- #1
MisterX
- 758
- 71
We seek stationary solutions to
\int_{x_0}^{x_1} F(x, y, y')dx
subject to the constraint
\int_{x_0}^{x_1} G(x, y, y')dx = c
where c is some constant.
I have read that this can be solved by applying the Euler Lagrange equations to
F(x, y, y') + \lambda G(x, y, y')
and then finding the appropriate value of \lambda when solving so that the constraint is satisfied.
Why does this work? I am not sure what reference to use.
Also, this may still work when the unconstrained integral has no stationary solutions, right?
\int_{x_0}^{x_1} F(x, y, y')dx
subject to the constraint
\int_{x_0}^{x_1} G(x, y, y')dx = c
where c is some constant.
I have read that this can be solved by applying the Euler Lagrange equations to
F(x, y, y') + \lambda G(x, y, y')
and then finding the appropriate value of \lambda when solving so that the constraint is satisfied.
Why does this work? I am not sure what reference to use.
Also, this may still work when the unconstrained integral has no stationary solutions, right?