Understanding Hamiltons Principle & the Variational Formulation of Mechanics

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The discussion centers on confusion regarding the variational formulation of mechanics as presented in a book. The user questions the introduction of the differential quantity δJ and its relation to the Hamiltonian integral being zero, suggesting that satisfying the Euler-Lagrange equation should suffice. There is a concern that the new notation complicates rather than clarifies the concepts, as it replaces partial derivatives with variations. A request for a detailed explanation of the underlying ideas is made, highlighting a need for clarity in the transition to this new notation. Understanding the rationale behind these concepts is essential for grasping the variational approach in mechanics.
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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?
 

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They just introduce new notation. So that instead of writing the partial derivatives, one could just write those variations.
 
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