Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2504.19430

Invariance of subbundles and nonholonomic trajectories

Andrew D. Lewis·Ahmed Gamal Shaltut
View original source ↗Read source document ↗Export metadata ↓

Abstract

In the comparison of nonholonomic mechanics and constrained variational mechanics, invariant affine subbundles arise in the determination of the initial conditions where the two methods yield the same trajectories. Motivated by this, differential conditions are considered for invariant affine subbundles as they arise in the comparison of nonholonomic and constrained variational mechanics. First of all, the formal integrability of the resulting linear partial differential equation is determined using Spencer cohomology. Second, iterative formulae are provided that permit the determination of the largest invariant affine subbundle invariant under an affine vector field. Finally, the problem of a disc rolling on an inclined plane with no-slip is considered as an example to illustrate the theory.

Explore related subjects

math.DGmath-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Andrew D. Lewis, Ahmed Gamal Shaltut. 2025-08-14. Invariance of subbundles and nonholonomic trajectories. https://arxiv.org/abs/2504.19430

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quantum propagation for Berezin-Toeplitz operators

We describe the asymptotic behaviour of the quantum propagator generated by a Berezin-Toeplitz operator with real-valued principal symbol. We also give precise asymptotics for smoothed spectral projectors associated with the operator in the autonomous case; this leads us to introducting quantum states associated with immersed Lagrangian submanifolds. These descriptions involve geometric quantities of two origins, coming from lifts of the Hamiltonian flow to the prequantum bundle and the canonical bundle respectively. The latter are the main contribution of this article and are connected to the Maslov indices appearing in trace formulas, as will be explained in a forthcoming paper.

math.DG↗
arXiv2026

Classification of compact manifolds with positive isotropic curvature

We show the following result: Let $(M,g_0)$ be a compact manifold of dimension $n\geq 12$ with positive isotropic curvature. Then $M$ is diffeomorphic to a spherical space form, or a quotient manifold of $\mathbb{S}^{n-1}\times \mathbb{R}$ by a cocompact discrete subgroup of the isometry group of the round cylinder $\mathbb{S}^{n-1}\times \mathbb{R}$, or a connected sum of a finite number of such manifolds. This extends previous works of Brendle and Chen-Tang-Zhu, and improves a work of Huang. The proof uses Ricci flow with surgery on compact orbifolds, with the help of the ambient isotopy uniqueness of closed tubular neighborhoods of an isolated singular point in an orbifold.

math.DG↗
arXiv2026

Willmore surfaces in 4-dimensional conformal manifolds

This paper is dedicated to the exploration of the conformal Willmore functional for surfaces within 4-dimensional conformal manifolds. We provide a detailed calculation of both the first and second variations, and present the Euler-Lagrange equation of this functional in a conformally invariant form. Utilizing the second variation formula we derived, we demonstrate that the Clifford torus in $\mathbb{C}P^2$ is strictly Willmore-stable. This finding strongly supports the conjecture proposed by Montiel and Urbano [J. reine angew. Math. 546 2002, 139-154], which posits that the Clifford torus in $\mathbb{C}P^2$ minimizes the Willmore functional among all tori. Moreover, by applying our formula to complex curves in $\mathbb{C}P^2$, we establish that the first nonzero eigenvalue of the Jacobi operator is at least 12. In the context of 4-dimensional locally symmetric spaces, we construct several holomorphic differentials to show that among all minimal 2-spheres, only those super-minimal ones can be Willmore.

math.DG↗