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I have some trouble understanding how my book introduces the variational formulation of mechanics. On the attached file I have stipulated with red a part, which I do not understand at all.

What is the idea behind introducing this differential quantity:

δJ = dJ/dα * dα and writing the requirement for the hamiltonian integral to be zero in the way described by the last equation. Surely all that is required is that the euler lagrange equation 2.11 is satisfied? For me it is just weird to start talking about these things. Can someone explain the idea behind it in some detail?

What is the idea behind introducing this differential quantity:

δJ = dJ/dα * dα and writing the requirement for the hamiltonian integral to be zero in the way described by the last equation. Surely all that is required is that the euler lagrange equation 2.11 is satisfied? For me it is just weird to start talking about these things. Can someone explain the idea behind it in some detail?