Analogy between Hamilton-Jacobi’s classical mechanics and geometrical optics.

## Abstract

It is well known that, by taking a limit of Schrödinger’s equation, we may recover Hamilton-Jacobi’s equation which governs one of the possible formulations of classical mechanics. Conversely, we may start from the Hamilton-Jacobi’s equation and, by using a lifting principle, we may reach a set of nonlinear generalized Schrödinger’s equations. The classical Schrödinger’s equation then occurs as the simplest equation among the set.

### Keywords

- Schrödinger’s equation
- Hamilton-Jacobi’s equation
- correspondence principle
- lifting principle

## 1. Introduction

Schrödinger’s equation is the fundamental equation of quantum mechanics. Using a correspondence principle, we may recover the classical limit of mechanics under the form of the Hamilton-Jacobi’s equation. This is a up-down process, from a general theory to a limit restricted theory, i.e. from quantum mechanics to classical mechanics. We may use another principle, that I call a lifting principle, which, starting from Hamilton-Jacobi’s equation allows one, through a bottom-up process, to reach a set of generalized Schrödinger’s equations, encompassing nonlinear terms. From this generalized set, we may turn back to a up-bottom process. In a first step, we recover the classical Schrödinger’s equation as, in some sense, the simplest equation in the set and, in a second step, we recover again classical mechanics from quantum mechanics, using again a correspondence principle.

The chapter is organized as follows. Section 2 recalls the Hamilton-Jacobi’s equation of classical mechanics which, in the present chapter, may be viewed as a turning equation, both the end of a up-bottom process and the beginning of a bottom-up process. Section 3 exemplifies a way to obtain Schrödinger’s equation by using an analogy relying on Hamilton-Jacobi’s equation. Section 4 expounds the bottom-up process from Hamilton-Jacobi’s equation to a set of generalized Schrödinger’s equations. Section 5 provides a complementary discussion while Section 6 is a conclusion.

## 2. Hamilton-Jacobi’s formulation of classical mechanics

We know that classical mechanics can be declined under four different formulations, which are mathematically and empirically equivalent. These are the Newton’s, Lagrange’s, Hamilton’s and Hamilton-Jacobi’s formulations. In the present chapter, we rely on the Hamilton-Jacobi’s formulation, see for instance Louis de Broglie [1], Blotkhintsev [2], Landau and Lifchitz [3], and Holland [4]. This formulation of nonrelativistic classical mechanics of a matter point relies on an equation, that I shall call Hamilton-Jacobi’s equation, reading as:

This equation allows one to study the motions of a particle of mass

in which

Inserting Eq. (4) into Eq. (1), we obtain:

We now consider the locus of the points for which

Eq. (6) shows that the locus is a time-independent surface. There is one surface, and only one, containing a point

Therefore,

Eq. (8) shows that the locus is still a surface but which now depends on time. When times goes on, the surface moves and, in general, experiences a deformation. For a given time

We now consider a fictitious point P, pertaining to the surface

in which

that is to say:

leading to:

But

We are therefore facing two different velocities (i) the velocity

We then remark that Newton’s formulation relies on the existence of trajectories while Hamilton-Jacobi’s formulation relies both on trajectories and on a field filling the space. Hamilton-Jacobi’s formulation is the first one in which the motion of a localized object has been associated with a space filling field. In other words, Hamilton-Jacobi’s formulation is nonlocal. This nonlocality actually anticipates the nonlocality of quantum mechanics and the space filling field

## 3. Guessing Schrödinger’s derivation

Strictly speaking, there is no derivation of Schrödinger’s equations but a variety of guessing approaches, with different flavors depending on the preferences of the authors. Basically, however, Schrödinger’s equation has been introduced in [7, 8] under its stationary form and in [9] under its time-dependent form. English translation is available from [10] and French translation from [11]. The derivation relies on an analogy between Hamilton-Jacobi’s formulation of classical mechanics and geometrical optics. As rather usual when something new is exposed for the first time, Schrödinger’s argument is more complicated than necessary. For instance, it relies on the use of non-Cartesian coordinates and on a non-Euclidean interpretation of the configuration space, requiring the use of covariant and contravariant components of vectors (more generally, of tensors), which may be unfamiliar to some readers. Feynman even commented that some arguments invoked by Schrödinger are erroneous [12]. Without showing any disrespect to Schrôdinger’s work, I prefer to present a more recent exposition extracted from Winogradski [13] who defended her thesis under the supervision of Louis de Broglie.

We begin with scalar wave optics and with the corresponding wave equation reading as:

in which

Because

leading to:

with:

In these expressions,

The wave-number

Inserting Eq. (16) into Eq. (15), we obtain:

Next, inserting Eq. (17) into Eq. (22), we obtain two equations relating the real amplitude

If the medium, besides being steady, is homogeneous (

in which

We are now equipped enough to turn to a discussion of geometrical optics which is an approximation to wave optics. This approximation is valid whenever the optical wave approximately behaves as a plane wave over a distance of the order of the wave-length

Furthermore, because

Also, from Eqs. (20) and (26), we have:

Now, similarly as for

Assembling the results obtained for the conservative Hamilton-Jacobi’s classical mechanics and for geometrical optics, we obtain a remarkable analogy exhibited in Table 1.

Classical mechanics | Geometrical optics |
---|---|

Trajectory | Ray |

This analogy has been discovered by Hamilton, about one century (!) before its use to the discovery of Schrödinger’s equations, see Refs. [14, 15], references therein and prior references from Hamilton. Formally, we may express the same structure by using a mechanical language or an optical language. Both languages may be translated, from one to the other, by using a dictionary D exhibited in Table 2, where the newly introduced constant

(a) | |

(b) | |

(c) | |

(d) | |

(e) | |

trajectory |

An analogy is not necessarily significant but any analogy should be, at least tentatively, taken seriously. If the analogy is fully meaningless, then the value of the constant * A contrario*, if the analogy is somehow meaningful, that is to say if the motion of a material point can be somehow associated with the propagation of a certain scalar field (the point of view taken

*by Louis de Broglie in his double solution), then the constant*very seriously

which we call de Broglie, or Einstein-de Broglie relations. Eq. (28) expresses an equivalence between momentum (mechanical language) and wave-number (optical language), while Eq. (29) expresses an equivalence between energy (mechanical language) and angular frequency (optical language).

The situation we are facing is now sketched in the Figure 1 below. First, we possess an analogy between Hamilton-Jacobi’s classical mechanics and geometrical optics, expressed by a dictionary D. Second, geometrical optics is an approximation to scalar wave optics. The Figure 1 then exhibits three filled rectangles, and we may feel intuitively but clearly that something is lacking, corresponding to the fourth empty rectangle. To fill this rectangle, we apply the dictionary D to wave optics. From the dictionary of Table 2, with

We may then translate Eq. (22) to:

which is exactly the time-independent (stationary) Schrödinger’s equation. Therefore, Eq. (16) is translated to:

and we readily establish that

Next, we can eliminate

leading to:

which is the general time-dependent Schrödinger’s equation. Invoking the “simplest” way to obtain Eq. (34) rules out awkward expressions such as the one obtained by deriving Eq. (32) twice with respect to time, i.e.:

## 4. Deriving a set of generalized Schrödinger’s equations

There are good reasons to believe that classical mechanics is suspicious. One of them is the existence of singularities in classical mechanics such as exhibited in the mechanical rainbow [16, 17]. If we trust a non-singularity principle stating that “local infinity in physics is not admissible” [18], we arrive to the conclusion that we must build a wave mechanics (nowadays better known as “quantum mechanics”). For this, we decide to start from what we know (actually what we are supposed to know), namely classical mechanics. We are looking for a wave mechanics based on a wave

in which

For the relationship between

Now, I invoke a principle that I call the lifting principle (later to be commented a bit more when the demonstration is completed). This principle tells us something very simple, even looking a bit like tautological, as follows: classical mechanics is an approximation to wave mechanics. Rather than simply using the argument

in which

The function

in which we used a subscript

This can be rewritten as:

which, relabelling, identifies with Eq. (38).

We are now looking for a differential equation satisfied by the wave

We begin with the assumption that, besides derivatives with respect to

The derivative

in which we again use a subscript

The set

We may also express the derivative

We rewrite Eq. (44) as:

or, invoking Eq. (42):

But, Hamilton-Jacobi’s equation (and the lifting principle) implies that the r.h.s. of Eq. (46) must contain a term with no derivative associated with

in which:

We therefore set, without any loss of generality:

in which

The evolution Eq. (42) then takes the form:

and our next task is to evaluate

To this purpose, we now return to Eq. (46) and insert in it Eqs. (49) and (47), leading to:

with:

In the classical limit (

which must identify with Hamilton-Jacobi’s equation. Under the proviso to be checked later that the r.h.s. of Eq. (52) must be vanishingly small, we then obtain, from the l.h.s.:

in which

We must now recall that the coefficient

From Eq. (55), we then have:

With

Inserting Eqs. (56) and (57) into Eq. (50), we then obtain:

Concerning the constant

Let

which is indeed

Eq. (58) is the main result of this subsection. It provides a set of generalized Schrödinger’s equations, being admitted that they are evolution equations (first derivative with respect to time), obtained by a deformation of Hamilton-Jacobi’s equation, according to the lifting principle. The classical Schrödinger’s equation is, in a certain sense, the simplest equation in the set. It is obtained by setting the nonlinear term

Let us note that the function

## 5. Complementary discussion

From the generalized Schrödinger’s Eq. (58) we may recover the classical Schrödinger’s equation, as we have commented, by setting

This is a first application of the correspondence principle. A second application of this correspondence principle afterward allows one to recover the classical Hamilton-Jacobi’s equation from Schrödinger’s equation, as discussed for instance by Blotkhintsev [2]. From the generalized Schrödinger’s equation, we therefore recover the classical Hamilton-Jacobi’s equation by a two-step up-bottom process, applying twice the correspondence principle. Another approach is to use Eq. (58) as an Ansatz under the form:

and to pursue the game with the correspondence principle to recover, using again a two-step approach, Hamilton-Jacobi’s equation. But the use of an Ansatz is less rigorous than the lifting principle because it contains the risk to make the Ansatz too simple, and therefore to omit significant terms. Note, however, that we have implicitly made the assumption that the state of the wave is defined by the wave

To clearly emphasize the difference between the correspondence and the lifting principles, let us consider two theories, denoted

For example, the lifting principle tells us that classical mechanics is an approximation to quantum mechanics. Therefore, quantum mechanics must indeed satisfy a correspondence principle, meaning that the correspondence principle is contained in the lifting principle. However, as we have seen, it does not identify with it. What we have done to use it is to start from

Another point of view may be taken by using a metaphor from Feynman [12] according to which the correspondence principle proceeds from one object to its shadow (and there is one shadow for one object) while the lifting principle proceeds from a shadow to objects (and there are several possible objects for a given shadow). Our results agree with this expectation. We did not reach Schrödinger’s equation, but rather a set of generalized Schrödinger’s equation. The derivation of Schrödinger, and all Schrödinger-like derivations, reach a single result because they used analogies, guesses and trials, with more or less implicit assumptions. Conversely, the use of the lifting principle simultaneously provides the whole set of admissible possibilities with a minimal number of assumptions (namely that we have to deal with an evolution equation). All candidates are reached in a single step.

## 6. Conclusion

The realm of nonlinear Schrödinger’s equations is very rich, with many applications such as to fluid mechanics, solitons, nonlinear optics and Bose-Einstein condensates. In the present chapter, we have demonstrated, using a lifting principle, that such equations occur naturally as a generalization of Hamilton-Jacobi’s formulation of classical mechanics, without however pretending that nonlinear equations obtained by the lifting process identify with nonlinear Schrödinger’s equations used in other different contexts (this would require another specific study outside of the scope of the present chapter). The material presented in this chapter is extracted from a book, namely [26]. It is here however presented under a single roof and might then attract the interest of other readers.

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