Classification of singular differential invariants in ()-dimensional space and integrability.

Area of Science:

• Mathematical Physics
• Differential Geometry
• Lie Algebras

Background:

• Singularities are crucial in physical phenomena and invariant differential equations.
• Understanding invariant structures requires classifying singularities.

Purpose of the Study:

• Investigate the classification of singular differential invariants in (1+3)-dimensional space.
• Analyze Lie algebras of dimension 4 and their associated systems of three second-order ordinary differential equations.
• Deduce canonical forms and categorize singular invariants.

Main Methods:

• Formulation of singular invariants for Lie algebras of dimension 4.
• Detailed study of systems of three second-order ordinary differential equations.
• Categorization based on conditional singularity, weak uncoupling, weak linearization, partial uncoupling, and partial linearization.
• Analysis of integrability for classified singular-invariant systems.

Main Results:

• Canonical forms for systems of three second-order ODEs associated with dimension 4 Lie algebras are deduced.
• Singular invariants are categorized based on specific properties.
• Cases not leading to singular invariant systems are identified.
• Integrability of classified singular-invariant systems is discussed.

Conclusions:

• The classification and categorization provide a framework for analyzing singular differential invariants.
• The study offers insights into the integrability of these systems.
• Physical applications in mechanics are demonstrated using two illustrative examples.