3.5 Ray methods
4.1.2 Poincar´e section
Ray dynamics 45
−15 −10 −5 0 5 10 15
4 3.5 3 2.5 2 1.5 1 0.5 0
ray angle [◦]
depth[km]
−10 −5 0 5 10 15 ray angle [◦]
Figure 4.2: Portraits of Poincar´e sections with Munk (left) and double duct profiles (right). Initial conditions – highlighted by markers – were: z0 = 1 km,−ϕ0 = 1, . . . ,12◦ – Munk, z0 = 1.3 km, −ϕ0 = 1, . . . ,15◦, z0 = 0.35 km, ϕ0 =−0.5,0◦, z0 = 0.325 km, ϕ0 =−7,−7.5,−8◦ and z0 = 0.2 km, ϕ0 = 1,2◦ – double duct
τn =τn−1−T. With this, due to the periodic property of f, a series of identical initial value problems can be generated:
˙
xn=f(xn, τn), x0n=xn(x0n,0), (4.17) certainly, with identical form of the sought-for functionsxn(x0n, τn) too. Now we utilize the recursive formulaτn=τn−1−T and write that
xn−1(x0n−1, τn+T) = xn(x0n, τn). (4.18) Ifτn is set to be zero here, the right hand side is found to be the general form of the initial condition under (4.17). Using this connection, and leaving the subscript of the sought-for function being invariant, the following map yields:
x0n =x(x0n−1, T). (4.19)
The above result is the Poincar´e return map itself. As intuition suggests, it is obtained by evaluating the sought-for function with respect to the independent variable at time T, the period of perturbation. The associated picture is that two stroboscopic points, with the measure ofT apart from one another along the time coordinate, are linked by a trajectory segment. In terms of the ray depth and ray tangent coordinates we write that
{zn+1, pn+1}=P(zn, pn), P ={Pz,Pp}. (4.20) Next, it is shown that the Poincar´e return map of a one-degree-of-freedom Hamil-tonian system is an are preserving mapping – a result attributed to Liouville. First the deformation gradient of the flowX, and the Jacobi determinant J0 are introduced as:
X = ∂x(x0, t)
∂x0 , (4.21)
and with this,
J0 = det(X). (4.22)
Here the subscript of J0 refers to the initial configuration at t = 0 with which the Jacobian is associated. This quantity is found to be the factor by which an infinitesimal volume of phase space is modified over time. It can be shown that the evolution of the Jacobian is governed by the following differential equation (B´eda n.d.):
J˙0 =J0div(f(x, t)). (4.23)
Ray dynamics 47 The divergence of a one-degree-of-freedom Hamiltonian flow can easily be shown to be zero due to the canonical form of the equations:
div(f) = ∂
∂q µ∂H
∂p
¶
− ∂
∂p µ∂H
∂q
¶
= 0, (4.24)
and hence the Jacobian is constantly unity. (Clearly, initially the Jacobian has to be unity since it gives the ratio of phase space volumes which are identical, as will later be shown in terms of the identity matrix.) Now, as the Poincar´e return map yields by taking the solutionx(x0, t) atT with respect to time, its Jacobian derives similarly from the Jacobian of the flow; i.e. the mapping inherits the area preserving property of the flow.
Extended phase space. Now we will look at the Poincar´e sections from a different perspective, in terms of sectioned trajectories in a bounded phase space. Henceforth, and continuing in the following subsection, we closely follow Lichtenberg & Lieberman (1982). To start we note that the Hamiltonian of the autonomous system is constant.
In order to show this, the Hamiltonian H(q, p, t) is differentiated as follows:
H˙ = ∂H
∂q q˙+∂H
∂pp˙+∂H
∂t . (4.25)
Using the the canonical form of the ray equations (4.3), the first two terms on the right cancel each other, and the equation reduces to ˙H = ∂H/∂t. If H does not depend explicitly on t, i.e. the system is autonomous, then ˙H = 0, and H is a constant depending on the initial conditions. In this caseH is said to be a constant of motion.
Suppose now that the one-degree-of-freedom system is non-autonomous. Then we introduce the following new variables of a two-degree-of-freedom system:
¯
q1 =q1, p¯1 =p1, q¯2 =t, p¯2 =−H. (4.26) If we parameterise the flow by time, so that the new independent variable τ = t, we find that the two-degree-of-freedom system retains the Hamiltonian structure by means of the canonical form of the equations:
˙¯
qi = ∂H¯
∂p¯i, p˙¯i =−∂H¯
∂q¯i, i= 1,2, (4.27) when the new Hamiltonian is implied as
H(¯¯ q,p) =¯ H(¯q1,p¯1,q¯2) + ¯p2. (4.28)
The equations for ¯q2 and ¯p2 respectively reproduce (4.25) and the identity of time and the new independent variable. Due to the definition of the new variables (4.26), ¯H is clearly constantly zero. This is an example of how a non-autonomous system can be transformed into an autonomous one of one greater degree of freedom, which conclusion can be extended to any N degree-of-freedom non-autonomous system. Also note that given a Hamiltonian periodic in range, ¯p2 can be defined by the time modulo the period T, thereby setting bounds on ¯p2 and so the phase space itself. For this reason it is commonly termed as a cylindrical phase space.
The above transformation suggests that the trajectories of a perturbed ray system live in a 4D phase space. Although, since the Hamiltonian ¯H(¯q,p) is constantly zero,¯ one of the new variables can be expressed as a function of the rest, say,
¯
p2 = ¯p2(¯q1,q¯2,p¯1), (4.29) which particular functional form stands for the original Hamiltonian H(q, p, t). This means that the trajectories, are, in fact, confined to a 3D volume in the 4D phase space. Furthermore, if the 3D volume is bounded we can conveniently set a surface of intersection at ¯q2 = 0 which the trajectory pierces through repeatedly, and plot the points of intersection onto the sectioning surface, which is now the original phase plane spanned byqandp. Note that for a general surface of intersection, only intersections of the same sense are counted, that is, for example, when the trajectory pierces through the surface from left to right. If there is a further constant of motion exists:
I(¯q,p) =¯ constant, (4.30)
the relation of the variables that it implies can be combined by (4.29) and ¯q2 = 0, so that it would yield the relation:
p=p(q), (4.31)
which suggests that the section of the trajectory is a curve on the sectioning surface.
In association with such, so-called invariant curves, it is common to speak about tori in the cylindrical phase space that trajectories wind around. Conversely we can say that if the section of the trajectory fills an area, there is no other constant of the motion in addition to the Hamiltonian ¯H. This provides a graphical means to determine the existence of constants of the motion, and hence – stability.
To make the link, it is noted that the points of intersection on the sectioning surface
¯
q2 = 0 are the stroboscopic points, and their plot is understood as the trajectory of the
Ray dynamics 49 Poincar´e return map. The above analysis concludes that the closed loops in Figure 4.2 refer to regular motion. So that we could discuss features in this figure in more details, we have to present some more theory.