# Cyclic coordinates

# Cyclic coordinates

# 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 does not contain a certain generalized coordinate *q* _{ρ}; Hamilton’s equations

*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 . Then, as explained in Chapter 2, Lagrange had shown (Lagrange 1788) that the equations of motion can be written (see eqn (2.13))

*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

*L*is defined by

*b*, §20) that one could use eqns (18.6) to express in terms of the constants

*c*

_{ρ}. If we substitute these expressions into the expression for

*L*we obtain a function whose partial derivatives with respect to and

*q*

_{ρ}for

*ρ*= 1, 2,…,

*k*can be written as

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

*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 as hidden, unobservable coordinates^{1}. Given a mechanical (or another type of physical) system that we describe by way of the coordinates . We can then, through geometric considerations, determine its apparent kinetic energy as a quadratic form in the If, moreover, we can determine a function such that the motion of the system is described by eqn (18.11), we will conclude that is the Lagrangian of the system, and hence that 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

*T*

_{1}is a quadratic form of 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

*c*

_{ρ}and by way of eqn (18.6) or equivalently by:

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

*T*

_{1}and potential energy Indeed, the latter is, from eqn (18.15), a function of the variables alone, independent of

# 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 2*l* passing through a pulley at *P*. We shall think of the rod, whose position is described by the angle , 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

The energy of the total system is given by

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

Thus, the modified Lagrangian is given by

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 d*Q* 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:

*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

*q*

_{i}is increased by an amount d

*q*

_{i}. The important fact about d

*Q*is that although it is not itself an exact differential of and

*θ*, the temperature

*θ*is an integrating denominator such that d

*Q/θ*is an exact differential of a function called the entropy

*S*would do the trick equally well. Indeed, if we put

*Q*, 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 d

*Q*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 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 *r* − *k* cyclic coordinates, i.e. 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:

*q*

_{ρ}are no longer constants but we have

*q*

_{ρ}is increased by d

*q*

_{ρ}can be written (see eqn (18.26)):

The assumption that the motion is adiabatic has several important consequences. First, the state of the system is described by and and since *L* does not depend on the momenta *p* _{ρ} will be functions of . 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}:

*q*

_{r}. In this case the energy transformation equation (36) can be written

*Q*

_{r}stored in the cyclic motion as a measure of heat, and let and

*p*

_{r}play the roles of temperature and entropy, respectively. As remarked above in the section on thermodynamics, an integrating denominator as 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

*S*such that

*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 having kinetic energy *T* and potential energy *U*. He then constructed a new system *S* _{1} described by and one new cyclic coordinate *q* _{r}. The new system is assumed to have only kinetic energy determined by

*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.