Simple derivations of the Hamilton–Jacobi equation and the eikonal equation without the use of canonical transformations Alex Smalla兲 and Kai S. Lam Department of Physics, California State Polytechnic University, Pomona, California 91768

共Received 13 November 2010; accepted 13 January 2011兲 The Hamilton–Jacobi equation in classical mechanics and the related eikonal equation in geometrical optics are often described as the “point of closest approach” between classical and quantum mechanics. Most textbook treatments of Hamilton–Jacobi theory are aimed at graduate students and derive the equation only after a long introduction to canonical transformations. Most treatments of the eikonal equation only emphasize its use in geometrical optics. We show that both the Hamilton–Jacobi equation and the eikonal equation can be derived by a common procedure using only elementary aspects of the Lagrangian and Hamiltonian formalisms introduced in undergraduate classical mechanics courses. Through this common approach, we hope to highlight to undergraduates the deep connections between classical mechanics, classical wave theory, and Schrödinger’s wave mechanics. © 2011 American Association of Physics Teachers. 关DOI: 10.1119/1.3553462兴 I. INTRODUCTION The deep connection between classical mechanics, classical wave theory, and quantum mechanics is a fascinating subject that has received little attention in the undergraduate curriculum. A focal point of these connections is the relation between the Hamilton–Jacobi theory of Hamiltonian dynamics and the short-wavelength limit of wave optics 共geometrical optics兲. Most advanced undergraduate classical mechanics textbooks include at least a short overview of the Hamiltonian formulation of mechanics. At the level of most undergraduate courses, the Hamiltonian formulation has few obvious advantages over the Lagrangian formulation as a means for solving problems. The most compelling reason to study Hamiltonian mechanics is not to directly solve specific mechanics problems, but rather to acquire deeper insights into the structure of classical mechanics and its relation to statistical mechanics, nonlinear dynamics, quantum mechanics, and field theory. Most of those insights require a level of sophistication that is beyond the typical undergraduate course. The main goal of this paper is to show how the Hamilton– Jacobi equation in Hamiltonian dynamics and the closely related eikonal equation in geometrical optics can be derived by a simple and common approach that assumes only the elements of Lagrangian and Hamiltonian mechanics presented in a typical upper-division undergraduate course at the level of, say, Taylor,1 or Marion and Thornton.2 We will then discuss how these equations provide the link between classical and quantum mechanics3 in a manner that is accessible to undergraduates. The usual approach to derive the Hamilton–Jacobi equation in advanced graduate textbooks is to first develop the theory of canonical transformations and show that by a suitable transformation of canonical coordinates and momenta, from the old coordinates qi and momenta p to new coordinates Qi and momenta Pi, the transformed Hamiltonian can be made identically zero.4–6 The generator of this transformation is a function S共qi , Pi , t兲 共where qi is a coordinate and Pi is the conjugate momentum兲, which is shown to obey the Hamilton–Jacobi equation, 678

Am. J. Phys. 79 共6兲, June 2011

http://aapt.org/ajp

冉

冊

⳵S ⳵S ,qi,t = 0, +H ⳵t ⳵ qi

共1兲

where H共pi , qi , t兲 is the Hamiltonian. Typically, S is then shown to be equal to the action integral of Lagrangian mechanics. The full theoretical machinery of canonical transformations, though useful in advanced problems, is a significant challenge for most undergraduates, and even with wellprepared undergraduates, it is difficult to fit this theory into the time constraints of a typical semester-long course. This lack of attention to Hamilton–Jacobi theory is unfortunate because straightforward arguments from the Hamilton– Jacobi equation show the deep connections between classical and quantum mechanics, namely, that S is the phase of the quantum mechanical wave function 共up to a factor of 1 / ប兲. With a little bit of algebra, it can be shown that the Hamilton–Jacobi equation can be obtained in the shortwavelength limit of the Schrödinger equation. In this paper, we first show that the Hamilton–Jacobi equation can be obtained as a direct consequence of the form of the action integral. Although we are not aware of a published derivation using our approach, our derivation is similar to the approach of Landau and Lifshitz.7 However, these authors derived the relation between the momentum and the action from the variational approach used to derive the Euler– Lagrange equations, which is more involved than our approach of working from the form of the action integral. Unlike the undergraduate-level derivation of Yan,8 we make no assumptions about the form of the velocity field and work strictly from Hamilton’s principle. After deriving the Hamilton–Jacobi equation, we solve it for a free particle and note the similarity to the phase of a free-particle wave function. This derivation only requires students to be familiar with the action integral from Lagrangian mechanics and to know that the Hamiltonian is given by H = pq˙ − L. To show that the relation between a wave theory and a theory of straight-line propagation subject to a variational principle is not a mere coincidence, we then derive the eikonal equation of geometrical optics. The eikonal equation is similar to the Hamilton–Jacobi equation, a point noted in graduate mechanics textbooks when discussing the relation between the Hamilton–Jacobi equation and wave mechanics. © 2011 American Association of Physics Teachers

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Most graduate students do not encounter the eikonal equation in a traditional optics course unless they take an advanced course on lens design or the foundations of geometrical optics, and thus the eikonal equation is unknown to many graduate students. We derive the eikonal equation in a manner analogous to our derivation of the Hamilton–Jacobi equation to show the connections between 共geometrical and physical兲 optics and 共classical and quantum兲 mechanics.

We begin with the action integral for a system with one degree of freedom,

冕

t

共2兲

L共q,q˙,t兲dt.

t0

We write the Lagrangian as L = pq˙ − H, where p = ⳵L / ⳵q˙, and obtain 关for a given trajectory q共t兲兴 S共q,t兲 =

冕 冕

t

p共t兲q˙共t兲dt −

t0

t

H共p共t兲,q共t兲,t兲dt

共3兲

t0

q

=

冕

pdq −

q0

冕

共4兲

t0

where q0 = q共t0兲 and Eq. 共4兲 follows from the fact that q˙dt = dq. Our task now is to interpret this pair of integrals. From Eq. 共4兲, the action is seen to be a line integral in the t-q plane. In general, any line integral S along some path from 共t0 , q0兲 to 共t , q兲 can be written as

冕

⳵S dq + S共q,t兲 = q0 ⳵ q q

冕

⳵S dt. t0 ⳵ t t

共5兲

By equating terms in Eqs. 共4兲 and 共5兲, we conclude that p=

⳵S , ⳵q

共6兲

冉 冊

⳵S ⳵S = − H共p,q,t兲 = − H ,q,t , ⳵t ⳵q

共7兲

where Eq. 共7兲 is the Hamilton–Jacobi equation. The generalization to a system with an arbitrary number of degrees of freedom is straightforward. We replace pq˙ by 兺piq˙i, where i indexes the degrees of freedom, and the single integral over q becomes a sum of integrals over qi,

兺i

冕

qi

qi共t0兲

pidqi −

冕

t

Hdt,

共8兲

t0

from which it follows that pi = ⳵S / ⳵qi. With the appropriate derivatives of S substituted for momenta in the Hamiltonian, we obtain the Hamilton–Jacobi equation from Eq. 共1兲.

H=−

III. CONSEQUENCES FOR A SINGLE PARTICLE Having derived a partial differential equation for the action S, we now elucidate a few consequences of the equation and hence an interpretation of S. From now on, we will work in the Cartesian coordinates so that for a single particle the momentum is p = ⵱S. We thus see that particle velocities 共in Am. J. Phys., Vol. 79, No. 6, June 2011

⳵S = E. ⳵t

共10兲

The fact that the partial derivative of S with respect to time is a constant means that we can write S as 共11兲

S = W共r兲 − Et, and the Hamilton–Jacobi equation becomes 1 兩ⵜS兩2 + V − E = 0. 2m

共12兲

兩ⵜS兩 = 兩ⵜW兩 = 冑2m共E − V兲 = p.

共13兲

For a particle in a piecewise-constant potential 共so that the linear momentum remains constant during some finite portion of the particle’s trajectory兲, the right-hand side of Eq. 共13兲 is a constant and S becomes 共14兲

S = p · r − Et.

At this point, astute students who have had an elementary introduction to quantum mechanics might notice that the relation between energy and the time derivative of S is similar to the relation between the energy of a particle and the time derivative of the wave function; the relation between momentum and the gradient of S is similar to the relation between the momentum of a particle and the gradient of the wave function; moreover, Eq. 共14兲 is similar to the phase of a free-particle wave function, except for a factor of 2␲ / h. Having noted these analogies between classical and quantum mechanics, we can determine whether these analogies are flukes or fundamental by investigating light, which is also known to exhibit wave-particle duality and obey a principle similar to Hamilton’s principle. IV. GEOMETRICAL OPTICS, FERMAT’S PRINCIPLE, AND THE EIKONAL EQUATION We know that in geometrical optics light traveling between points A and B follows a path that minimizes the total time elapsed. The elapsed time T, much like the action, can be written as a line integral,

冕 冕 B

T=

679

共9兲

We can also rewrite Eq. 共12兲 as

t

Hdt,

1 ⳵S 兩ⵜS兩2 + V + = 0, 2m ⳵t

where we used the fact that the kinetic energy term in the Hamiltonian is p2 / 2m and p = ⵱S. We are only considering systems for which the energy is conserved and therefore

II. DERIVATION OF THE HAMILTON–JACOBI EQUATION

S=

the Cartesian coordinates兲 are perpendicular to surfaces of constant action, which provides a context for S as a quantity that governs particle motion. The Hamilton–Jacobi equation in the Cartesian coordinates is

B

dt =

A

A

n dl, c

共15兲

where n共x , y , z兲 is the refractive index of an inhomogeneous medium, c is the speed of light, and dl is the differential element of displacement along the path. We write cT = S and dl = 冑x˙2 + y˙ 2 + z˙2dt and convert the line integral in Eq. 共15兲 to the time integral, A. Small and K. S. Lam

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S=

冕

t

n共x,y,z兲冑x˙2 + y˙ 2 + z˙2dt.

共16兲

t0

The problem is to find the path 共x共t兲 , y共t兲 , z共t兲兲 joining A and B that minimizes S. This problem resembles the variational problem in classical mechanics with the Lagrangian function given by L共x,y,z,x˙,y˙ ,z˙兲 = n共x,y,z兲冑x˙2 + y˙ 2 + z˙2 .

共17兲

The canonical momenta px, py, and pz corresponding to this Lagrangian are px =

⳵L nx˙ = 2 2 2, 冑 ˙ ⳵x x˙ + y˙ + z˙

共18a兲

py =

⳵L ny˙ = 2 2 2, ⳵ y˙ 冑x˙ + y˙ + z˙

共18b兲

pz =

⳵L nz˙ = 2 2 2. ⳵ z˙ 冑x˙ + y˙ + z˙

共18c兲

It follows immediately that p2x + p2y + pz2 = n2

共19兲

pxx˙ + py y˙ + pzz˙ = n冑x˙2 + y˙ 2 + z˙2 = L.

共20兲

and

The Hamiltonian H = pxx˙ + py y˙ + pzz˙ − L thus vanishes identically. Hence, we have S=

冕

t

Ldt =

t0

=

冕

冕

t

共piq˙i − H兲dt =

t0

冕

t

piq˙idt

共21a兲

t0

B

共21b兲

pxdx + pydy + pzdz.

A

It follows that 关see Eq. 共5兲兴

⳵S = p x, ⳵x

⳵S = py, ⳵y

⳵S = p z, ⳵z

⳵S =0 ⳵t

共22兲

or, as in our derivation of the Hamilton–Jacobi equation, p = ⵱S共r兲. The eikonal equation in geometrical optics,

冉 冊 冉 冊 冉 冊 ⳵ S共r兲 ⳵x

2

+

⳵ S共r兲 ⳵y

2

+

⳵ S共r兲 ⳵z

2

= 关n共r兲兴2 ,

共23兲

follows directly from Eqs. 共19兲 and 共23兲. Note that Eq. 共23兲 is formally the same as the time-independent Hamilton– Jacobi equation 关Eq. 共12兲兴, in which 2m关E − V共r兲兴 plays the role of 关n共r兲兴2. Because of this similarity, the Hamilton– Jacobi equation is also commonly referred to as the eikonal equation in the mathematical physics literature.9 Given a solution S共x , y , z兲 to the eikonal equation, a ray of light will have a momentum parallel to the gradient of S. The momentum is proportional to the ray’s velocity r˙ 关according to Eq. 共17兲兴 and is also equal to the gradient of S 关according to Eq. 共13兲兴. Because the gradient of S is perpendicular to surfaces of constant S, it follows that the direction of travel is perpendicular to surfaces of constant S. 680

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The eikonal equation is usually mentioned in graduate texts comparing classical mechanics and geometrical optics. Most of those texts do not derive the equation and do not elaborate on the physical context. Most undergraduate optics texts develop the principle of least time in a more concrete form that does not require the eikonal equation, and thus students are left to accept the assertion that the eikonal equation is a fact of geometrical optics.10 We note the following analogies between classical mechanics and wave optics: 共1兲 The eikonal equation is similar in form to the Hamilton– Jacobi equation and involves the gradient of the quantity being minimized 共time in optics and action in mechanics兲. 共2兲 Just as the action S is proportional to the phase of the quantum mechanical wave function, the line integral in Eq. 共15兲 is proportional to the phase shift due to a light wave following a path, up to a factor of 2␲ / ␭. 共3兲 The proportionality factor in the phase is, in both cases, very small except in the limit where wave mechanics is applicable. Classical mechanics is valid only in the limit h → 0 and geometrical optics is valid only in the limit ␭ / L → 0, where L is the characteristic length scale of the phenomena being studied. At this point, it is clear that some of the key features of Hamilton–Jacobi theory are not unique to classical mechanics, but are common in the short-wavelength limit of wave phenomena.

V. THE CLASSICAL ACTION S AND QUANTUM MECHANICS To complete the connection to quantum mechanics, it is appropriate to discuss with students two issues that have been carefully explained elsewhere and which we merely summarize here for completeness. The close similarities between the Hamilton–Jacobi equation and the Schrödinger equation are a direct consequence of the latter.11 To see these similarities, we write the wave function in the form ␺ = ReiS/ប, where R is a function of position and time. By substituting this form for ␺ into the Schrödinger equation and equating the real and imaginary parts of the resulting expression, two equations are obtained. One of those equations expresses conservation of probability. The other equation is the same as the Hamilton–Jacobi equation, with the exception of one term that is proportional to ប 共and is also nonlocal, in keeping with the nature of quantum mechanics兲. In the classical limit, this equation becomes the Hamilton–Jacobi equation.4 Given the correspondence between the classical action and the quantum mechanical phase, it is necessary to explain why the classical path corresponds to stationary action. As explained by Feynman et al.,12 if the phase does not change 共to first order兲 when moving to adjacent paths, there is constructive interference between the classical path and adjacent paths because all of the contributions from different paths are in phase. This interference results in a large probability amplitude for propagation along the classical path. For more discussion of the relation between the Hamilton–Jacobi equation and paths in the classical limit of quantum mechanics, the reader is referred to the work of Landauer.13 A. Small and K. S. Lam

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VI. CONCLUSION We have shown here that the Hamilton–Jacobi equation and the eikonal equation can be derived in a straightforward manner from the form of the action integral without using the full theoretical apparatus of canonical transformations. Although some insights of Hamilton–Jacobi theory require a thorough understanding of canonical transformations, the connection between Hamilton–Jacobi theory and quantum mechanics can be easily established once students have seen a derivation of the Hamilton–Jacobi equation and the connection between action and momentum. ACKNOWLEDGMENTS The authors thank Peter Siegel and John Mallinckrodt for very useful discussions of this problem. a兲

Electronic mail: [email protected] J. R. Taylor, Classical Mechanics 共University Science Books, Sausalito, CA, 2005兲. 2 S. T. Thornton and J. B. Marion, Classical Dynamics of Particles and Systems, 5th ed. 共Brooks/Cole, New York, 2004兲. 3 Historically, the Hamilton–Jacobi and the eikonal equations played a crucial role in Schrödinger’s discovery of the equation that bears his name. See, for example, his 1926 seminal paper “Quantization as a problem of 1

proper values, Part II,” an English translation of which can be found in Collected Papers on Wave Mechanics by E. Schrödinger, Together with His Four Lectures on Wave Mechanics 共Chelsea Publishing Company, New York, 1982兲. The “four lectures” in this volume are much less technical than the original paper and can be read with profit by undergraduate students. 4 H. Goldstein, Classical Mechanics, 2nd ed. 共Addison-Wesley, Reading, MA, 1980兲. See the treatment on pp. 484–492 for the connection between wave mechanics and Hamilton–Jacobi theory. 5 J. V. Jose and E. J. Saletan, Classical Dynamics: A Contemporary Approach 共Cambridge U. P., Cambridge, 1998兲. 6 L. N. Hand and J. D. Finch, Analytical Mechanics 共Cambridge U. P., Cambridge, 1998兲. 7 L. D. Landau and E. M. Lifshitz, Mechanics 共Butterworth-Heinemann, Oxford, 1976兲, pp. 138–139, 147–149. 8 C. C. Yan, “Simplified derivation of the Hamilton-Jacobi equation,” Am. J. Phys. 52, 555–556 共1984兲. 9 R. Courant and D. Hilbert, Methods of Mathematical Physics 共Interscience Publishers, London, 1962兲, Vol. 2. 10 The eikonal equation follows from Maxwell’s equations in the limit of short wavelengths. For a derivation see, for example, M. Born and E. Wolf, Principles of Optics, 7th ed. 共Cambridge U. P., Cambridge, 1999兲. 11 P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part II 共McGraw-Hill, Boston, 1953兲. 12 R. P. Feynman, R. B. Leighton, and M. L. Sands, The Feynman Lectures on Physics 共Addison-Wesley, Reading, MA, 1964兲, Vol. 2, p. 19-9. 13 R. Landauer, “Path concepts in Hamilton-Jacobi theory,” Am. J. Phys. 20, 363–367 共1952兲.

Superb Calculation of the Pressure in a Fluid Bob Panoff, Shodor Foundation Tune: Supercalifragilisticexpialidocious Bernoulli knew he had a rule he used for wings in air For fluid incompressible he’d never have a scare. The density of energy’s the same at every spot A caveat is cavitation in which case it’s not! Oh, Superb calculation of the pressure in a fluid Is simple so that anyone with any sense can do it. We all deserve a force conserved among the objects paired. Just add to pressure rho gee aitch then add half rho vee squared A water tower tower’s o’er a town so water goes Through every pipe, and when you turn the faucet on it flows. The pressure head is now instead a steady stream, you see, The pipe’s diameter determines stream velocity. The sum at every point’s a constant, check it if you care Each term can change within a range for water or for air. The key’s to keep the units straight and don’t have any gap Or else your fluid starts to leak and then you’ll just get Oh... .

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