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If I have a variational problem of the form:

$$\min \int_{a}^{b} F(x, y(x), y'(x)dx\quad\text{subject to}\quad \int_{a}^{b} G(x, y(x), y'(x))dx=C\tag{1}$$

The solution can be found by transforming the problem into an unconstrained problem:

$$\min \int_a^b H(x, y(x), y'(x), \lambda)dx\tag{2}$$

Where, $$H(x, y(x), y'(x), \lambda)=F(x, y(x), y'(x))-\lambda G(x, y(x), y'(x))\tag{3}$$

WHERE $\lambda$ is a Lagrange multiplier and solve then the Euler-Lagrange equation:

$$\frac{d}{dx}H_{y'}-H_y=0\tag{4}$$ together with the constraint.

Now, suppose my constraint becomes $$\int_{c}^{d} G(x, y(x), y'(x))dx=C\tag{5}$$ instead of $$\int_{a}^{b} G(x, y(x), y'(x))dx=C .\tag{6}$$ If $a\neq c$ or $b\neq d$ I cannot use the same trick, how can I solve the problem?

Qmechanic
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Juan Imbet
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  • You can use a linear change of variables to change the interval over which you are integrating on. – Jonathan Davidson Jul 21 '17 at 22:25
  • In particular, use the u substitution $u = \frac{d-c}{b-a}(x-a)+c$ – Jonathan Davidson Jul 21 '17 at 22:42
  • @JonathanDavidson. I also had that idea but then realized that it will modify $y$ in the constraint integral in a way that I am not sure how to handle. – md2perpe Jul 21 '17 at 22:45
  • Thanks for the suggestion! If one of the limits is $\infty$, instead of a linear change of variables can I use an isomorphism? E.g if $a=-\infty, b=\infty$ and $c=0, d=\infty$ can I use the transformation $u=exp(x)$ ? – Juan Imbet Jul 21 '17 at 23:05

1 Answers1

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Hints:

  1. Extend $F$, $G$, and $y$ in an arbitrary smooth fashion to the whole real $x$-axis $\mathbb{R}$. (Less is really needed; we are just illustrating the method.)

  2. Replace the $H$-function in eq. (3) with $$\begin{align}H(x, y(x), y'(x), \lambda) ~=~&1_{[a,b]}(x)~ F(x, y(x), y'(x))\cr ~-~&\lambda~1_{[c,d]}(x)~G(x, y(x), y'(x)),\end{align}\tag{3'}$$ and replace eq. (2) with $$ \min_{y} \int_{\mathbb{R}} \!\mathrm{d}x~ H(x, y(x), y'(x), \lambda) \tag{2'}$$ with pertinent boundary conditions (BCs). Here $x\mapsto 1_{[a,b]}(x)$ denotes the indicator function for the interval $[a,b]$.

Qmechanic
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