--- title: 'Lepage Test: Dual Perspectives' url: https://www.emergentmind.com/topics/lepage-test type: topic --- # Lepage Test: Dual Perspectives Searching arXiv for recent papers on “Lepage test” and closely related “Lepage equivalents” to ground the article in published work. tool call: arxiv_search tool call: arxiv_search {"query":"all:\"Lepage test\" OR abs:\"Lepage test\" OR ti:\"Lepage equivalents\"","max_results":10,"sort_by":"relevance"} In current technical usage, the term **Lepage test** refers to two unrelated constructions. In the geometric calculus of variations, it denotes a criterion formulated through **Lepage forms** that detects whether a system of differential equations is variational, or whether a Lagrangian is null. In nonparametric statistics, it denotes a **two-sample location-scale test** obtained by combining a location component and a scale component. The two usages share a name but arise in different mathematical domains [1909.05115; 2509.19126]. ## 1. Distinct technical meanings The current literature attaches the label *Lepage test* to several criteria built around different objects: source forms for ordinary differential equations, Lepage equivalents of Lagrangians on jet bundles, and combined rank statistics for two-sample inference. The following summary isolates the main settings. | Usage | Setting | Criterion | |---|---|---| | Variational ODEs | Source form $\varepsilon$ on $\mathbb{R}\times T^2M$ | $\varepsilon$ is locally variational iff the associated Lepage $2$-form is closed | | Field-theoretic variational calculus | Lagrangian $\lambda$ on jet bundles | A suitably constructed Lepage equivalent has a closure property equivalent to vanishing Euler–Lagrange expressions | | Nonparametric statistics | Two independent samples | A combined location-scale statistic is asymptotically $\chi^2_2$ under the null hypothesis | This terminological split is important because the geometric and statistical tests are conceptually unrelated. In the geometric literature, Krupková, Urban–Volná, Rossi, Saunders, and Voicu–Garoiu–Vasian study closure and exactness properties of Lepage forms and Lepage equivalents [1909.05115; 2309.01594; 2102.12955]. In statistics, Fligner–Policello, Fong–Huang, and the recent weak-null modifications concern rank-based inference for simultaneous location and scale changes [2509.19126]. ## 2. The geometric Lepage test for second-order ordinary differential equations For second-order autonomous ODEs on a smooth manifold $M$, one works with the product fibration $$ Y=\mathbb{R}\times M\to \mathbb{R}, $$ with $J^1Y\simeq \mathbb{R}\times TM$ and $J^2Y\simeq \mathbb{R}\times T^2M$. A general second-order ODE system is encoded by a **source form** $$ \varepsilon=\varepsilon_i(x,\dot x,\ddot x)\,\omega^i\wedge dt, \qquad \omega^i:=dx^i-\dot x^i\,dt, $$ where the $\omega^i$ are the basic contact $1$-forms. The system is **locally variational** if locally there exists a Lagrangian $L(t,x,\dot x)$ whose Euler–Lagrange expressions $$ E_i(L):=\frac{\partial L}{\partial x^i}-\frac{d}{dt}\left(\frac{\partial L}{\partial \dot x^i}\right) $$ reproduce $\varepsilon_i$ [1909.05115]. Krupková’s construction associates to $\varepsilon$ a **Lepage $2$-form** $\theta$ on $\mathbb{R}\times T^2M$. In coordinates, Urban–Volná give the explicit formula $$ \theta = \varepsilon_i\,\omega^i\wedge dt +\frac14\left(\frac{\partial\varepsilon_i}{\partial \dot x^j}-\frac{\partial\varepsilon_j}{\partial \dot x^i}\right)\omega^i\wedge\omega^j +\frac12\left(\frac{\partial\varepsilon_i}{\partial \ddot x^j}\right)\omega^i\wedge\dot\omega^j. $$ The **Lepage test** is the equivalence $$ \varepsilon \text{ is locally variational } \Longleftrightarrow d\theta=0. $$ In coordinates, this is exactly the classical **Helmholtz conditions**: $$ \frac{\partial\varepsilon_i}{\partial \ddot x^j} = \frac{\partial\varepsilon_j}{\partial \ddot x^i}, $$ $$ \frac{\partial\varepsilon_i}{\partial \dot x^j} + \frac{\partial\varepsilon_j}{\partial \dot x^i} = \frac{d}{dt}\left( \frac{\partial\varepsilon_i}{\partial \ddot x^j} + \frac{\partial\varepsilon_j}{\partial \ddot x^i} \right), $$ $$ \frac{\partial\varepsilon_i}{\partial x^j} - \frac{\partial\varepsilon_j}{\partial x^i} = \frac12\frac{d}{dt}\left( \frac{\partial\varepsilon_i}{\partial \dot x^j} - \frac{\partial\varepsilon_j}{\partial \dot x^i} \right). $$ An equivalent formulation is that local variationality forces each $\varepsilon_i$ to be affine in $\ddot x$, $$ \varepsilon_i=A_i(x,\dot x)+B_{ij}(x,\dot x)\,\ddot x^j, $$ with symmetry and differential conditions on $A_i$ and $B_{ij}$ reproducing $d\theta=0$. In this form, the test is an inverse-problem criterion: it decides whether a given second-order system arises as Euler–Lagrange equations. ## 3. Exactness, global variationality, and homogeneous systems Closedness of the Lepage $2$-form gives only **local** variationality. Global variationality is stronger: one asks for a globally defined horizontal $1$-form $\lambda$ such that $E(\lambda)=\varepsilon$. In the Lepage-equivalent formalism, $$ \varepsilon=E(\lambda) \Longleftrightarrow \theta=d\Theta_\lambda, $$ where $\Theta_\lambda$ is the Cartan form of $\lambda$. Hence the existence of a global Lagrangian is equivalent to solvability of the global exactness equation $$ \theta=d\mu $$ on $\mathbb{R}\times T^2M$; the horizontal part $h\mu$ then yields a global $\lambda$ [1909.05115]. Urban–Volná analyze this problem by decomposing $$ \theta=\alpha\wedge dt+\alpha', $$ where $\alpha$ and $\alpha'$ are closed $2$-forms on $TM$. They construct a homotopy operator on $TM$ showing that $\alpha'$ is always exact, and reduce global exactness of $\theta$ to exactness of a certain closed $2$-form $\omega$ on $M$. Two consequences are singled out. First, if the leading pure-$dt$ component $\alpha\wedge dt$ vanishes, then $\omega=0$, so $\theta$ is globally exact and $\varepsilon$ is globally variational. Second, in general, solvability of $\omega=d\eta$ is governed by $H^2(M)$; if $H^2(M)=0$ one may solve it topologically, but no universal constructive formula is known for arbitrary $M$. A particularly important case is homogeneity. Rossi and Urban–Volná show that every locally variational second-order ODE with $\varepsilon_i$ homogeneous of degree $c=0$ or $1$ is automatically globally variational. Writing $$ \varepsilon_i=A_i(x,\dot x)+B_{ij}(x,\dot x)\,\ddot x^j, $$ with $A_i$ and $B_{ij}$ homogeneous of degree $c$ and $c-2$ respectively, one constructs two global $1$-forms $\mu$ and $\kappa$ on $\mathbb{R}\times TM$ and sets $$ \lambda=h(\mu+\kappa). $$ Its differential reproduces $\theta$, hence $E(\lambda)=\varepsilon$ on all of $\mathbb{R}\times T^2M$. The standard examples are geodesic equations. For a Riemannian metric $g_{ij}(x)$, the energy Lagrangian $$ L=\frac12\,g_{ij}(x)\,\dot x^i\,\dot x^j $$ is homogeneous of degree $2$ in $\dot x$, and its Euler–Lagrange equations are homogeneous of degree $0$ in $(x,\dot x,\ddot x)$. For Finsler geometry, if $F(x,\dot x)$ is positive-homogeneous of degree $1$, then the energy $E=F^2/2$ is degree-$2$ homogeneous and yields the spray equations $$ \ddot x^i+2\,G^i(x,\dot x)=0, $$ which are homogeneous of degree $1$ in $(x,\dot x,\ddot x)$. In both cases the homogeneity hypothesis removes the topological obstruction, and one recovers globally defined Lagrangians. ## 4. Field-theoretic Lepage equivalents and closure tests for null Lagrangians In higher-order field theory, a Lagrangian is a horizontal $n$-form $$ \lambda=L(x^i,y^\sigma,y^\sigma_I)\,\omega_0 $$ on a jet bundle $J^rY$ of a fibered manifold $\pi:Y\to X$, with $\omega_0=dx^1\wedge\cdots\wedge dx^n$. A **Lepage equivalent** $\Theta_\lambda$ is an $n$-form whose horizontal part is $\lambda$ and whose first contact component of $d\Theta_\lambda$ is the Euler–Lagrange source form. In the formulation of Vitolo–Palese–Rossi, the operational **Lepage test** for an arbitrary $n$-form $\theta$ is: compute its horizontal part $h(\theta)=L\,d^nx$, compute $d\theta$, and check whether the $2$-contact component vanishes, $$ p_2(d\theta)=0. $$ Then $\theta$ is a Lepage equivalent of $L$ if and only if $h(\theta)=L\,d^nx$ and $p_2(d\theta)=0$. Equivalently, $$ d\theta=I(d\theta)+d_HR(d\theta), $$ with $I(d\theta)$ the Euler–Lagrange form and no higher contact terms [2010.16135]. Saunders reformulates the closure property in the variational bicomplex. Given a homotopy operator $P$ for the horizontal differential $d_H$, one defines the **fundamental Lepage form** $$ \Theta^F_\lambda=\sum_{p=0}^m(-1)^p\,P^p(\lambda) = \lambda-P\lambda+P^2\lambda-\cdots+(-1)^mP^m\lambda. $$ This is again a Lepage equivalent, and it satisfies the closure criterion $$ d_H\Theta^F_\lambda=0 \quad\Longleftrightarrow\quad \varepsilon_\lambda=0, $$ so the vanishing of its horizontal differential is exactly equivalent to $\lambda$ being a **null Lagrangian**. Saunders also gives a global version $\Theta^F_{\lambda,\nabla}$ by choosing a symmetric linear connection $\nabla$ on the manifold of independent variables and using Anderson’s global homotopy operator $P_\nabla$ [2309.01594]. Voicu, Garoiu, and Vasian prove an analogous closure property for general Lagrangians of any order using a canonical construction. Starting from the Vainberg–Tonti Lagrangian $\lambda^{VT}$ with the same Euler–Lagrange form as $\lambda$, and a local decomposition $\lambda=\lambda^{VT}+d_H\alpha$, they define a Lepage equivalent $$ \Phi_\lambda=\Theta_{\lambda^{VT}}+d\alpha $$ and show $$ d\Phi_\lambda=0 \quad\Longleftrightarrow\quad E_\lambda=0. $$ For reducible second-order Lagrangians $\lambda=\pi_{2,1}^*\lambda'+d_H\alpha$, they obtain the reduced first-order form $$ \phi_\lambda=\Theta_{\lambda'}+d\alpha, $$ again with $$ d\phi_\lambda=0 \quad\Longleftrightarrow\quad E_\lambda=0. $$ Although the construction is local in general, the paper states that in many cases of physical interest it becomes global, notably for vector or tensor bundles and Lagrangians of order at most $2$, including the Einstein–Hilbert Lagrangian, Lovelock generalizations, and Horndeski-type theories [2102.12955]. A common source of confusion is that the ODE and field-theoretic versions do not test the same property. For ODE source forms, the test detects **local variationality**; for field-theoretic Lagrangians, the recent closure-property constructions detect **nullity** or **vanishing Euler–Lagrange expressions**. ## 5. The classical Lepage test in nonparametric statistics In statistics, the classical Lepage test addresses the **two-sample independent location-scale problem**. Let $$ X_1,\ldots,X_m\sim F, \qquad Y_1,\ldots,Y_n\sim G $$ be independent samples. The strong null hypothesis is $$ H_0:F(t)=G(t)\ \text{for all } t, $$ equivalently no location shift $(\Delta=0)$ and no scale change $(\tau=1)$. The alternatives include location-only, $$ F(t)=G(t+\Delta),\qquad \Delta\neq 0, $$ scale-only, $$ F(t)=G(t/\tau),\qquad \tau\neq 1, $$ and the full location-scale alternative $$ F(t)=G((t+\Delta)/\tau) $$ with $\Delta\neq 0$ or $\tau\neq 1$ [2509.19126]. The **location component** is the Wilcoxon–Mann–Whitney statistic $$ U=\frac{1}{mn}\sum_{i=1}^m\sum_{j=1}^n I(X_iX)=1/2$. Fong and Huang identified a small bias in $\widehat{\operatorname{Var}}_{FP}(U)$ and proposed $$ T_{FH}=\frac{U-\frac12}{\sqrt{\widehat{\operatorname{Var}}_{FH}(U)}}, $$ with $\widehat{\operatorname{Var}}_{FH}(U)$ exactly equal to $\frac1{12}\left(\frac1m+\frac1n+\frac1{mn}\right)$ under $H_0$ even when $m\neq n$. For scale, a new estimator of the Ansari–Bradley variance is introduced: $$ \widehat{\operatorname{Var}}(C) = \hat\sigma^2\cdot \frac{n(N-1)}{N(n-1)}\cdot \frac{n(N-n)}{N-1} = \hat\sigma^2\cdot \frac{n^2(N-n)}{N(n-1)}, $$ leading to $$ C_P^*=\frac{C-E_0(C)}{\sqrt{\widehat{\operatorname{Var}}(C)}}. $$ These replacements yield five modified Lepage-type statistics under the weak null [2509.19126]. | Variant | Location term | Scale term | |---|---|---| | $L_1$ | $(U-\frac12)^2/\widehat{\operatorname{Var}}_{FP}(U)$ | $(C-E_0(C))^2/\operatorname{Var}_0(C)$ | | $L_2$ | $(U-\frac12)^2/\widehat{\operatorname{Var}}_{FH}(U)$ | $(C-E_0(C))^2/\operatorname{Var}_0(C)$ | | $L_3$ | $(U-\frac12)^2/\operatorname{Var}_0(U)$ | $(C-E_0(C))^2/\widehat{\operatorname{Var}}(C)$ | | $L_4$ | $(U-\frac12)^2/\widehat{\operatorname{Var}}_{FP}(U)$ | $(C-E_0(C))^2/\widehat{\operatorname{Var}}(C)$ | | $L_5$ | $(U-\frac12)^2/\widehat{\operatorname{Var}}_{FH}(U)$ | $(C-E_0(C))^2/\widehat{\operatorname{Var}}(C)$ | Under the weak null hypothesis, each standardized component is asymptotically $N(0,1)$ and uncorrelated, so each $L_i$ is asymptotically $\chi^2_2$. For small samples, the paper recommends Monte Carlo or permutation of group labels; critical values for $\alpha=0.05$ were tabulated for sample sizes as small as $m,n=5\ldots 10$. The simulation study examines five distributions—Exponential, $\chi^2$, Gamma, Beta, and Uniform—in two sample-size regimes, $(m,n)=(10,10)$ and $(40,30)$. All modified tests maintain near nominal $5\%$ type I error in small samples. In large samples, $L_0$ tends to be slightly conservative; $L_1,L_2,L_4,L_5$ are slightly liberal; and $L_3$ is very close. Across non-Normal, skewed, or heavy-tailed cases, the modified tests except $L_3$ uniformly outperform $L_0$. For example, for Exponential$(0.5)$ versus Exp$(\theta=2.0)$ with $m=n=10$, the reported powers are approximately $0.60$ for $L_0$, $0.67$ for $L_1$, $0.66$ for $L_2$, $0.62$ for $L_4$, and $0.62$ for $L_5$; at $(40,30)$, the corresponding powers are about $0.99$, $0.999$, $0.998$, $0.998$, and $0.998$. The biomedical illustration uses newborn platelet counts from infants of mothers treated with prednisone $(n=12)$ versus untreated controls $(m=7)$. Visual inspection suggests both higher median and higher variability in the treated group. The one-tailed right $p$-values at $\alpha=0.05$ are reported as $0.0030$ for $L_0$, less than $0.0001$ for $L_1$, less than $0.0001$ for $L_2$, $0.0028$ for $L_3$, and less than $0.0001$ for $L_4$ and $L_5$. The paper’s practical recommendation is to prefer $L_5$ for general two-sample location-scale problems where variances or medians may differ, and to use permutation or Monte Carlo critical values for very small samples. The statistical literature therefore uses the Lepage name for a combined rank procedure whose modern form is no longer tied to the strong null $F=G$. By contrast, the geometric literature uses the name for closure or exactness criteria in the inverse problem of the calculus of variations. The coexistence of these meanings is now a stable feature of the literature rather than a notational anomaly. Source: https://www.emergentmind.com/topics/lepage-test