3.2 Ray equations

Ray equations in a plasma slowly varying in space and time are defined by the condition (??) = 0 along the ray trajectory. Therefore, at all points along the ray trajectory, the condition

(25)

must be satisfied. Let define τ a dimensionless parameter, which is a measure of distance along the trajectory. Then (??) can be rewritten as

(26)

where Ẋ =dX∕dτ, =dk∕dτ, ṫ=dt∕dτ and = dω∕dτ. The wave vector k is expressed in coordinates that are canonically conjugate to those of the position vector X

(27)

where is Poisson bracket, δ_ij is the Kronecker delta symbol, X_i and k_j are the coordinate of X and k respectively. The frequency ω and the time t being also canonically conjugate, the relations

(28)

result from the Hamiltonian nature of , and Eq. ?? is automatically satisfied. These four equations are therefore the ray equations. From the first and the third equations,

(29)

meaning that the ray is directed along the group velocity v_g and thus indicates the direction of energy flow since v_g∥S [13]. In a constant plasma ∂∕∂t = 0 and ω is a constant of the ray dynamics. The time t is considered as the evolution parameter of the ray and the two remaining equations are

(30)

which are the usual forms found in the literature. However, one may also keep the simple form

(31)

if the determination of the group velocity from ∂∕∂ω is not required. This reduces the number of derivative to calculate.

We define Y the 8-dimension vector that represents the phase-space coordinates

(32)

The ray equations (??-??) require to evaluate the derivatives of the dispersion relation (??) with respect to the coordinates of Y, explicitly

(33)

Since X^s = X^s,

In this equation the derivatives of the susceptibility tensor depend upon the plasma dispersion model (i.e. cold, kinetic, etc) while the derivatives with respect to Y are function of the equilibrium and the coordinate system.