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

We present a concise variational framework for planar elastic rods, applicable to both intrinsically two-dimensional rods and three-dimensional rods constrained to planar deformation. This framework enables the modular inclusion of multiple distributed energy contributions, such as gravitational loading, elastic bending, and internal magnetic torques. While mathematically straightforward, the approach systematically handles several energy terms without relying on the cumbersome nested-integral constructions typical of classical methods. All loading contributions are expressed directly at the level of the energy functional, bypassing traditional force-balance derivations (e.g., Wang, 1986), which simplifies the derivation of governing equations and allows for the transparent and systematic incorporation of additional physical effects. We demonstrate that, under appropriate conditions, this framework exactly reproduces the governing equations for hard-magnetic elastic rods deforming in a plane and naturally accommodates gravitational effects. Likewise, it precisely recovers the classical heavy-elastica model in the corresponding limit. The resulting planar energy functional provides compact, analytically tractable governing equations while maintaining a clear separation of contributions from different loads. Overall, this approach offers a versatile, modular, and conceptually simple tool for analyzing planar elastic rods, providing clarity, flexibility, and ease of extension to complex loading scenarios.