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  "body": "\n## Summary\n\nThe Yang–Mills equations are the field equations of a gauge theory: they are nonlinear partial differential equations for a [[connection|NJ21PN5ZGJ]] on a [[principal bundle|8E37E74XVP]].\nThey are the Euler–Lagrange equations of the Yang–Mills functional, the integral of the squared norm of the [[curvature|ZAJFW1S078]].\nFor the circle group they reduce to Maxwell's equations in vacuum, and for non-abelian groups they describe the strong and weak forces.\nIn dimension four their self-dual solutions, the instantons, led to deep results on the topology of four-manifolds.\n\n## Explanations\n\n### L1 human\n\nIn electromagnetism the field is described by a potential, and the field strength measures how the potential twists.\nGauge theories replace numbers in this picture by elements of a matrix group, so twists in different directions no longer commute.\nThe Yang–Mills equations say that the field strength is as balanced as possible, in the sense that it is a critical point of its total energy.\nThe simplest solutions are flat connections, whose field strength is zero everywhere.\n",
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