Variational problem is equivalent to minimizing the energy functional

The variational problem Poisson's equation reads "For given $$f \in L^2(\Omega)$$ find $$u \in H_0^1(\Omega)$$ such that $$\int_{\Omega} \nabla u \cdot \nabla v \, \mathrm dx= \int_{\Omega}fv \, \mathrm dx \quad \forall v\in H_0^1(\Omega).$$

Very often I read that it is equivalent to minimizing the energy functional $$J(v):=\frac{1}{2}a(v,v)-F(v)$$ where $$a(u,v)=\int_{\Omega} \nabla u \cdot \nabla v \, \mathrm dx$$ and $$F(v)=\int_{\Omega}fv \, \mathrm dx.$$ I get the proof, but what is the added value of introducing such equivalence, e.g. what relevance does the energy functional have? Sometimes I just see it randomly squeezed in and not being mentioned further in the literature.