Why not just replace a coordinate by its adjacent force?
We will have
a closer look on a 1D example of a function \(y(x)\).
Let
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| \begin{equation*} z:=dy/dx \end{equation*} | (1.19) |
and
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| \begin{equation*} x=x(z) \qquad . \end{equation*} | (1.20) |
We find
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| \begin{equation*} y = y(x(z)) = f(z) \qquad , \end{equation*} | (1.21) |
consequently
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| \begin{equation*} y = f( dy/dx ) \qquad , \end{equation*} | (1.22) |
respectively
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| \begin{equation*} dy/dx = f^{-1}(y) \qquad . \end{equation*} | (1.23) |
This differential equation has the solution \(y(x)\); but there exist more solutions
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| \begin{equation*} y = y(x+const.) \qquad . \end{equation*} | (1.24) |
Simply substituting a coordinate \(x\) by the adjacent force \(z\) leads to a loss of information, since the original function is known only up to a constant. For a function with several parameters this is even more critical since the ”constant” may by a function of all the remaining parameters.
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© J. Carstensen (Stat. Meth.)