A Unified Variational Integral-Reduction Framework for Planar Elastica under Magnetic and Gravitational Loading

We present a simple variational framework for planar elastica that enables distributed energies, such as gravitational loading or magnetic body torques, to be incorporated in a modular and unified manner. The formulation is based on expressing all load induced contributions directly at the level of the energy functional, which avoids the force balance constructions used in classical treatments such as Wang (1986) and makes the inclusion of additional physical effects straightforward. The resulting planar energy functional yields compact governing equations in which the contributions of individual load types remain clearly separated. We demonstrate that the framework reproduces the classical heavy elastica equations exactly and naturally accommodates magnetic energy terms commonly used in hard magnetic rod models. Although mathematically elementary, the formulation provides a clean and extensible structure for describing planar rod deformations under general distributed loads.