2000 character limit reached
Integration of first-order ODEs by Jacobi fields (2404.14352v2)
Published 22 Apr 2024 in math.CA and math.DG
Abstract: A new class of vector fields enabling the integration of first-order ordinary differential equations (ODEs) is introduced. These vector fields are not, in general, Lie point symmetries. The results are based on a relation between 2-dimensional Riemannian manifolds and the integrability of first-order ODEs, which was established in a previous work of the authors. An integration procedure is provided, together with several examples to illustrate it. A connection between integrating factors of first-order ODEs and Schr\"odinger-type equations is highlighted.
- N. H. Ibragimov. A Practical Course in Differential Equations and Mathematical Modelling: Classical and New Methods, Nonlinear Mathematical Models, Symmetry and Invariance Principles. World Scientific, Beijing, 2010.
- H. Stephani. Differential Equations: Their Solutions Using Symmetry. Cambridge University Press, New York, 1989.
- P. J. Olver. Applications of Lie groups to differential equations, volume 107. Springer-Verlag, New York, 1986.
- J. Sherring and G. Prince. Geometric aspects of reduction of order. Trans. Amer. Math. Soc., 334(1):433–453, 1992.
- 𝒞∞superscript𝒞\mathcal{C}^{\infty}caligraphic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT-structures in the integration of involutive distributions. Phys. Scr., 98(8):085222, jul 2023.
- Antonio J. Pan-Collantes and José A. Álvarez García. Surfaces associated with first-order odes, 2023.
- Z. O. Bayrakdar and T. Bayrakdar. Burgers’ Equations in the Riemannian Geometry Associated with First-Order Differential Equations. Advances in Mathematical Physics, 2018:1–8, 2018.
- Lectures on differential geometry, volume 1. World Scientific Publishing Company, 1999.
- S. Morita. Geometry of Differential Forms. American Mathematical Society, Rhode Island, 2001.
- M. P. Do Carmo. Riemannian geometry, volume 6. Springer, 1992.
- Geodesic and Newtonian vector fields and symmetries of mechanical systems. Symmetry, 15(1):181, jan 2023.
- S. Deshmukh and V. Azam Khan. Geodesic vector fields and Eikonal equation on a Riemannian manifold. Indagationes Mathematicae, 30(4):542–552, 2019.
- Geodesic vector fields on a riemannian manifold. Mathematics, 8(1):137, 2020.
- J. M. Lee. Introduction to Riemannian manifolds, volume 2. Springer, 2018.
Sponsor
Paper Prompts
Sign up for free to create and run prompts on this paper using GPT-5.
Top Community Prompts
Collections
Sign up for free to add this paper to one or more collections.