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There seems to be a slight confusion about the meaning of solution: The principle of least action leads to the equation of motion (Euler-Lagrange equation), which correspond to a minimum of the action functional. These equations can have multiple solutions, so there is no contradiction in the formalism. There can multiple solutions that minimize the energy, ...


It's Stokes's theorem. Consider a field $F = dA + A \wedge A$ such that $A$ is pure gauge at infinity, that is, $\lim_{x\to\infty} A(x) = \omega\, d \omega^{-1}$ for some $\omega : S^3 \to SU(2) \sim S^3$ where $\omega$ is a function on the 3-sphere because the limit can depend on the direction out to infinity. In differential forms the first expression is ...

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