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The Hamilton-Jacobi Equation: an Intuitive Approach

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The Hamilton-Jacobi Equation: an Intuitive Approach The Hamilton-Jacobi Equation : an intuitive approach. Bahram Houchmandzadeh. CNRS, LIPHY, F-38000 Grenoble, France Univ. Grenoble Alpes, LIPHY, F-38000 Grenoble, France The Hamilton-Jacobi equation (HJE) is one of the most elegant approach to Lagrangian systems such as geometrical optics and classical mechanics, establishing the duality between trajectories and waves and paving the way naturally for the quantum mechanics. Usually, this formalism is taught at the end of a course on analytical mechanics through its technical aspects and its relation to canonical transformations. I propose that the teaching of this subject be centered on this duality along the lines proposed here, and the canonical transformations be taught only after some familiarity with the HJE has been gained by the students. I. INTRODUCTION. anything to the engineer and very little to the physicist”. Indeed, many examples of HJE treated in the above men- tioned textbooks of analytical mechanics can be as easily There are three different formalization of classical me- treated by the Lagrangian and Hamiltonian approach. chanics : the Lagrangian, the Hamiltonian and the Hamilton-Jacobi formalism. Usually, textbooks on me- chanics (see for example [1–5] ) begin with the La- II. GEOMETRICAL OPTICS. grangian formalism and the variational principle, where students discover the beauty of post-Newtonian mechan- Eighteen century physics saw a raging debate between ics. Historically, this formalism was developed in analogy the particle theory and wave theory of light[10]. In the with optics and the principle of Fermat[1]. Then, after a first description, light is made of particles whose trajecto- Legendre transform, the Hamiltonian approach is intro- ries can be followed and are called the “ray paths”. In the duced where students discover the beauty of the phase second description, light is made of waves, and the “wave space and the geometry herein. The mathematics be- front” can be followed exactly as we follow waves on the hind these two methods is fairly standard and more or surface of a liquid or sounds. This second approach was less easily digested by students. Finally, students come developed first by Huygens around 1680 AD[7]. In the to the Hamilton-Jacobi equation (HJE). The HJE is usu- limit of geometrical optics, when the wave length can be ally introduced after a heavy passage through canonical considered small, these two approaches are equivalent : transformations to uncover a first-order non-linear partial knowing the wave fronts, one can deduce the ray paths differential equation that does not seem any more useful and vice versa. We will detail this derivation below, but to students at first glance than the former approaches. let us first define more precisely what a wave front is in The aim of this short note is to make an intuitive ap- optics. proach to the HJE by reversing how it is generally taught. Consider light emitted from a point r0 at time t0. The The beauty of the HJ approach is to uncover the du- boundary t,t0 of the domain that the light has covered ality between trajectories and wavefronts. This duality at time t isC called the “wave front” (figure 1) at time t. If was known in optics[6] where light could be either in- the propagation medium is homogeneous, the wave front vestigated by rays and geometric optics (Fermat’s prin- is a sphere given by the equation ciple) or by wavefront (Huygens principle)[7], much be- c fore interference and the electromagnetic nature of light r r0 = (t t0) was discovered. Hamilton showed that this duality can k − k n − be extended to any system described by a Lagrangian where c is the speed of light and n the index of the prop- formalism, including and foremost, mechanics. I believe arXiv:1910.09414v1 [physics.class-ph] 21 Oct 2019 agating medium. We can rewrite this equation as that this duality and its various extensions, specifically to quantum mechanics,are what should be taught first S(r,t)= (c/n)t0 and foremost to students , studied in depth. Only when − the students are familiarized with these concepts, one Where the function S(r,t) = r r0 (c/n)t. The k − k − should introduce the canonical transformations and the relation S(r,t)= (c/n)t0 defines the collection of points − technical aspects that make this approach, in the words that the light (emitted at r0,t0) has reached at time t. of Arnold[8], “[...] the most powerful method known We don’t need to suppose that the light is emitted by for the exact integration [of Hamilton equations]”. At a single point, we can as well describe the wave front of the undergraduate level, specifically to physics students, the light emitted by a line or a surface (or any at most these technical aspects seem less relevant : Arnold[8] n 1 dimensional object). In fact, Huygens discovered quotes Felix Klein, who had great respect for the work that− the wave front at time t can be described by the of Hamilton[9], about HJ method “that does not bring light emitted by the wave front at time t t . This is − α 2 Figure 1. Wave fronts C (in red) of light emitted at point r0 at Figure 3. The trajectory chosen by an object (solid line) op- time t0. Blue lines are the rays path. The Huygens principle timizes the action compared to all other possible trajectories states that the wave front at time t can be seen as the wave (dashed lines) (1) front of light emitted at time t − tα by the wave front at this time (red dashed lines). jects linked together through an orthogonality. Even if the medium is not isotropic, we can still com- pute the wave front from the rays, and vice versa. All we need is a relation between the tangent to the ray path (let’s call it q˙ ) at a point and the normal to the wave front (call it p) at the same point. We will come to this subject in more general detail in the next sections. III. BASIC NOTIONS OF ANALYTICAL MECHANICS. Figure 2. In geometrical optics in isotropic media, trajectories Very soon after the publication of Principia by Newton Pt of the light rays and wave fronts are orthogonal. Therefore, (1684), Bernoulli challenged (1696) the scientific commu- trajectories can be recovered from the wave front: from the nity to find the fastest path that, under gravity, brings a C point Pt on the wave front t, draw the orthogonal to the mass from point A to point B. The analogy with optics wave front and recover the point Pt+dt at which it intercepts and the Fermat’s principle was not lost on the mathe- the wave front Ct+dt. Proceeds by recurrence. maticians who responded to the challenge[11]. This anal- ogy was then fully developed in subsequent years [12] and took its definitive form under the name of Euler-Lagrange called the Huygens principle. Finally, note that if r0 r, ≫ equation. r r0 r0 (r0/r0).r and we can approximate the The foundation of analytical mechanics is based on a sphericalk − k ≈wave− by a plane one of the form S(r,t)= u.r − variational principles: Given a Lagrangian (q,q,t ˙ ), an (c/n)t where u = (r0/r0) is the direction of the plane object (be it a particle or a ray of light) Lchooses the wave propagation.− trajectory q(t) that makes the action If the medium is not homogeneous (n = n(r) ), the wave fronts are not spherical any more. The principle of t1,q1 Fermat states that the path taken by a ray to go from S = (q,q,t ˙ )dt (1) ˆ L a point A to a point B is the one that minimizes the t0,q0 traveling time : stationary (figure 3). The action depends on the end points (t0, q0) and (t1, q1) and the trajectory q(t) must 1 B T = nds obey the Euler-Lagrange equation c ˆ A d ∂ ∂ L L =0 (2) where ds is the element of arc length along a path. In dt ∂q˙ − ∂q order to compute a wave front now, one has to compute For a classical particle, the Lagrangian is the difference the ray paths and collect points along the path that have between the kinetic and the potential energy = T been reached at a given time t. If the medium is isotropic V , while for geometrical optics, the LagrangianL is the− (i.e. not like a crystal with particular directions of prop- traveling time. agation), it can be shown that ray paths and wave fronts We can reformulate equation (2) by making a Legendre are orthogonal (see below). In this case, deducing the transform. Defining the momentum wave fronts from the ray paths is simple. On the other hand, if we knew the wave fronts, we could compute the ∂ p = L (3) ray paths (figure 2). Paths and wave fronts are dual ob- ∂q˙ 3 t1 d ∂ = Lδq dt ˆ dt ∂q˙ t0 ∂ t1 ∂ = Lδq = L dq ∂q˙ ∂q˙ t0 t1 As we have kept the final time fixed, δS = (∂S/∂q) dq and therefore ∂S ∂ = L = p(t1) (7) ∂q ∂q˙ t1 If we vary the end point q1, the relative variation in S is Figure 4. Varying the end points of a movement. the momentum p at the end point. To compute the variation of S as a function of the end point’s time, consider letting the original trajectory to expressing q˙ as a function of p and defining H(p,q,t) = continue along its optimal path.
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