Noether’s calculus of invariant variations yields exact identities from functional symmetries. The standard application to an action integral allows to identify conservation laws. Here we rather consider generating functionals, such as the free energy and the power functional, for equilibrium and driven many-body systems. Translational and rotational symmetry operations yield mechanical laws. These global identities express vanishing of total internal and total external forces and torques. We show that functional differentiation then leads to hierarchies of local sum rules that interrelate density correlators as well as static and time direct correlation functions, including memory. For anisotropic particles, orbital and spin motion become systematically coupled. The theory allows us to shed new light on the spatio-temporal coupling of correlations in complex systems. As applications we consider active Brownian particles, where the theory clarifies the role of interfacial forces in motility-induced phase separation. For active sedimentation, the center-of-mass motion is constrained by an internal Noether sum rule.

Noether 的不变变化演算从函数对称性中产生了精确的恒等式。动作积分的标准应用允许识别守恒定律。在这里，我们宁愿考虑为平衡和驱动多体系统生成泛函，例如自由能和功率泛函。平移和旋转对称操作产生机械定律。这些全局恒等式表示总内力和总外力和扭矩的消失。我们表明，功能差异会导致局部总和规则的层次结构，这些规则将密度相关器以及静态和时间直接相关函数（包括记忆）相互关联。对于各向异性粒子，轨道和自旋运动变得系统耦合。该理论使我们能够阐明复杂系统中相关性的时空耦合。作为应用，我们考虑活性布朗粒子，其中理论阐明了界面力在运动诱导相分离中的作用。对于主动沉积，质心运动受内部 Noether 和规则约束。