A Note on Singular Boundary Regularisation in Hamiltonian Systems

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 Schrödinger equation blow-up.