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arXiv · 2607.09952

A Note on Singular Boundary Regularisation in Hamiltonian Systems

Cristina Stoica
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Abstract

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 Schr\"odinger equation blow-up.

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Cristina Stoica. 2026-07-10. A Note on Singular Boundary Regularisation in Hamiltonian Systems. https://arxiv.org/abs/2607.09952

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