AI 中文总结
该数学研究推导切触拉格朗日2-网中萨缪尔森障碍的施瓦茨残差表达式，揭示其与等仿曲率的关系，解释萨缪尔森条件局部失效的原因并推导其重参数化变换律。
AI 中文摘要
萨缪尔森条件是平面拉格朗日2-网的经典面积比条件，在局部网坐标(u,v)下等价于∂_u∂_v log|J_F(u,v)|=0，其中F是逆网坐标映射，J_F是其雅可比行列式。对于切线族L_t:y=tx+h(t)，令Ψ(u,v)（u≠v）为L_u与L_v的交映射，J_Ψ为其雅可比行列式，定义S_h(u,v):=∂_u∂_v log|J_Ψ(u,v)|。在每个满足h''(t₀)≠0的对角点(t₀,t₀)附近，障碍可分解为S_h(u,v)=1/(v-u)²+R_h(u,v)，其中R_h在(t₀,t₀)附近可光滑延拓至对角线上。我们称R_h(t,t)为施瓦茨残差，计算得R_h(t,t)=h⁽⁴⁾(t)/(3h''(t)) - (4/9)(h'''(t)/h''(t))²=(1/2){σ,t}=-κ_aff(σ)(dσ/dt)²，此处{σ,t}表示σ关于t的施瓦茨导数，σ是包络γ(t)=(-h'(t),h(t)-th'(t))的等仿弧长参数，κ_aff是其等仿曲率，约定γ_σσσ + κ_affγ_σ=0。因此，普适极点解释了萨缪尔森条件的局部失效，而对角有限部分携带包络的仿射-射影信息，我们还推导了施瓦茨残差在切线参数重参数化下的变换律。
英文摘要
The Samuelson condition, a classical area-ratio condition for planar Lagrangian $2$-webs, is equivalent in local web coordinates $(u,v)$ to $\partial_u\partial_v\log|J_F(u,v)|=0$, where $F$ is the inverse web-coordinate map and $J_F$ is its Jacobian. For the tangent-line family $L_t:y=tx+h(t)$, let $Ψ(u,v)$, for $u\neq v$, be the intersection map of $L_u$ and $L_v$, write $J_Ψ$ for its Jacobian, and set $S_h(u,v):=\partial_u\partial_v\log|J_Ψ(u,v)|$. Near each diagonal point $(t_0,t_0)$ with $h''(t_0)\neq0$, the obstruction admits the decomposition $S_h(u,v)=1/(v-u)^2+R_h(u,v)$, where $R_h$ extends smoothly across the diagonal near $(t_0,t_0)$. We call $R_h(t,t)$ the Schwarzian residue and compute $R_h(t,t)=h^{(4)}(t)/(3h''(t))-(4/9)(h'''(t)/h''(t))^2=(1/2)\{σ,t\}=-κ_{\mathrm{aff}}(σ)(dσ/dt)^2$. Here $\{σ,t\}$ denotes the Schwarzian derivative of $σ$ with respect to $t$, where $σ$ is the equi-affine arclength parameter of the envelope $γ(t)=(-h'(t),h(t)-th'(t))$, and $κ_{\mathrm{aff}}$ denotes its equi-affine curvature, with the convention $γ_{σσσ}+κ_{\mathrm{aff}}γ_σ=0$. Thus the universal pole accounts for the local failure of the Samuelson condition, while the diagonal finite part carries affine-projective information about the envelope. We also derive the transformation law of the Schwarzian residue under reparametrization of the tangent-line parameter.
Comments13 pages, no figures. Revised the Introduction and Sections 3 and 5 to add the Hess connection and bi-Lagrangian interpretation of the Samuelson obstruction and to clarify and directly derive the equi-affine curvature--Schwarzian identity. Main results unchanged
