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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 kspherical− 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. Then dS = dt. On the pq˙ , we obtain the Hamilton equations other hand L − L dq ∂H dp ∂H ∂S ∂S = ; = (4) dS = dt = dq + dt dt ∂p dt − ∂q L ∂q ∂t which allows us to move to the phase space and have a Using our previous result (7), we have more geometrical view of the trajectories. One conse- quence of the above equation is the variation of H as a ∂S ∂S function of time along a trajectory: dt = pdq + dt = pq˙ + dt L ∂t ∂t ∂H ∂H ∂H ∂H dH = dp + dq + dt = dt (5) ∂p ∂q ∂t ∂t and therefore ∂S Therefore, if the Hamiltonian does not depend explicitly = pq˙ = H (8) on time, the Hamiltonian is conserved along a trajectory: ∂t L− − H = E. In the above two formulation of analytical mechanics, Relation (7,8) are very general results of variational cal- the action S() itself plays little explicit role; what is im- culus with varying end points and are not restricted to portant is the differential equations (2) or (4) whose so- mechanics. The contact angle of a liquid droplet on a lution determines the trajectory. However, Let us have a solid surface is obtained for example by these compu- closer look at the action itself. By action S here we mean tations. Note also that even though we derived these the integral expression (1) when the particle moves along equations in one dimension of space, they are trivially the optimal path. Even though the absolute value of S generalized to any dimension. can be hard to compute analytically, we can compute its variation if we vary the end points (figure 4). We will IV. GENERAL WAVE FRONTS. keep here the initial point fixed and vary the final end point either by dt or dq. We begin by keeping the final time fixed at t1 but move In geometrical optics, we had used the traveling time to the final position by dq (figure 4). The trajectory q(t) define the wave front. But the traveling time is just one will vary by δq(t) where δq(t0)=0 and δq(t1)= dq. The example of action and variational principles. In analogy variation in S is with optics, let us define the function Sq0,t0 (q,t) as the action of a particle that arrives at (q,t) after leaving t1 ∂ ∂ δS = Lδq˙ + Lδq dt (6) (q0,t0), following its optimal path. By this function, we ˆ ∂q˙ ∂q q t0 can associate to each point ( ,t) a value in space-time. Then, S(q,t)= C defines an n 1 dimensional surface , However, the trajectories obey the Euler-Lagrange equa- t i.e. the collection of points q that− have the same valueCC tion (2) and we must have of action at time t. Figure 1 that illustrated wave front ∂ d ∂ in optics illustrates similarly the general wavefronts of L = L ∂q dt ∂q˙ action. Consider for example a classical free particle, whose On the other hand, δq˙ = d(δq)/dt. Using these relations, trajectories are straight lines with constant speed v = we can rewrite equation (6) as q q0 /(t t0). The action is therefore k − k − t1 ∂ d (δq) d ∂ δS = L + L δq dt m 2 m 2 ˆ ∂q˙ dt dt ∂q˙ S(q,t)= v (t t0)= q q0 /(t t0) t0 2 − 2 k − k − 4 and the curves are spheres of radius proportional to Ct 2(t t0)/m. If the initial point is far away from the − region of interest ( t t0 , q q0), we can develop the pabove expression and| | ≪ write | | it,≪ to the first order in q,t : m 2 S(q,t) q0 2q0.q ( 1 t/t0) ≈ 2t0 − − − = S0 + p.q Et (9) − where we have defined the constants p = mq0/t0 and 2 2 E = (1/2)mq0/t0. In this case, the action is a plane wave. We have defined the wave front as the collection of Figure 5. From known wave fronts Ct(in red) to trajectories points q at time t for which S(q,t)= const. To compute : at each point, the normal to the wave front p = ∂S/∂q the wavefronts however, we have relied on the knowl- (in blue) can be computed ; knowing p, we can compute the edge of trajectories. To go further, we need to derive an tangent to the trajectory q˙ ( in green ) and find a trajectory independent equation from which S() can be computed following a given line of tangents. The procedure is trivially directly, without any a priori knowledge of trajectories. generalized to higher dimensional space where q collects the For this purpose, we just have to recall from the last sec- coordinates of many particles. tion (7,8) that we can compute the variation of S as a function of the variation of its end points:
∂S ∂S = p ; = H (10) ∂q ∂t − where ∂S/∂q = (∂q1 S, ∂q2 S, ...). Note that this a gener- alization of the free particle case where (according to 9), dS = pdq Edt. Now, we know that H = H(q, p,t), therefore combining− the above two expressions, we have
∂S ∂S Figure 6. An illustration of the wave front in a two dimen- + H q, ,t =0 (11) q ∂t ∂q sional space where the function W ( ) is represented as a sur- face in three dimension. The wave front CS is the contour plot which is a first order PDE and called the Hamilton- of the function W (q). At any given point q,the momentum Jacobi equation (HJE). If we can solve this equation and is given by p = ∇W find the wave fronts, then we can deduce the trajectories from the wavefronts. The procedure is similar to what we did in geometrical optics : At each time t, we know the where the function W () (often called Hamilton principal wave front S, and therefore, we can compute the momen- function) obeys the relation tum at points q: p(q) = ∂S/∂q (figure 5). This vector ∂W is related to the tangent to a trajectory q˙ through the H q, = E ∂q relation ∂ Once W () is solved for, we can find the wave fronts by p = L ∂q˙ slicing the function W () at different “heights” : at a given time t, we collects all points q such that W (q) = Et + By resolving the above relation, we can compute q˙ at const. into the wave front (figure 6). each point of space at each time : Ct q˙ = f(q,t) (12) V. EXAMPLES. If we knew the wave fronts, the second order differen- tial equations of Euler-Lagrange (equation 2) are trans- A. One particle. formed into ordinary first order differential equations (12) as above. For the simplest mechanical systems with one particle and a potential V (q,t), p and ˙q are co-linear and Consider one classical free particle with the Lagrangian q2 p q p2 the construction is really similar to optics. = (1/2)m ˙ , = m ˙ and H = /2m where we L u2 u u We can further simplify the HJE (eq. 11) if the func- use the square of a vector as a shorthand: = . . tion H does not contain t explicitly. In this case, we can Therefore, the HJE is simply separate the function S into 2 ∂S 1 ∂S + =0 (14) S(q,t)= W (q) Et (13) ∂t 2m ∂q − 5
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