Methodological Equivalence in Potential Derivation for Discrete Gravitation Systems

A rigorous multimethod analysis of the gravitational potential generated by a discrete distribution of \( N \) point masses is presented. The work systematically demonstrates the existence of a scalar potential \( U(\mathbf{r}) \) such that \( \mathbf{F}(\mathbf{r}) = -\nabla U(\mathbf{r}) \) using four independent yet equivalent mathematical approaches. The first method solves the system of partial differential equations \( \partial U / \partial x_i = F_i \) through direct integration using trigonometric substitution. The second employs the fundamental theorem for line integrals, establishing path independence and recovering the potential via \( \int_C \mathbf{F} \cdot d\ell = U(\mathbf{r}) - U(\mathbf{r}_0) \). The third method demonstrates the irrotational nature of the field \( \nabla \times \mathbf{F} = 0 \) through Stokes' theorem, confirming zero circulation over arbitrary closed paths. The fourth approach connects the potential with orbital dynamics, deriving the relation \( v_c = \sqrt{R|d\Phi/dR|} \) for circular orbits in radial configurations. All derivations converge to the canonical result \( \Phi(\mathbf{r}) = -G \sum_k M_k / \|\mathbf{r} - \mathbf{r}_k\| \), offering a complete pedagogical synthesis of vector calculus, differential equations, and classical gravitational theory.