AI 中文总结
研究哈密顿系统变量奇异变化时边界附近诱导辛二形式的问题，通过基本有限维准则记录各向异性障碍，并用麦吉hee正则化及聚焦非线性薛定谔方程爆破的调制几何举例说明。
AI 中文摘要
哈密顿系统中的变量奇异变化，如天体力学中的麦吉hee坐标或色散偏微分方程爆破中的重整化变量，旨在将运动方程扩展到奇异边界。然而，在边界附近，诱导的辛二形式可能会按奇异尺度的不同幂次重新缩放不同的几何方向。如果主导加权部分退化，则不存在单个共形因子使该形式扩展为光滑的非退化二形式。这种各向异性障碍是对bm-辛几何中研究的各向同性奇点的补充。我们将其记录在一个基本的有限维准则中，并用齐次中心力碰撞的麦吉hee正则化以及形式上用聚焦非线性薛定谔方程爆破的调制几何来举例说明。
英文摘要
Singular changes of variables in Hamiltonian systems, such as McGehee coordinates in celestial mechanics or renormalised variables in dispersive PDE blow-up, are designed to extend the equations of motion to a singular boundary. In contrast, it may be that near the boundary, the the induced symplectic two-form may rescale distinct geometric directions by distinct powers of the singular scale, and if the leading weighted part is degenerate, no single conformal factor makes the form extend as a smooth non-degenerate two-form. This anisotropic obstruction is complementary to the isotropic singularities studied in bm-symplectic geometry. We record it in an elementary finite-dimensional criterion and illustrate it with McGehee regularisation of homogeneous central-force collisions and, formally, with the modulation geometry of focusing nonlinear Schrödinger equation blow-up.
