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II. Classical Dynamics of the Harmonic Oscillator

As long as the Classical Mechanics is involved - and we shall limit ourselves only to this theoretical environment - the measurements with the harmonic oscillator are expected to be best described in relation to the classical equation of motion:

(6)

We prefer this equation from many points of view, but mainly for the fact that its coefficients - physical parameters characterizing the harmonic oscillator - had always have in history, sometimes explicitly, an interaction explanation. R here is a damping coefficient, always related to an external force acting proportionally to the velocity of the oscillator. K - the stiffness - although is usually supposed to mean something intrinsic to the oscillator, is actually a material property belonging to the system whose part is the harmonic oscillator - the local environment - thus also being an interaction property. The same can be said of the mass M: the history of Physics has considerations of different types of mass as determined by different types of interactions that the particle may encounter (electromagnetic mass, gravitational mass, inertial mass, etc.). We hold it true that the mass is an expression of the global environment. The circumstances always left unspecified for this kind of problem are then summed up in the interaction explanation of parameters. Usually we are not interested but in their values and these values explain the motion as described by equation (6). However, there are situations requiring an interaction explanation and, when Physics come back for it, it is, without exception, either downright unsatisfactory or partial at best. We desire to put the things in order at least regarding the place of this interaction explanation.

The equation (6) can be written in the form of a system

(7)

The second equation here is obviously a mere definition of the momentum. Equation (7), however, is not a Hamiltonian system, for its matrix is not an involution. To make it more obvious, we write the system in the matrix form

(8)

As long as the parameters are constants this equation can be put in a form exhibiting the position of the energy, therefore of the Hamiltonian (in the particular instances when the energy can be identified with the Hamiltonian). Indeed, using (8) we readily arrive at

(9)

proving that energy (the quadratic form in the right hand side) is the rate of variation of physical action represented as the elementary area of the phase plane. This was always the case in Physics, so equation (9) is nothing new. The point we want to make here is that the energy needs not conservation in order to be taken as the rate of physical action; however that action has to be properly defined as the area in phase plane. The equation (9) is actually a Riccati equation for a frequency, for it can be put into the form

(10)

Whether this frequency has a physical meaning or not does not concern us here. We just notice that the solution of the above equation is provided by the ratio of the solutions of the Hamiltonian system corresponding to equation (8) viz.

(11)

This is a general aspect of the relation between Riccati equation and Hamiltonian Dynamics (Zelikin 2000). We recover equation (9) above by simply constructing from (11) the differential 1-form of the area in phase plane. As to the equation (10), it can be easily integrated to show that the energy is not conserved, but we have the more complicated conservation law discovered by H. H. Denman (Denman 1968)

(12)

It is seen here that the energy is conserved in the classical sense only in case where the damping is zero. This fact is usually associated with the absence of dissipation forces. While this makes plainly sense, our contention here will be the identity of these dissipation forces. The focus is now the meaning of Riccati equation (10) and its associated Hamiltonian system (11).

What is the apriori possibility of ‘tuning’? The fact of the matter is that the classical dynamical equation (6) is expression of a variational principle related to the Lagrangian

(13)

representing an oscillator with variable parameters. Integral over any finite time interval [t₀, t₁] of this function is the physical action of the oscillator over that time interval. In obtaining the equation (6) a mandatory condition is that the variation of coordinate at the time ends is zero

(14)

but in order to obtain a closed trajectory we have to ask at least one more condition, namely

(15)

Furthermore, if we are to discuss the closed trajectory in the phase plane, the equality of speeds at the time ends is also a mandatory condition. These circumstances allow us to define the measurement process with the oscillator in a more precise manner.

Indeed, let us think along the following lines: from the point of view of the variational principle and the resulting equation of motion, the Lagrangian (13) is defined up to an additive function of time which is an exact time derivative of some other function. This fact is used in different branches of Theoretical Physics to define the so-called gauge transformations. Then let us define a gauge where the Lagrangian is a perfect square. The fact is well known and largely exploited as such in Control Theory (Zelikin 2000), so we walk along known route. Namely, add to the Lagrangian (13) the term

(16)

where w is a continuous function of time, and ask that the final Lagrangian be a perfect square. The variation of function under derivative is zero at the time ends by conditions from equation (14), so the equation of motion is not changed. The new Lagrangian turns out to be, in terms of generic coordinates

(17)

provided the function w satisfies to the following Riccati differential equation:

(18)

The Lagrangian (17) will then be taken to represent the energy of the results of measurement. As before, there is a relationship of Riccati equation (18) with the Hamiltonian dynamics. We find the analog of equation (11) as

(19)

which is plainly a Hamiltonian system. So we can, at rigor, conveniently identify the factors of w as coordinates in the phase plane. One can say that the measurement Lagrangian (17) represents an ensemble of oscillators along a Hamiltonian evolution in phase plane, as given by equation (19). The subsequent problem is the meaning of this ensemble.

The cautious reader may diligently ask: what makes a perfect square Lagrangian so special in order to represent the result of measurement? There is no answer to this question, other than the philosophical one deduced from our theoretical experience. First, is worth recalling that L. D. Landau and E. M. Lifschitz start their Mechanics by considering the Lagrangian of the free particle and then building on it by generalization (Landau, Lifschitz 1966). Why not use the inverse process?! Namely, in order to be able to say “this oscillator made that measurement” we have to be able to identify the oscillator as an individual structure. Whenever we refer, in Classical Mechanics, to an individual particle we acknowledge that its potential freedom is to be recognized in the form of its Lagrangian, which is simply the kinetic energy - a perfect square. Analogously we can accept that, in arbitrary environmental conditions the expression of apriori identity of a particle is a perfect square Lagrangian. Thus such a Lagrangian will be taken as representing the measurement with the harmonic oscillator. This purely philosophical fact will eventually be rationally proved to be a consequence of the Stoka Theorem.

Equation (18) shows that w is a dissipation coefficient more specifically a rate of mass variation (in case the mass is not constant), and this mass variation refers to the harmonic oscillator performing measurement. It is therefore important, for obvious physical reasons, to find the most general solution of that equation. J. F. Carinena and A. Ramos give a brief but pertinent survey of the integrability of the Riccati equation (Carinena, Ramos 1998). For our present needs we just notice that the complex numbers

(20)

are two particular (constant) solutions of the Riccati equation (18). So, we first perform the transformation

(21)

It is then easily seen that z is a solution of the linear homogeneous equation

(22)

Now, when conveniently expressing the initial condition z (0), we can write the general solution of the equation (18) by inverting (21) to the effect that

(23)

where r and t_r are two real constants characterizing the solution. Using equation (20) we can put this general solution in real terms:

(24)

revealing a modulation of frequency by a Stoler transformation (Stoler 1970, 1971) to the complex form of this parameter. Furthermore, if we denote

(25)

equation (24) becomes

(26)

where h is given by

(27)

The meaning of this complex quantity will become clear later. For the moment we note that the measurement process has been characterized here as a frequency modulation process. More specifically this process is a scaling of the difference between kinetic and potential energy - the Lagrangian - bringing this quantity to a perfect square. The physical meaning of the perfect square Lagrangian is that it describes a fundamental physical unit inside a complicated system. As expected, the Lagrangian actually corresponds to a set of such fundamental physical units: a set of oscillators having the same frequency.

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