Lagrange Equation: Solution for J(q_1,...,q_n)

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The discussion centers on the correct representation of the functional J in the context of the Lagrange equation. It emphasizes that J should be expressed as a functional of the trajectory variables, specifically J[q_1(t), ..., q_n(t)], rather than as a function of the generalized coordinates and their velocities. The confusion arises from the notation used in the integral, which leads to misunderstandings about the nature of J. The importance of distinguishing between functionals and functions in this context is highlighted. Clarifying this notation is essential for accurate application of the Lagrange equation.
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For functional

J(q_1,...,q_n)=\int^{t}_{t_0}L(q_1,...,q_n;\dot{q}_1,...,\dot{q}_n;t)

Why isn't J(q_1,...,q_n;\dot{q}_1,...,\dot{q}_n;t)?
 
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I had an answer for you, but the TeX system stumped me, and I was unable to write it out. Sorry about that.
 
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J is a functional, and not a function of the q's. It should be written as:
<br /> J[q_1(t), \ldots, q_n(t)]<br />
 

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