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In the Calculus of Variations book by Gelfand and Fomin it says to consider the transformation $$x^{*} = \Phi(x,y,y')$$ $$y^{*} = \Psi(x,y,y').$$ Here it seems that $y'$ is the derivative of $y$ with respect to $x$.

This doesn't make any sense to me. Normally when we change coordinates we have that $(x,y)$ is a fixed coordinate system and we change to say $(x^{*},y^{*})$ where $x^* = f(x,y), y^* = g(x,y)$. Should I consider the variable $y$ inside $\Phi$ to be a function. If I just consider (x,y) as fixed basis then $y'$ makes no sense.

Joe
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  • I usually assume this is because we are implicitly working on some jet-space where $y'$ is not really the derivative of $y$ with respect to $x$. This is relevant to variational calculus where we often take derivatives with respect to $y'$ and for example $f(x,y,y') = y$ would have partial derivative w.r.t. $y'$ of zero; $\frac{\partial f}{\partial y'} = 0$. Sort of like, but even more frustratingly in my experience, the independence of $z$ and $\bar{z}$ when those are used as a complex notation for real derivatives... – James S. Cook Jul 26 '15 at 19:17
  • You have to get used to the notations in CoV. The integrand is considered to be a function of three variables $f(z_1,z_2,z_3)$, where $z_1$ is then set to $x$, $z_2$ to $y(x)$ and $z_3$ to $y'(x)$. To make things shorter it is a common practice to write $f(x,y,y')$ instead of $z_k$, but here $y'$ is just a name for the third variable, independent on other two. – A.Γ. Jul 26 '15 at 19:27
  • As @A.G. stated, the issue is a bit abuse of notation. This occurs in applications in physics. For example, in classical mechanics, the Lagrangian $L$ is a function of Generalized coordinates $q$ and their derivatives $q'$ with $L(q,q',t)$. But Lagrange's equation, which results from minimization of the action functional, is $$\frac{d}{dt}\frac{\partial L}{\partial q'}-\frac{\partial L}{\partial q}=0$$and we see that $q'$ is viewed first as an independent variable, and then, the derivative of $q$ with respect to time. – Mark Viola Jul 26 '15 at 19:41

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