18.4). As pointed out by Routh, this transformation corresponds to a partial use of Hamilton’s transformation of Lagrange’s equations, the constants c ρ being precisely the momenta conjugate to q ρ.

18.2 Hidden cyclic motion. J.J. Thomson

Routh used his modified Lagrangian in a study of ‘stability of motion’ (Routh 1877 b) (see also (Routh 1877 a)), but with Helmholtz ((Helmholtz 1884) and (Helmholtz 1886)) and J.J. Thomson ((Thomson 1886), (Thomson 1888)) the technique got a new twist in that they considered the cyclic coordinates                    Cyclic coordinates                as hidden, unobservable coordinates1. Given a mechanical (or another type of physical) system that we describe by way of the coordinates                    Cyclic coordinates                . We can then, through geometric considerations, determine its apparent kinetic energy                    Cyclic coordinates                as a quadratic form in the                    Cyclic coordinates                If, moreover, we can determine a function                    Cyclic coordinates                such that the motion of the system is described by eqn (18.11), we will conclude that                    Cyclic coordinates                is the Lagrangian of the system, and hence that                    Cyclic coordinates                is the potential energy of the system that in turn describes the forces on the system. However, we may have been deceived by our lack of knowledge of a certain number of hidden cyclic coordinates so that what we believed was the Lagrangian, was in fact only the modified Lagrangian. What we mistook for potential energy may therefore have been the result of kinetic energy due to the hidden coordinates.

To be more precise, let us with J.J. Thomson (Thomson 1888) consider a complete system as in eqns (18.4)–(18.11), which has only kinetic energy L = T, and such that T can be divided into

(18.12)                    Cyclic coordinates
where T 1 is a quadratic form of                    Cyclic coordinates                and T cycl is a quadratic form in the remaining cyclic velocities. According to Euler’s fundamental theorem about (p.211) homogeneous functions we have
(18.13)                    Cyclic coordinates
and thus, according to eqns (18.10) and (18.12), the modified Lagrangian takes the form
(18.14)                    Cyclic coordinates
where                    Cyclic coordinates                indicates that                    Cyclic coordinates                have been expressed as functions of c ρ and                    Cyclic coordinates                by way of eqn (18.6) or equivalently by:
(18.15)                    Cyclic coordinates

By Routh’s modified Lagrangian equation (18.11) we therefore have

(18.16)                    Cyclic coordinates
This is the equations of motion of a free mechanical system described by the coordinates                    Cyclic coordinates                and with kinetic energy T 1 and potential energy                    Cyclic coordinates                Indeed, the latter is, from eqn (18.15), a function of the variables                    Cyclic coordinates                alone, independent of                    Cyclic coordinates

18.3 A simple standard example

In order to illustrate Thomson’s ideas let me apply them to a simple mechanical system that it will also be useful to keep in mind, when discussing Hertz’s approach to mechanics. Consider a rod of length l and moment of inertia I swinging around its one end O in the plane of the paper (Fig. 18.1), and let its other end A be connected to a cyclical system similar to a centrifugal regulator consisting of a point mass m rotating around the point P (a position of the endpoint A) in a plane perpendicular to OP. The point mass m is connected to A by a massless string of length 2l passing through a pulley at P. We shall think of the rod, whose position is described by the angle                    Cyclic coordinates                , as a visible system, and the mass m as a hidden system. The position of this hidden system is described by an angle ϕ (see Fig. 18.1), and the distance x = Pm, or alternatively by ϕ and ω, where x is expressed in terms of ω by the formula

(18.17)                    Cyclic coordinates

(p.212)

                   Cyclic coordinates

Fig. 18.1. A simple standard example

The energy of the total system is given by

(18.18)                    Cyclic coordinates
(18.19)                    Cyclic coordinates
We see that 1. ϕ is a cyclic coordinate of the system since it does not enter into the expression of the energy and 2. the energy can be divided as assumed by Thomson into a quadratic form                    Cyclic coordinates                of                    Cyclic coordinates                and a quadratic form                    Cyclic coordinates

Following Routh and Thomson we shall express the last term in terms of the conserved momentum                    Cyclic coordinates                conjugate to ϕ:

(18.20)                    Cyclic coordinates
so that
(18.21)                    Cyclic coordinates

Thus, the modified Lagrangian is given by

(18.22)                    Cyclic coordinates
(p.213) and the visible system will move as though it had kinetic energy given by
(18.23)                    Cyclic coordinates
and potential energy given by
(18.24)                    Cyclic coordinates
From his general considerations, Thomson concluded (somewhat rashly):

Thus we may look on the potential energy of any system as kinetic energy arising from the motion of systems connected with the original system – the configurations of these systems being capable of being fixed by kinesthetic or speed coordinates. [Thomson’s terms for cyclic coordinates]

Thus from this point of view all energy is kinetic, and all terms in the Lagrangian function express kinetic energy, the only thing doubtful being whether the kinetic energy is due to the motion of ignored or positional coordinates; this can however be determined at once by inspection. (Thomson 1888, p. 14)

In this way, the widespread British feeling (see Chapters 3 and 4) that action at a distance, or equivalently potential energy, ought to be explainable through mechanical systems in motion (i.e. kinetic energy), was combined with Routh’s mathematical formalism of cyclic coordinates and modified Lagrangians to yield an abstract method for explaining forces kinetically, sidestepping thereby the problem of describing the hidden motions in all mechanical detail. It was precisely this approach Hertz chose in his Mechanics.

18.4 Helmholtz on adiabatic cyclic systems

However, although Hertz was obviously well aware of the general tendencies in British physics he claimed in the preface to his book that he did not know of Thomson’s particularly clear anticipation of his approach to mechanics until his own research was well advanced. Instead, he built on Helmholtz’s two papers (Helmholtz 1884), (Helmholtz 1886). Helmholtz did not suggest that all energy might turn out to be kinetic energy, but he introduced the idea of adiabatic processes, i.e. processes where the non-cyclic observable coordinates change very slowly in comparison with the cyclic coordinates. This idea that came to play a central role in Hertz’s mechanics, was important to Helmholtz’s account of thermodynamics.

Helmholtz’ aim was to display a mechanical model of thermodynamics. The heat dQ gained by a thermodynamical system when its temperature θ and its external parameters qi (e.g. the volume of a gas) are changed by infinitesimal amounts, is described by the equation:

(18.25)                    Cyclic coordinates
(p.214) where E is the total energy of the system, and Pi denotes the external force tending to increase the parameter q i.. This means that the expression
(18.26)                    Cyclic coordinates
measures the energy gained by the system when q i is increased by an amount dq i. The important fact about dQ is that although it is not itself an exact differential of                    Cyclic coordinates                and θ, the temperature θ is an integrating denominator such that dQ/θ is an exact differential of a function                    Cyclic coordinates                called the entropy
(18.27)                    Cyclic coordinates
Helmholtz emphasized that any function                    Cyclic coordinates                of S would do the trick equally well. Indeed, if we put
(18.28)                    Cyclic coordinates
we will have
(18.29)                    Cyclic coordinates
Thus, the temperature is not uniquely determined as the integrating denominator of dQ, but, according to Helmholtz, it is the only choice of η that satisfies the important criterion: when two systems of equal value of θ are put into contact with each other no heat dQ will flow from the one to the other.

It is important to note that in this type of equilibrium thermodynamics, we consider the system determined entirely by q,…,qk and θ. That means that we consider                    Cyclic coordinates                as negligible so that the kinetic energy, which would be associated with the changes in the parameters, is neglected in comparison with the potential energy corresponding to these parameters and the energy stored as heat. Moreover, it is assumed that θ changes so slowly that the system can be considered as having the same temperature everywhere. Under these circumstances we say that the system changes adiabatically.

Now, Helmholtz showed that one can mimic eqns (18.25) and (18.26) by assuming that the energy stored as heat is associated with a hidden cyclic motion. So, let us as above (eqns (18.6)–(18.16)) consider a mechanical system with r degrees of freedom and rk cyclic coordinates, i.e. coordinates                    Cyclic coordinates                that do not appear in the Lagrangian. However, as opposed to the system described by eqn (18.4), Helmholtz assumed that in addition to conservative inner forces described by the potential energy V, the system is influenced by external (not necessarily conservative) forces P i tending to increase q i. Then eqn (18.4) must be replaced by the more general Lagrangian equation:

(18.30)                    Cyclic coordinates
(p.215) The momenta
(18.31)                    Cyclic coordinates
conjugate to the cyclic coordinates q ρ are no longer constants but we have
(18.32)                    Cyclic coordinates
Thus, the energy gained by the system when q ρ is increased by dq ρ can be written (see eqn (18.26)):
(18.33)                    Cyclic coordinates
In conformity with the thermodynamic situation Helmholtz now assumed that the non-cyclic coordinates                    Cyclic coordinates                change adiabatically, i.e. that                    Cyclic coordinates                and                    Cyclic coordinates                are negligible in comparison with                    Cyclic coordinates                This will happen if the external forces are very close to the values that will keep                    Cyclic coordinates                constant. For example (as mentioned by Helmholtz in (Helmholtz 1886, p. 148) but not in (Helmholtz 1884)) if the kinetic energy can be divided as in eqn (18.12) and if the external forces                    Cyclic coordinates                are zero, then there are motions of the system for which                    Cyclic coordinates                are constant.

The assumption that the motion is adiabatic has several important consequences. First, the state of the system is described by                    Cyclic coordinates                and                    Cyclic coordinates                and since L does not depend on                    Cyclic coordinates                the momenta p ρ will be functions of                    Cyclic coordinates                . Secondly, for the non-cyclic coordinates the first term of eqn (18.30) vanishes compared with the second term so that the Lagrangian works as a sort of potential function for the external forces P ρ:

(18.34)                    Cyclic coordinates
Thirdly, since the kinetic energy is approximately a quadratic form in the cyclic velocities alone, the other velocities being approximately zero, we have
(18.35)                    Cyclic coordinates
Finally, the total energy
(18.36)                    Cyclic coordinates
is a function of                    Cyclic coordinates                whose differential can, according to eqn (18.32), be written
(18.37)                    Cyclic coordinates
(p.216) Helmholtz considered, in particular, such systems for which the cyclic motions are all determined by one parameter. He called such systems monocyclic. Let us consider the simplest case where there is only one cyclic motion described by q r. In this case the energy transformation equation (36) can be written
(18.38)                    Cyclic coordinates
where
(18.39)                    Cyclic coordinates
This is in complete accordance with eqns (18.25) and (18.27) if we interpret the work dQ r stored in the cyclic motion as a measure of heat, and let                    Cyclic coordinates                and p r play the roles of temperature and entropy, respectively. As remarked above in the section on thermodynamics, an integrating denominator as                    Cyclic coordinates                is not unique, and in the case treated here, the kinetic energy of the system is itself an integrating denominator. Indeed from eqn (18.35) we have
(18.40)                    Cyclic coordinates
and therefore we obtain
(18.41)                    Cyclic coordinates
if we choose S such that
(18.42)                    Cyclic coordinates
or
(18.43)                    Cyclic coordinates
or
(18.44)                    Cyclic coordinates
where A is a constant. This is, according to Helmholtz, the choice of integrating denominator that will have the property that two systems having the same value of T will remain in equilibrium if coupled with each other. It is therefore the kinetic energy that plays the role of the temperature in Helmholtz’s mechanical model, in conformity with the kinetic theory of gases as developed by Boltzmann. One may also remark that L (according to eqn (18.36)) plays the role of Gibbs’s free energy.

As mentioned above, Helmholtz did not suggest that heat was in reality due to one hidden cyclic coordinate. Rather, he thought that it was probably a result of an elimination of many so-called adiabatic coordinates (see Helmholtz 1886, p. 157). Such eliminations can be done in a way similar to Routh’s elimination of cyclic coordinates (see (Helmholtz 1884, pp. 130–131) and (Helmholtz 1886, pp. 148–149)).

However, my research on combined monocyclic systems [systems with many cyclic motions described by one parameter (Helmholtz 1884)] have shown that also more complicated systems (p.217) in motion, which will be more similar to the inner molecular motions of hot bodies, can lead to the same theorems. (Helmholtz 1886, p. 157)

18.5 What is new in Hertz’s Mechanics?

In his book Hertz acknowledged his debt to Helmholtz’s papers:

Both in its broad features and in its details my own investigation owes much to the above-mentioned papers (Helmholtz 1884, 1886); the chapter on cyclical systems is taken almost directly from them. (Hertz 1894, Preface)

In view of Helmholtz’s and J.J. Thomson’s works we may ask: what was new in Hertz’s Mechanics? As far as Helmholtz is concerned Hertz answered the question himself:

Apart from matters of form, my own solution differs from that of von Helmholtz chiefly in two respects. Firstly I endeavour from the start to keep the elements of mechanics free from that which von Helmholtz only removes by subsequent restriction from the mechanics previously developed. Second, in a certain sense I eliminate less from mechanics, inasmuch as I do not rely upon Hamilton’s principle or any other integral principle. (Hertz 1894, Vorwort/Preface)

What Hertz says here about Helmholtz can be said of J.J. Thomson’s explicit, and other British physicists’ more implicit idea that all energy might be kinetic so that potential energy (and thus forces) are merely apparent fictions due to our lack of knowledge of the finer details of nature. No one had tried to build mechanics up from the ground on this assumption. This was left to Hertz. Moreover, in Hertz’s Mechanics the cyclic coordinates are not just mathematical parameters, they present coordinates of a system of hidden point masses coupled with the visible system under consideration. Hertz did not emphasize this distinctive character of his account of forces, perhaps because the contemporary physicists had implicitly made similar assumptions about the cyclic, hidden motions. However, the great difference between just introducing a new cyclic coordinate and insisting that it be a real coordinate of a hidden system, is made quite clear by a simple observation by Roger Liouville from 1892.

18.6 R. Liouville: One cyclic coordinate suffice

R. Liouville (Liouville 1892) considered a conservative mechanical system S described by the coordinates                    Cyclic coordinates                having kinetic energy T and potential energy U. He then constructed a new system S 1 described by                    Cyclic coordinates                and one new cyclic coordinate q r. The new system is assumed to have only kinetic energy                    Cyclic coordinates                determined by

(18.45)                    Cyclic coordinates
(p.218) and no potential energy. The motion of                    Cyclic coordinates                will be governed by Lagrange’s equations (seeeqn (18.4))
(18.46)                    Cyclic coordinates
and
(18.47)                    Cyclic coordinates
From eqn (18.47) we conclude that                    Cyclic coordinates                is a constant, and we consider those motions for which it is equal to 1. This means that the remaining eqns (18.46) are reduced to
(18.48)                    Cyclic coordinates
which are precisely the same as Lagrange’s equations (18.3) for the system S.

Thus, by an analysis, that is the reverse of J.J. Thomson’s argument, R. Liouville showed that in any conservative mechanical system one can assume that the potential energy is due to kinetic energy associated with just one hidden cyclic coordinate.

Hertz almost certainly did not learn about R. Liouville’s brief note, and even if he did, it would probably not have helped him much; indeed it is not clear how the new cyclic coordinate could be thought of as a coordinate of a hidden material system coupled with the system S. In fact, as pointed out by Hertz, it can only approximately be interpreted in this way (see Section 20.4).

Notes:

1 As Topper pointed out in his analysis of Thomson’s early commitment to mechanistic philosophy (Topper 1971) Thomson’s ideas about hidden cyclic motion went back to his thesis.