Lecture notes
LECTURE EXAPMLES APPM LAGRANGIAN MECHANICS
- Module
- Institution
Examples of common questions in Lagrangian mechanics and Applied Mathematics
[Show more]
Seller
Follow
camarafantaj1
Content preview
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
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller camarafantaj1. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for £6.74. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
75057 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy revision notes and other study material for 14 years now