Lagrange Multiplier
المؤلف:
Arfken, G
المصدر:
"Lagrange Multipliers." §17.6 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press
الجزء والصفحة:
...
21-9-2018
2784
Lagrange Multiplier
Lagrange multipliers, also called Lagrangian multipliers (e.g., Arfken 1985, p. 945), can be used to find the extrema of a multivariate function 
subject to the constraint 
, where 
and 
are functions with continuous first partial derivatives on the open set containing the curve 
, and 
at any point on the curve (where 
is the gradient).
For an extremum of 
to exist on 
, the gradient of 
must line up with the gradient of 
. In the illustration above, 
is shown in red, 
in blue, and the intersection of 
and 
is indicated in light blue. The gradient is a horizontal vector (i.e., it has no 
-component) that shows the direction that the function increases; for 
it is perpendicular to the curve, which is a straight line in this case. If the two gradients are in the same direction, then one is a multiple (
) of the other, so
 |
(1)
|
The two vectors are equal, so all of their components are as well, giving
 |
(2)
|
for all 
, ..., 
, where the constant 
is called the Lagrange multiplier.
The extremum is then found by solving the 
equations in 
unknowns, which is done without inverting 
, which is why Lagrange multipliers can be so useful.
For multiple constraints 
, 
, ...,
 |
(3)
|
REFERENCES:
Arfken, G. "Lagrange Multipliers." §17.6 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 945-950, 1985.
Lang, S. Calculus of Several Variables. Reading, MA: Addison-Wesley, p. 140, 1973.
Simmons, G. F. Differential Equations. New York: McGraw-Hill, p. 367, 1972.
Zwillinger, D. (Ed.). "Lagrange Multipliers." §5.1.8.1 in CRC Standard Mathematical Tables and Formulae, 31st Ed. Boca Raton, FL: CRC Press, pp. 389-390, 2003.
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