Second-Order Variational Function Analysis

• Second-order variational functions are quadratic response measures that describe the behavior of functions near stationary points using epi-derivatives.
• They employ techniques such as second subderivatives, generalized Hessians, and coderivatives to extend classical curvature concepts to nonsmooth analysis.
• Applications include stability analysis, Newton-type methods, and geometric mechanics, illustrating their impact in optimization and advanced variational problems.

Searching arXiv for recent and foundational papers on second-order variational analysis, second variation, and second-order variational functions. A second-order variational function is a second-order object that records the quadratic response of a function, functional, or variational system around a reference primal-dual pair or stationary trajectory. In modern nonsmooth variational analysis the term is used explicitly for the second-order epi-derivative $d^2 f(\bar x\mid \bar v)$, defined as the epi-limit of the scaled difference quotients

$$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$

while in the smooth calculus of variations the analogous role is played by the second variation $\delta^2J$, a quadratic form on admissible perturbations (Hang et al., 2017, Hounkonnou et al., 2010). Across current research, the same second-order viewpoint appears through second subderivatives, coderivatives of subgradient mappings, generalized Hessians, quadratic bundles, and second-order directional or shape derivatives.

1. Core definitions and conceptual scope

The foundational nonsmooth definition starts from a proper extended-real-valued function $f\colon \mathbb R^n\to(-\infty,+\infty]$, a reference point $\bar x\in\operatorname{dom} f$, and a subgradient $\bar v\in\partial f(\bar x)$. The second-order epi-derivative $d^2f(\bar x\mid \bar v)$ is the epi-limit of the family $w\mapsto \Delta_t^2 f(\bar x\mid \bar v)(w)$ as $t\downarrow0$; equivalently,

$$d^2f(\bar x\mid \bar v)(w) = \liminf_{\substack{t\downarrow0\ w'\to w}} \Delta_t^2 f(\bar x\mid \bar v)(w').$$

Twice epi-differentiability means that these quotients epi-converge to a proper extended-real-valued limit (Hang et al., 2017).

A parallel construction is the second subderivative

$$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$0

which, for composite problems $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$1, yields

$$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$2

when $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$3 is generalized conic-quadratic, i.e.

$$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$4

for a symmetric matrix $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$5 and a closed convex cone $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$6 (Ouyang et al., 2023).

For first-order generalized differentiation, the second-order object most often used is the second-order subdifferential

$$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$7

When $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$8 is $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$9, this collapses to the classical Hessian action $\delta^2J$0 (Mordukhovich et al., 2015). This suggests that “second-order variational function” is best understood not as a single universal formula, but as a family of equivalent or complementary second-order models whose exact form depends on whether the ambient problem is smooth, nonsmooth, set-valued, or geometric.

A more recent primal reformulation is generalized twice differentiability. A proper lower semicontinuous $\delta^2J$1 is generalized twice differentiable at $\delta^2J$2 for $\delta^2J$3 if it is twice epi-differentiable at $\delta^2J$4 and its second-order subderivative has the generalized quadratic form

$\delta^2J$5

with $\delta^2J$6 symmetric and $\delta^2J$7 a linear subspace (Khanh et al., 3 Jan 2025). This moves second-order information from coderivative language into a primal quadratic representation.

2. Classical second variation in the calculus of variations

For a $\delta^2J$8-th order variational integral

$\delta^2J$9

with $f\colon \mathbb R^n\to(-\infty,+\infty]$0, the second variation at an admissible $f\colon \mathbb R^n\to(-\infty,+\infty]$1 in direction $f\colon \mathbb R^n\to(-\infty,+\infty]$2 is

$f\colon \mathbb R^n\to(-\infty,+\infty]$3

In jet-space language this is a symmetric quadratic form $f\colon \mathbb R^n\to(-\infty,+\infty]$4, where $f\colon \mathbb R^n\to(-\infty,+\infty]$5 is the Hessian of $f\colon \mathbb R^n\to(-\infty,+\infty]$6 with respect to the jet coordinates (Hounkonnou et al., 2010).

If $f\colon \mathbb R^n\to(-\infty,+\infty]$7 is a weak local minimizer, then necessarily $f\colon \mathbb R^n\to(-\infty,+\infty]$8 for all admissible $f\colon \mathbb R^n\to(-\infty,+\infty]$9. From this one obtains the Legendre–Hadamard condition on the principal block: $\bar x\in\operatorname{dom} f$0 A sufficient condition for strict local minimality is a coercive lower bound on the full quadratic form, namely positivity of the jet-space Hessian against all admissible jets, which yields

$\bar x\in\operatorname{dom} f$1

for sufficiently small $\bar x\in\operatorname{dom} f$2 (Hounkonnou et al., 2010).

In the one-dimensional first-order calculus of variations, the second variation can be reduced to

$\bar x\in\operatorname{dom} f$3

or, after eliminating the mixed term,

$\bar x\in\operatorname{dom} f$4

Under $\bar x\in\operatorname{dom} f$5, several classical sufficient conditions are equivalent: existence of a positive solution of the Jacobi equation $\bar x\in\operatorname{dom} f$6, existence of a bounded Riccati solution, coercivity of the second variation, and absence of conjugate points. A particularly practical equivalent test is to solve

$\bar x\in\operatorname{dom} f$7

and check that $\bar x\in\operatorname{dom} f$8 on $\bar x\in\operatorname{dom} f$9 (Hager, 2024).

This classical theory fixes the smooth template for later nonsmooth work: the second-order variational function is the curvature carrier that separates mere stationarity from minimality, coercivity, or instability.

3. Generalized Hessians and explicit nonsmooth formulas

For convex piecewise linear functions, second-order behavior is nontrivial despite piecewise affinity. If $\bar v\in\partial f(\bar x)$0 is CPL, its epigraph is a convex polyhedron, and it admits a max-plus-domain representation with slope vectors $\bar v\in\partial f(\bar x)$1 and domain normals $\bar v\in\partial f(\bar x)$2. At $\bar v\in\partial f(\bar x)$3, the domain of the second-order subdifferential is

$\bar v\in\partial f(\bar x)$4

where $\bar v\in\partial f(\bar x)$5 are the positive-multiplier active index sets. Under the affine-independence qualification, $\bar v\in\partial f(\bar x)$6 is given exactly by a sum of spans and cones generated by active slope differences and active domain normals (Mordukhovich et al., 2015).

In the canonical example $\bar v\in\partial f(\bar x)$7 at $\bar v\in\partial f(\bar x)$8, $\bar v\in\partial f(\bar x)$9, one gets

$d^2f(\bar x\mid \bar v)$0

and

$d^2f(\bar x\mid \bar v)$1

Thus the generalized Hessian detects curvature concentrated on the crease where active slopes switch, even though the function is affine on each side (Mordukhovich et al., 2015).

For CPWL functions in composite optimization, the second-order object also admits a finite-dimensional exact chain rule. If $d^2f(\bar x\mid \bar v)$2 with $d^2f(\bar x\mid \bar v)$3 and $d^2f(\bar x\mid \bar v)$4 $d^2f(\bar x\mid \bar v)$5, the second-order qualification condition

$d^2f(\bar x\mid \bar v)$6

is equivalent to partial nondegeneracy. Under this condition the multiplier is unique and one has the exact coderivative chain rule

$d^2f(\bar x\mid \bar v)$7

This formula is the main bridge from explicit polyhedral second-order geometry to stability theory for composite models (Mordukhovich et al., 2015).

A plausible implication is that the term “function” in “second-order variational function” should often be read structurally rather than literally: in nonsmooth problems the relevant object is frequently set-valued, with domain restrictions encoding active manifolds, faces, or critical cones.

4. Variational convexity, quadratic bundles, and strong regularity

For prox-regular functions, second-order variational analysis has been extended from local curvature tests to full characterizations of variational convexity. A function $d^2f(\bar x\mid \bar v)$8 is variationally $d^2f(\bar x\mid \bar v)$9-convex at $w\mapsto \Delta_t^2 f(\bar x\mid \bar v)(w)$0 for $w\mapsto \Delta_t^2 f(\bar x\mid \bar v)(w)$1 if, locally and below a truncation level $w\mapsto \Delta_t^2 f(\bar x\mid \bar v)(w)$2, its subgradient graph agrees with that of a model $w\mapsto \Delta_t^2 f(\bar x\mid \bar v)(w)$3 such that $w\mapsto \Delta_t^2 f(\bar x\mid \bar v)(w)$4 is convex. Equivalently, the truncated graph of $w\mapsto \Delta_t^2 f(\bar x\mid \bar v)(w)$5 is strongly monotone with modulus $w\mapsto \Delta_t^2 f(\bar x\mid \bar v)(w)$6: $w\mapsto \Delta_t^2 f(\bar x\mid \bar v)(w)$7 Under prox-regularity this is equivalent to the coderivative inequality

$w\mapsto \Delta_t^2 f(\bar x\mid \bar v)(w)$8

and also to corresponding statements in terms of $w\mapsto \Delta_t^2 f(\bar x\mid \bar v)(w)$9-attentive tangent cones and SC-derivatives. The exact variational convexity bound is

$t\downarrow0$0

(Gfrerer, 2024).

Generalized twice differentiability sharpens this by requiring the second-order subderivative itself to be a generalized quadratic form. For prox-bounded, $t\downarrow0$1-level prox-regular functions, this property is equivalent to classical twice differentiability of the Moreau envelope $t\downarrow0$2 at $t\downarrow0$3 for every $t\downarrow0$4, with the exact identity

$t\downarrow0$5

The associated quadratic bundle

$t\downarrow0$6

is nonempty for prox-regular functions (Khanh et al., 3 Jan 2025).

For decomposable nonsmooth composite problems, these second-order objects are tied directly to Newton-type regularity. If

$t\downarrow0$7

the strong second-order sufficient condition is

$t\downarrow0$8

Under structural properties of $t\downarrow0$9, this SSOSC is equivalent to uniform positive definiteness of the generalized Jacobians of the normal map, to CD-regularity and BD-regularity, to strong metric regularity of the subdifferential, normal map, and natural residual, and to uniform quadratic growth or tilt-stability (Ouyang et al., 2023).

In this regime the second-order variational function is not merely diagnostic. It governs semismooth Newton convergence, identifies strong local stability, and determines when subgradient-based stationarity maps admit single-valued Lipschitz inverses.

5. Geometric, directional, and supremal forms

A distinct geometric manifestation appears in the second-order directional derivative along level sets. Let $$d^2f(\bar x\mid \bar v)(w) = \liminf_{\substack{t\downarrow0\ w'\to w}} \Delta_t^2 f(\bar x\mid \bar v)(w').$$0, and on a regular level curve $$d^2f(\bar x\mid \bar v)(w) = \liminf_{\substack{t\downarrow0\ w'\to w}} \Delta_t^2 f(\bar x\mid \bar v)(w').$$1 let $$d^2f(\bar x\mid \bar v)(w) = \liminf_{\substack{t\downarrow0\ w'\to w}} \Delta_t^2 f(\bar x\mid \bar v)(w').$$2 denote the positively oriented unit tangent and $$d^2f(\bar x\mid \bar v)(w) = \liminf_{\substack{t\downarrow0\ w'\to w}} \Delta_t^2 f(\bar x\mid \bar v)(w').$$3 the Hessian quadratic form in the tangent direction. If $$d^2f(\bar x\mid \bar v)(w) = \liminf_{\substack{t\downarrow0\ w'\to w}} \Delta_t^2 f(\bar x\mid \bar v)(w').$$4 is compact, connected, free of critical points, and bounded by simple closed level curves, then

$$d^2f(\bar x\mid \bar v)(w) = \liminf_{\substack{t\downarrow0\ w'\to w}} \Delta_t^2 f(\bar x\mid \bar v)(w').$$5

where $$d^2f(\bar x\mid \bar v)(w) = \liminf_{\substack{t\downarrow0\ w'\to w}} \Delta_t^2 f(\bar x\mid \bar v)(w').$$6 records the enclosure orientation of the level curves. The proof combines a co-area parametrization, the curvature identity

$$d^2f(\bar x\mid \bar v)(w) = \liminf_{\substack{t\downarrow0\ w'\to w}} \Delta_t^2 f(\bar x\mid \bar v)(w').$$7

and the total signed curvature formula $$d^2f(\bar x\mid \bar v)(w) = \liminf_{\substack{t\downarrow0\ w'\to w}} \Delta_t^2 f(\bar x\mid \bar v)(w').$$8 (Ding, 2021).

The same paper derives a companion identity for the normal-direction second derivative $$d^2f(\bar x\mid \bar v)(w) = \liminf_{\substack{t\downarrow0\ w'\to w}} \Delta_t^2 f(\bar x\mid \bar v)(w').$$9: $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$00 If $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$01 is harmonic, this simplifies to

$$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$02

Extensions are given for finitely many critical points, critical levels of measure zero, disconnected slabs, and noncompact ends (Ding, 2021).

A different higher-order variational setting is the second-order $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$03 supremal functional

$$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$04

with $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$05 depending on Hessian and lower-order terms. Under Carathéodory regularity, coercivity-growth in the Hessian, level-convexity in $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$06, and continuity in lower-order variables, global minimizers exist under first-order Dirichlet data; when $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$07, absolute minimizers also exist (Dutton et al., 2024).

For smooth absolute minimizers, the Aronsson-type third-order PDE is

$$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$08

Because this equation is third-order, non-elliptic, and the natural class is only $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$09, the paper introduces generalized D-solutions via diffuse third-order Young measures and proves existence for the Dirichlet problem (Dutton et al., 2024).

These examples show that second-order variational functions need not be restricted to local optimization tests. They also encode curvature transport along level sets, govern supremal energies, and generate nonlinear PDEs whose order exceeds the order of the original variational functional.

6. Computational, geometric, and dynamical applications

In celestial mechanics, second-order variational equations are obtained by differentiating the flow map $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$10 of the Newtonian $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$11-body problem twice with respect to parameters in the initial data. If $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$12 is the first variation and $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$13 the second, then

$$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$14

For gravity this leads to explicit componentwise equations for $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$15 and $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$16, with a linear part driven by the same Jacobian as the first variation and a quadratic source in the first variations. The method is implemented in REBOUND with IAS15, supports analytic initialization for masses and orbital elements, and is used for Newton’s method, RMHMC and Langevin sampling, trajectory optimization, asteroid deflection, and radial-velocity or transit-timing-variation fitting (Rein et al., 2016).

In geometric mechanics on Lie algebroids, second-order constrained variational problems are formulated on the admissible second-order bundle

$$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$17

with a constraint submanifold $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$18 and action

$$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$19

Using the Skinner–Rusk presymplectic formalism on

$$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$20

the dynamics is given by $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$21. Under the regularity condition

$$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$22

the Gotay–Nester algorithm stops at the first step and the restricted form becomes symplectic, producing a unique solution. The framework specializes to second-order Euler–Poincaré and Lagrange–Poincaré equations and is applied to optimal control of mechanical systems (Colombo, 2017).

In shape optimization, second-order shape derivatives admit a variational representation. For

$$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$23

with smooth convex $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$24, the second-order derivative along a deformation field $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$25 exists and has the representation

$$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$26

where $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$27 is expressed as an infimum of a quadratic functional involving the state $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$28, the deformation, and a quadratic form $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$29. For $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$30-torsional rigidity, $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$31, this yields an explicit boundary term plus a weighted quadratic minimization involving

$$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$32

(Bouchitté et al., 2015).

Conformal geometry provides another instance in which the second-order variational function becomes an operator. For

$$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$33

the second variation at a critical metric takes the form

$$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$34

with

$$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$35

For Einstein manifolds with $$\Delta_t^2 f(\bar x\mid \bar v)(w) = \frac{f(\bar x+t w)-f(\bar x)-t\langle \bar v,w\rangle}{\tfrac12 t^2},$$36 and positive scalar curvature, this yields strict local maximality within the conformal class unless the metric is the round sphere up to scaling (Guo et al., 2010).

Taken together, these applications show that second-order variational functions occupy a common structural role across optimization, stability theory, PDE, geometry, and dynamics: they are the carriers of quadratic sensitivity, the objects from which coercivity, curvature, regularity, and higher-order evolution laws are extracted.