The Three Minimal Techniques of Variational Method

The variational method, as the core tool of functional extremum analysis, has a wide and profound impact on mathematical theory and engineering applications through its three basic operations - chain rule, partial integration, and variable substitution. This article systematically reviews the historical development of these three methods and their key roles in classical mechanics and partial differential equations. The research results indicate that the chain rule provides an effective approach for the variation of complex functionals by handling composite dependency relationships; Divisional integration is not only indispensable in high-order derivative reduction and boundary condition treatment, but also lays the foundation for weak solution theory of partial differential equations; Variable substitution reveals the conservation laws and structural characteristics of physical systems (such as Noether's theorem) through symmetry utilization and dimensionality reduction simplification. Through typical cases such as Dirichlet energy functionals, this paper verifies the synergistic effect of compulsion, convexity, and weak lower semi continuity in proving the existence and uniqueness of minima.