Comment on “On the Lagrangian and Hamiltonian description of the damped linear harmonic oscillator” [J. Math. Phys. 48, 032701 (2007)]

It seems unlikely that the equation for the damped classical harmonic oscillator

a dissipative system, could be derived from a Hamiltonian. This is because $E= 1 2 x ̇ 2 + 1 2 ω 2 x 2$⁠, the standard expression for the total energy, is not conserved for γ≠0. Indeed, it satisfies the equation

showing that it decreases with time.

Nonetheless, Bateman¹ made the remarkable observation that if one appends the time-reversed oscillator equation with undamping (gain) instead of damping,

then even though the two oscillators are independent and noninteracting [the two evolution equations (1) and (3) are not coupled], the equations of motion (1) and (3) can be derived from the time-independent quadratic Hamiltonian

involving the two pairs of canonical variables (x, p) and (y, q). To derive the original Equation (1) we use the pair of Hamilton’s equations

Substituting for q in (6) gives $q ̇ =−γ x ̇ − ω 2 x$⁠, and then differentiating (5) with respect to t gives (1). The partner Equation (2) can be similarly obtained from the remaining Hamilton’s equations

It is worth noting that the Hamiltonian (4) is $PT$ symmetric² if we define the action of $P$ as interchanging the two oscillators

while $T$ reverses the signs of the momenta,

The PT symmetry of H in (4) and the success of Bateman’s strategy depend crucially on the gain and loss terms in (1) and (3) being exactly balanced. As a consequence of the gain/loss balance, the system possesses a conserved quantity, namely the value of H. However, the energy has the complicated form (4) and is not a simple sum of kinetic and potential energies.

Recently it was shown³ that the equation of motion (1) of the damped oscillator can be derived from a (nonquadratic) time-independent Hamiltonian depending on only a single canonical pair (x, p). This remarkable result was proved by using a modification⁴ of the Prelle-Singer approach⁵ to identify integrals of motion of dynamical systems. This paper is a comment on the interesting work in Refs. 3 and 4.

In Ref. 3 different forms of the Hamiltonian were given depending on whether the system was overdamped, underdamped, or critically damped. In particular, for the overdamped case (γ > ω) the Hamiltonian takes the unconventional form

where A, B, and δ are constants that we will specify later. Different functional forms were given for the other cases in order to have a real Hamiltonian. However, since this is not a concern for us, the functional form of (11) serves for all cases (apart from an obvious modification in the case of critical damping).

In this paper we show in Sec. II how the Hamiltonian (11) can be derived by elementary means and we identify the coefficients therein in terms of the (generally complex) eigenfrequencies of the problem. The conserved quantity, written in terms of $x ̇$ and x, is similarly identified. In Sec. III we go on to apply the procedure to the third-order linear equation with constant coefficients, a particular example of which describes the nonrelativistic self-acceleration of a charged oscillating particle.⁶ The additional conceptual problems that arise in this case are discussed in detail. In Sec. IV we generalize the procedure to an arbitrary nth-order constant-coefficient equation. Section V discusses the difficult problem of quantizing Hamiltonians of this type. Finally, Sec. VI gives a brief summary.

We begin with (1) and substitute x(t) = e^−iνt. This gives a quadratic equation for the frequency ν,

which factors as

where

with

The generic form of a Hamiltonian H(x, p) that can generate the evolution equation (1) is given in (11). There are two such Hamiltonians, corresponding to the two eigenfrequencies in (15). The first is

A second and equally effective Hamiltonian is obtained by interchanging ω₁ and ω₂,

As mentioned in Sec. I, Hamiltonians of this form appear in Ref. 3 for the case of over-damping (γ² > ω²), where they are real since in that case ω₁ and ω₂ are purely imaginary, but they apply equally well when γ² < ω² if we are not concerned with the reality of the Hamiltonian. Indeed, the Hamiltonian is no longer the standard real energy, which is not conserved. Rather, it is a complex quantity which is conserved and from which the equations of motion can be derived in the standard way.

For the Hamiltonian H₁, Hamilton’s equations read

We then take a time derivative of (18) and simplify the resulting equation first by using (19) and then by using (18),

Thus,

which reduces to (1) upon using (14).

The evolution equation (1) has one conserved (time-independent) quantity, which can be expressed in terms of x(t) only. To find this quantity, we begin with (18) and solve for p,

We then use this result to eliminate p from H₁. Since H₁ is time independent, we conclude that

is conserved. Had we started with the Hamiltonian H₂ we would have obtained the conserved quantity

but this is not an independent conserved quantity because C₂ = 1/C₁. These conserved quantities were also found in Ref. 3 for the case of over-damping.

When γ = 0, these results reduce to the familiar expressions for the simple harmonic oscillator. In that case we let ω = ω₁ = − ω₂, so that H₁ becomes

which is related to the standard Hamiltonian for the simple harmonic oscillator by the change of variable p → p − iωx. The conserved quantities C₂ and C₁ become simply $( x ̇ 2 + ω 2 x 2 ) ± ω$⁠, in which we recognize the usual conserved total energy.

In this section we show how to construct a Hamiltonian that gives rise to the general third-order constant-coefficient evolution equation

where D ≡ d/dt. The Hamiltonian that we will construct has just one degree of freedom.

A physical example of such an equation is the third-order differential equation

that describes an oscillating charged particle subject to a radiative back-reaction force.⁶ Following Bateman’s approach for the damped harmonic oscillator, Englert⁷ showed that the pair of noninteracting Equation (27) and

can be derived from the quadratic Hamiltonian

This Hamiltonian contains the four degrees of freedom (x, p), (y, q), (z, r), and (w, s). An interacting version of this model was studied in Ref. 8. In fact, we find that the two equations of motion (27) and (28) can be derived from the simpler quadratic Hamiltonian

which has only the three degrees of freedom (x, p), (y, q), and (z, r). A similar three-degree of freedom Hamiltonian was also found in Ref. 7.

Our objective here is to find a one-degree-of-freedom Hamiltonian that can be used to derive the third-order differential equation (26). Note that the general solution to (26) is

where a_k are arbitrary constants. If we form (D + iω₂)(D + iω₃) x, that is, $x ̈ +i ( ω 2 + ω 3 ) x ̇ − ω 2 ω 3 x$⁠, we obtain

in which the constants a₂ and a₃ do not appear. Similarly, we have

So, assuming that the frequencies ω_k are all different, there are two distinct conserved quantities, namely

These expressions and the equation of motion can be derived from the Hamiltonian

where b₂ and b₃ are arbitrary constants. Thus, $p ̇ ≡−∂H/∂x=i ω 1 p$⁠. This means that p ∝ e^iω₁t, so that 1/p is directly related to the combination in (31).

Then, from Hamilton’s equation $x ̇ ≡∂H/∂p$ and from further differentiation with respect to t, we obtain

This further differentiation is crucial because it is required to eliminate the constants b₂ and b₃, and thus to obtain a differential equation that only contains x(t) and is independent ofb₂andb₃. Combining these equations and performing some simplifying algebra, we obtain

We emphasize that the constants b₂ and b₃ have disappeared in this combination and we have reconstructed the equation of motion (26). These constants are reminiscent of Lagrange multipliers, but they are unlike Lagrange multipliers in that we do not vary the Hamiltonian with respect to them. Rather, we require that the equations of motion be independent of these constants.

Using only derivatives up to the second order, we can find expressions for b₂p^{−ω₂/ω₁} and b₃p^{−ω₃/ω₁}, namely

in which we recognize two of the quantities in square brackets that appear in (33). The third such quantity, namely $x ̈ +i ( ω 1 + ω 2 ) x ̇ − ω 1 ω 2 x$⁠, is closely related to H,

We conclude that

Similarly, we have

Thus C₂ and C₃ are constants of the motion because they are both proportional to the Hamiltonian, with proportionality constants given by powers of b₂ and b₃, respectively.

To summarize, by eliminating the parameters b₂ and b₃ the evolution equation (26) can be derived from the unusual time-independent Hamiltonian (34) containing the single coordinate variable x and its conjugate momentum p. This Hamiltonian is a conserved quantity, which can be expressed as

The conserved quantities C₂ and C₃ are both proportional to H.

Before moving on, we must explain how a Hamiltonian with a single degree of freedom can give rise to a differential equation whose order is greater than two. The problem is as follows. Our Hamiltonian has the generic form

Therefore, the equations of motion are simply

where g(p) = f′(p), and

First, we solve (41)

where C is an arbitrary constant. Next, we return to (40), which becomes

after we eliminate p by using (42). This is a first-order equation. Thus, its solution has only two arbitrary constants:

We obtain the higher-order differential equation (36) by the sequence of differentiations in (35) that were required to eliminate the constants b₂ and b₃. Of course, the solution to an nth-order equation can incorporate n pieces of data such as n initial conditions: x(0), $x ̇ ( 0 )$⁠, $x ̈ ( 0 )$⁠, $x ⃛ ( 0 )$⁠, and so on. How is it possible to incorporate n pieces of data with only two arbitrary constants C and D? There appear to be n − 2 missing arbitrary constants.

The answer is that the n − 2 pieces of initial data determine n − 2 parameters b_k multiplying each of the fractional powers of p in H. (One parameter can always be removed by a scaling.) We can incorporate the initial data into the Hamiltonian in the form of these parameters. These parameters specify an ensemble of Hamiltonians, all of which give a unique nth-order field equation that is independent of these parameters and is capable of accepting n pieces of initial data.

For the triple-dot equation we can see from (37) that the ratio $b 2 1 / ω 2 / b 3 1 / ω 3$ is related to the initial conditions. So, for the case of the third-order equation, the three arbitrary constants are C, D, and $b 2 1 / ω 2 / b 3 1 / ω 3$⁠. We emphasize that the Hamiltonian gives the higher-order equations of motion precisely because of the requirement that the parameters b_k drop out from the equation of motion. The parameters b_k in the Hamiltonian are crucial because they incorporate the initial data and are determined by the initial data. The nonzero parameter b₃ forces the evolution equation to be third order. Without b₃ the Hamiltonian does not know about the third frequency ω₃. Indeed, if b₃ = 0, (34) reduces to (16) (with b₂ = 1). [This is consistent with (37) because setting b₃ = 0 there implies that $x ̈ +i ( ω 1 + ω 2 ) x ̇ − ω 1 ω 2 x=0$⁠.]

Finally, one may ask what would happen if we followed the standard procedure for deriving the equations of motion for x(t) from the Hamiltonian equations of motion $p ̇ =−∂H/∂x$ and $x ̇ =∂H/∂p$⁠. This would mean solving the second equation for p in terms of x and $x ̇$ and then substituting back in the first to obtain a second-order equation for x(t). In our case that would mean solving the first of Equations (35) for p, which is not possible for general values of the parameters. However, it is instructive to see how this procedure works if we choose the parameters so that an explicit solution is possible. For example, if we choose ω₁ = 1, ω₂ = − 2, ω₃ = 4, b₂ = 2, and b₃ = 1, the equation can be solved to give $p 2 =−1+ x ̇ + i x + 1$⁠. Substituting back into the equation $p ̇ =ip$ gives, after some algebra, the nonlinear second-order equation

Further manipulation shows this to be equivalent to the constancy of C₂/C₃.

So, in the cases where the standard procedure can be followed in practice, the resulting nonlinear second-order equation is equivalent to an equation for a constant of the motion (which of course depends on the parameters b₂ and/or b₃).

It is straightforward to generalize to the case of an arbitrary nth-order linear constant-coefficient evolution equation

whose general solution is

For simplicity, we assume first that the frequencies ω_r are all distinct; at the end of this section we explain what happens if some of the frequencies are degenerate.

Corresponding to (32) and (33), we have

Thus, the quantity

is proportional to e^−it for all s. Hence, the n − 1 independent ratios Q_s/Q₁ (s > 1) are all conserved. Any other conserved quantities can be expressed in terms of these ratios.

The equation of motion and the conserved quantities can be derived from the Hamiltonian

which is the nth order generalization of (34) for the cubic case. In this expression the n − 1 coefficients b_r are arbitrary, and must be eliminated to give the nth-order equation of motion. Note that in constructing the Hamiltonian H there is nothing special about the subscript “1” and it may be replaced by the subscript “s” (1 < s ≤ n).

Until now, we have assumed that the frequencies ω_r are all distinct. However, if some of the frequencies are degenerate, there is a simple way to construct the appropriate Hamiltonian: If the frequencies ω₁ and ω₂ are equal, we make the replacement

(In making this replacement we are shifting the Hamiltonian by an infinite constant.) Thus, for ω₁ = ω₂ the Hamiltonian H₁ in (16) reduces to

Hamilton’s equations for this Hamiltonian immediately simplify to (21) with ω₁ = ω₂. Similarly, for the case ω₁ = ω₂ the Hamiltonian (34) reduces to

and Hamilton’s equations for this Hamiltonian readily simplify to (36) with ω₁ = ω₂.

Also, if the frequencies are triply degenerate, ω₁ = ω₂ = ω₃ = ω, the Hamiltonian in (34) is replaced by

where b and c are two parameters that are determined by the initial data. Once again, Hamilton’s equations for this Hamiltonian combine to give (36) with ω₁ = ω₂ = ω₃ = ω.

The obvious question to be addressed next is whether it is possible to use the Hamiltonians that we have constructed to quantize classical systems that obey linear constant-coefficient evolution equations. Let us begin by discussing the simple case of the quantum harmonic oscillator (QHO), whose Hamiltonian H₁ is given in (25).

One possibility is to quantize the Hamiltonian in p-space by setting x = id/dp. Then by shifting E by ω we see that the time-independent Schrödinger eigenvalue equation is

whose solution is

In this way of doing things we derive the quantization condition by demanding that $ψ ̃$ be a well defined, nonsingular function, which requires that E = nω, where n is a non-negative integer.⁹ However, these “momentum-space” eigenfunctions are problematical because p has no clear physical interpretation as a momentum. The p-space eigenfunctions are certainly not orthonormal in any simple sense because they do not solve a Sturm-Liouville boundary-value problem.¹⁰ 

However, we can calculate the corresponding x-space eigenfunctions by Fourier transform using the formula¹¹ 

We find that

where φ_n(x) is the nth eigenfunction of the QHO. This is consistent with our remark above that H₁ is related to the standard QHO Hamiltonian by the transformation p → p − iωx. This transformation is achieved at the operator level by the similarity transformation p → e^−ω2x2/2pe^ω2x2/2.¹² Because of this additional factor, our eigenfunctions are orthonormal with respect to the metric η = e^ω2x2. As an alternative approach, we can cast (25) in x-space as

from which we can obtain the ψ_n(x) directly.

To summarize, the quantized version of (25) corresponds to a transformed version of the QHO, where the x-space eigenfunctions are simply related to the standard eigenfunctions and are orthonormal with respect to an additional weight function. The p-space eigenfunctions can be written down but their interpretation is not obvious (the operator p corresponds to the conventional raising operator a^†) and are not orthogonal in any simple way. Moreover, p, represented as −id/dx, is not Hermitian because the overlap integral between two wave functions has to be calculated with the inclusion of the metric η(x), so integration by parts is no longer simply a matter of a minus sign. Instead, p is pseudo-Hermitian^13,14 with respect to η, i.e., p^† = ηpη⁻¹. In p space the weight function e^ω2x2 becomes the highly nonlocal operator e^−ω2d2/dp2.

If we generalize to the damped harmonic oscillator, we can still find a solution $ψ ̃ ( p )$ to the time-independent Schrödinger equation, namely³ 

but even if we take E = nω₁ to make the prefactor nonsingular, we are still left with a nonintegral, and in general complex, power of p in the exponential. (See, also, the comments in Ref. 1.) Thus, in addition to the previously discussed problems with $ψ ̃ ( p )$⁠, we would now have to consider it to be a function in a cut plane. Moreover, there is no simple formula like (55) whereby one could obtain the x-space eigenfunctions. Furthermore, if we cast the equation in x-space we obtain

in which the difficulty associated with a fractional derivative is manifest.

Evidently, quantizing Hamiltonians of the form in (39) is nontrivial. The problem of quantizing the cubic equation describing the back-reaction force on a charged particle was solved in Ref. 8. However, the system that was actually quantized was a pair of coupled cubic equations in the unbroken $PT$-symmetric region. Thus, it may be that the most effective approach for quantizing a Hamiltonian of the form (39) is to introduce a large number of additional degrees of freedom.

We have shown that any nth-order linear constant-coefficient evolution equation can be derived from a nonconventional but simple Hamiltonian of the form (39). Remarkably, this Hamiltonian has only one degree of freedom, that is, one pair of dynamical variables (x, p). Furthermore, we have shown that for such a system there are n − 1 independent constants of the motion and we have constructed these conserved quantities in terms of x(t) and its time derivatives. However, we find that it is not easy to formulate a general procedure to quantize the system described by the Hamiltonian, and this remains an extremely interesting but difficult open problem.