On The Dricihlet Variational Principle Formulation of the Poisson Equation

We present a novel analytical mechanics framework for solving Poisson's equation by unifying the Dirichlet Principle with the Udwadia-Kalaba (UK) formulation of constrained motion. The classical Dirichlet Principle recasts the boundary value problem as a constrained quadratic minimization, where the solution minimizes an energy functional associated with the Laplace operator. This naturally leads to a quadratic programming (QP) problem in the discrete setting. Instead of relying on traditional optimization solvers, We apply the Udwadia Kalaba formulation to obtain an explicit solution of the quadratic programming problem. The resulting evolution drives the system toward the minimizer of the energy functional, while strictly enforcing Dirichlet boundary conditions using the Moore-Penrose inverse. This approach bridges variational principles, optimization, PDEs and analytical mechanics. Numerical results demonstrate the feasibility and accuracy of the method, providing a flexible alternative to conventional PDE solvers.