Background fields of electromagnetic and gravitational type emerge in the low kinetic energy limit of any regular Lagrangian system and, in particular, in the corresponding limit of any spacetime theory in which the free motion of test particles is described by an unspecified regular Lagrangian. Electromagnetic and gravitational type interactions are therefore a universal feature of low kinetic energy Lagrangian systems. These background interactions can be consistently turned into dynamic Einstein–Maxwell fields by promoting the Lagrangian function to a dynamic scalar field on the tangent bundle of the configuration space. Accordingly, Einstein–Maxwell theory can be deduced from the assumption that the motion of elementary test particles in spacetime is described by Lagrangian mechanics. For higher kinetic energy-type values, identified with the square of the invariant mass of the particle, the Lagrangian induces higher rank interactions that seem however too weak to have been detected in spacetime physics, but which might prove relevant at the Planck scale.

电磁和引力类型的背景场出现在任何正则拉格朗日系统的低动能极限中，特别是在任何时空理论的相应极限中，其中测试粒子的自由运动由未指定的正则拉格朗日函数描述。因此，电磁和引力型相互作用是低动能拉格朗日系统的普遍特征。通过将拉格朗日函数提升为配置空间切丛上的动态标量场，这些背景相互作用可以一致地转化为动态爱因斯坦-麦克斯韦场。因此，爱因斯坦-麦克斯韦理论可以从基本测试粒子在时空中的运动由拉格朗日力学描述的假设中推导出来。对于更高的动能类型值，