Second-Jet Formulation in Dynamics

• Second-Jet Formulation is a framework that incorporates second-order derivative information to extend modeling in dynamics, turbulence, PDE control, and robotics.
• It enables improved statistical closures and accurate high-order numerical solvers by transforming systems into jet-space for enhanced energy conservation and solution fidelity.
• Applications span fields from turbulence modeling via CE2, boundary control in PDEs, and semi-Lagrangian solvers to CR geometry and advanced robotic control.

The second-jet formulation refers to the inclusion and systematic exploitation of second-order derivative information in the analysis, simulation, or control of dynamical systems, partial differential equations, geometric mappings, and statistical closures for turbulent flows. The terminology originates in jet theory, where the "k-jet" encodes all derivatives of a map up to order k, but its contemporary applications span cumulant expansions in turbulence, PDE control theory, high-order numerical solvers, and CR geometry. This article surveys key second-jet constructs, methodologies, and their roles across fields, focusing on representative arXiv literature.

1. Second-Jet in Turbulent Jet Statistics and Cumulant Expansions

In direct statistical simulation of barotropic jets on a β-plane, the second-jet formulation appears as truncation at the second cumulant in the hierarchy of statistical closures. The vorticity field $\zeta(x,y,t)$ is decomposed into its zonal mean and fluctuation: $$\zeta(x,y,t) = \overline{\zeta}(y,t) + \zeta'(x,y,t)$$ The first cumulant $c_\zeta(y, t) = \overline{\zeta}(y, t)$ captures mean vorticity; the second cumulant encapsulates two-point spatial correlations: $$c_{\zeta\zeta}(y, y', \xi, t) = \langle \zeta'(x, y, t) \zeta'(x+\xi, y', t) \rangle_x$$ Truncation at the second order — the "CE2" closure — yields a set of evolution equations for $c_\zeta$ and $c_{\zeta\zeta}$, neglecting all third and higher cumulants and thus all nontrivial eddy-eddy interactions (Tobias et al., 2012).

The second-jet closure preserves conservation properties (energy/enstrophy) and admits symmetry reductions via Fourier modes, yet exhibits limitations for out-of-equilibrium flows with small zonostrophy parameter $R_\beta\sim1$ due to the suppressed eddy–eddy nonlinearity. Inclusion of the third cumulant in CE3 is required to restore eddy–eddy–eddy scattering mechanisms, improving fidelity in the meandering jet regime.

2. Second-Jet Formulation in Boundary Control of Second-Order PDEs

For abstract second-order boundary control systems, the second-jet formulation re-expresses dynamics in terms of both state and its first spatial derivative, facilitating realizations in jet-space. Given

$\ddot{x}(t) + D \dot{x}(t) + Sx(t) = 0$

the standard first-order in time formulation is

$$\dot{z}(t) = \mathcal{A} z(t), \quad \mathcal{A} = \begin{pmatrix} 0 & I \ -S & -D \end{pmatrix}$$

Factorization $S = T^* T$ enables a state transformation to jet-space,

$$w(t) = \begin{pmatrix} T x(t) \ \dot{x}(t) \end{pmatrix}$$

with the new generator

$$\widetilde{\mathcal{A}} = \begin{pmatrix} 0 & T \ -T^* & -D \end{pmatrix}$$

Concrete examples include

• The wave equation: $T = \nabla$, $T^* = -\mathrm{div}$, yielding the strain–momentum form $\left[0\;\;\nabla;\;-\mathrm{div}\;-D\right]$;
• Maxwell's equations: $T = c\,\mathrm{curl}$, $T^* = -c\,\mathrm{curl}$ (Preuster et al., 11 Oct 2025).

This jet-based representation streamlines characterization of well-posed boundary control systems, boundary triplets, energy norm, and passivity, and is foundational for impedance and scattering passive realizations.

3. Second-Jet in High-Order Semi-Lagrangian Solvers

In numerical analysis, specifically for eikonal and high-frequency Helmholtz equations, second-jet formulations involve explicit marching of second partial derivatives (the Hessian) alongside the solution and its gradient. On each mesh cell,

$$\text{Second-jet at } x: \{ u(x), \nabla u(x), D^2 u(x) \}$$

enables construction of bicubic (2D) or tricubic (3D) Hermite interpolants that are locally C¹ and provide curvature data essential for transport equations and WKB amplitude laws (Potter et al., 2020). Key update steps include:

• Hermite interpolation using first and second derivatives at cell corners,
• Extraction of $\partial_{xy}u(x)$ via central differences and extrapolation,
• Interpolation-based value, gradient, and Hessian update for newly accepted nodes.

Second-jet methods maintain compactness (local stencils) and second-order accuracy for the Hessian on regular grids or affine speed profiles, making them suitable for paraxial ray-tracing and geometric optics transport.

4. Second-Jet Determination in CR Geometry

In several complex variables and CR geometry, the second-jet determines the local uniqueness of CR diffeomorphisms. For a smooth CR map $\phi: M \to M'$, the $2$-jet $j^2_p\phi$ records all partial derivatives up to order two at $p$ (or up to weighted order two in CR weights). The main theorem states that for generic Levi nondegenerate submanifolds $M, M' \subset \mathbb{C}^n$, every germ of a $\mathcal{C}^5$ CR diffeomorphism is uniquely determined by its standard second jet if the Lie algebra $\mathfrak{g}_3 = 0$ (Tumanov, 2022).

Weighted jet formalism refines the notion further via weight assignments reflecting the geometry of the model quadric. Strongly pseudoconvex, D-nondegenerate, or low-codimension quadrics guarantee two-jet uniqueness, while higher codimension may break determination.

5. Second-Jet in Collision Problems for Axially Symmetric Jets

In hydrodynamic jet collision theory, the "second-jet" pertains to the mathematical formulation for two impinging incompressible, inviscid, axially symmetric fluid jets. The problem is formulated by considering fluid variables $(U_2, V_2, P_2)$ for the second jet, within a semi-infinite nozzle geometry. Boundary conditions enforce mass-flux, slip, and pressure continuity on interfaces, while the stream-function reduction leads to a linear elliptic PDE for the scaled stream-function $\psi$,

$\Delta \psi - \frac{1}{r} \psi_r = 0$

with piecewise boundary and matching conditions on free-stream and contact-discontinuity surfaces (Du et al., 2021). The existence theorem confirms axisymmetric solutions with analytic free-boundaries and prescribed asymptotic conical behavior.

6. Second-Order Jet Modeling in Robotics and Predictive Control

In jet-powered humanoid robotics, second-jet modeling refers to the use of second-order nonlinear actuator dynamics for jet engines in the predictive control framework. Each jet is modeled by states $(T_i, \dot{T}_i)$ and dynamic input $v_i$, with the continuous-time system

$$\dot{T}_i = x_{2,i}, \quad \ddot{T}_i = h(T_i, \dot{T}_i) + g(T_i, \dot{T}_i) v_i$$

capturing spool inertia, aerodynamic drag, and nozzle nonlinearity (Gorbani et al., 22 May 2025). The second-order model is embedded directly into a multi-rate MPC alongside centroidal momentum dynamics, with synchronization of high-frequency joint control and slow-acting jet throttles, leading to feasible real-time control of flight maneuvers.

7. Significance and Inter-Disciplinary Implications

Second-jet formulations systematically extend control, simulation, and analysis beyond first-order frameworks by incorporating curvature, covariance, or second-derivative information:

• In turbulence, CE2 and higher cumulants in second-jet closures underpin the fidelity of statistical predictions for jets, particularly under strong forcing (Tobias et al., 2012).
• In PDE boundary control, jet-state representations yield Hamiltonian forms, passivity criteria, and transparent boundary trace constructions (Preuster et al., 11 Oct 2025).
• In high-order numerics for eikonal-type problems, second-jet marching achieves superior accuracy for field gradients and curvatures critical to transport and wave amplitude computation (Potter et al., 2020).
• In CR geometry, second-jet uniqueness criteria control local rigidity and classification of mappings between smooth submanifolds (Tumanov, 2022).
• In hydrodynamic collisions and robotic propulsion, second-jet constructs encode essential dynamics (discontinuities, actuator inertia, nonlinear coupling) for global existence and control (Du et al., 2021, Gorbani et al., 22 May 2025).

The second-jet paradigm thus serves as a unifying technical concept, enabling advanced modeling, analysis, and computation in domains ranging from fluid mechanics and geometric analysis to robotics and applied mathematics.