Understanding Hamiltons Principle & the Variational Formulation of Mechanics

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The discussion focuses on the variational formulation of mechanics, specifically Hamilton's Principle. The user expresses confusion regarding the introduction of the differential quantity δJ = dJ/dα * dα and its relation to the Hamiltonian integral being zero. They question the necessity of this notation when the Euler-Lagrange equation (Equation 2.11) appears sufficient. The conversation emphasizes the importance of understanding the underlying concepts of variations and their notation in mechanics.

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  • Understanding of Hamilton's Principle in classical mechanics
  • Familiarity with the Euler-Lagrange equation
  • Basic knowledge of differential calculus and variations
  • Concept of Hamiltonian mechanics
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Students and professionals in physics, particularly those studying classical mechanics, as well as educators seeking to clarify the variational formulation of 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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