Parts V, VI and VII of this study are dedicated to the computation of paraxial rays and dynamic characteristics along the stationary rays obtained numerically in Part III. In this part (Part V), we formulate the linear, second-order, Jacobi dynamic ray tracing equation. In Part VI, we compare the Lagrangian and Hamiltonian approaches to the dynamic ray tracing in heterogeneous isotropic and anisotropic media; we demonstrate that the two approaches are compatible and derive the relationships between the Lagrangian's and Hamiltonian's Hessian matrices. In Part VII, we apply a similar finite-element solver, as used for the kinematic ray tracing, to compute the dynamic characteristics between the source and any point along the ray. The dynamic characteristics in our study include the relative geometric spreading and the phase correction factor due to caustics (i.e., the amplitude and the phase of the Green's function of waves propagating in 3D heterogeneous anisotropic media). The basic solution of the Jacobi equation is a shift vector of a paraxial ray in the plane normal to the ray direction at each point along the central ray. A general paraxial ray is defined by a linear combination of up to four basic vector solutions, each corresponds to specific initial conditions related to the ray coordinates at the source. We define the four basic solutions with two pairs of initial condition sets: point-source and plane-wave. For the proposed point-source ray coordinates and initial conditions, we derive the ray Jacobian and relate it to the relative geometric spreading for general anisotropy. Finally, we introduce a new dynamic parameter, referred as the normalized geometrical spreading, considered a measure of complexity of the propagated wave/ray phenomena, and hence, we propose using a criterion based on this parameter as a qualifying factor associated with the given ray solution.

中文翻译：

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3D 异质各向异性介质中的特征射线：第一部分 - 运动学

本研究的第 V、VI 和 VII 部分致力于计算近轴射线和沿第 III 部分数值获得的静止射线的动态特性。在这部分（第五部分）中，我们制定了线性二阶 Jacobi 动态光线追踪方程。在第六部分，我们比较了拉格朗日方法和哈密顿方法在非均质各向同性和各向异性介质中动态光线追踪；我们证明这两种方法是兼容的，并推导出拉格朗日矩阵和哈密顿 Hessian 矩阵之间的关系。在第 VII 部分，我们应用了一个类似的有限元求解器，用于运动学射线追踪，来计算源和沿射线的任何点之间的动态特性。我们研究中的动态特性包括相对几何扩展和由于焦散引起的相位校正因子（即波在 3D 非均质各向异性介质中传播的格林函数的幅度和相位）。雅可比方程的基本解是在垂直于射线方向的平面内沿中心射线的每一点处的近轴射线的位移矢量。一般的近轴射线由最多四个基本矢量解的线性组合定义，每个解对应于与光源处的射线坐标相关的特定初始条件。我们用两对初始条件集定义四个基本解：点源和平面波。对于建议的点源射线坐标和初始条件，我们推导出光线雅可比矩阵并将其与一般各向异性的相对几何扩展相关联。最后，我们引入了一个新的动态参数，称为归一化几何扩展，考虑了传播波/射线现象的复杂性，因此，我们建议使用基于此参数的标准作为与给定射线相关的限定因素解决方案。