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LECTURE EXAPMLES APPM LAGRANGIAN MECHANICS
- Institution
- University Of The Witwatersrand (wits)
Examples of common questions in Lagrangian mechanics and Applied Mathematics
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Section 7.4: Lagrange Multipliers andConstrained Optimization
A constrained optimization problem is a problem of the form
maximize (or minimize) the function F (x, y) subject to the
condition g(x, y) = 0.
1
From two to one
In some cases one can solve for y as a function of x and then find
the extrema of a one variable function.
That is, if the equation g(x, y) = 0 is equivalent to y = h(x), then
we may set f (x) = F (x, h(x)) and then find the values x = a for
which f achieves an extremum. The extrema of F are at (a, h(a)).
2
, Example
Find the extrema of F (x, y) = x2 y − ln(x) subject to
0 = g(x, y) := 8x + 3y.
3
Solution
We solve y = −8 −8 −8 3
3 x. Set f (x) = F (x, 3 x) = 3 x − ln(x).
Differentiating we have f 0 (x) = −8x2 − x1 . Setting f 0 (x) = 0, we
must solve x3 = −1 −1
8 , or x = 2 . Differentiating again,
f 00 (x) = −16x + x12 so that f 00 ( −1
2 ) = 12 > 0 which shows that 2
−1
is a relative minimum of f and ( −1 4
2 , 3 ) is a relative minimum of F
subject to g(x, y) = 0.
4
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