Find lagrangian (please check work)

In summary: No: you need to solve the equations correctly! I think maybe you meant to say "regardless of which form of Lagrangian you use". Then the answer would be yes, if you make no errors during solving.
  • #1
939
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Homework Statement



F(x, y) = 96xy - 4x
subject to constraint of 11 = x + y

Form the lagrangian.

Homework Equations



F(x, y) = 96xy - 4x
subject to constraint of 11 = x + y

The Attempt at a Solution



My only question is solving 11 = x + y

My book says the answer is:
L = 96xy - 4x + λ(11 - x - y)

But I got:
L = 96xy - 4x + λ(x + y - 11)

Just to confirm, both are correct because it just depends how you solve the constraint, no?
 
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  • #2
939 said:

Homework Statement



F(x, y) = 96xy - 4x
subject to constraint of 11 = x + y

Form the lagrangian.

Homework Equations



F(x, y) = 96xy - 4x
subject to constraint of 11 = x + y

The Attempt at a Solution



My only question is solving 11 = x + y

My book says the answer is:
L = 96xy - 4x + λ(11 - x - y)

But I got:
L = 96xy - 4x + λ(x + y - 11)

Just to confirm, both are correct because it just depends how you solve the constraint, no?

It makes no difference: one λ will just have the opposite sign of the other.

However, if does matter when you are doing post-optimality analysis. For example, you can use the value of λ to find the approximate change in the optimal value of F when the constraint changes to x+y = 11.1, for example. In that case you need to understand exactly which form of Lagrangian to use, or at least, how to apply either +λ or -λ in the analysis. It also matters whether you are maximizing or minimizing, and in your post you did not say which you were doing.
 
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  • #3
Ray Vickson said:
It makes no difference: one λ will just have the opposite sign of the other.

However, if does matter when you are doing post-optimality analysis. For example, you can use the value of λ to find the approximate change in the optimal value of F when the constraint changes to x+y = 11.1, for example. In that case you need to understand exactly which form of Lagrangian to use, or at least, how to apply either +λ or -λ in the analysis. It also matters whether you are maximizing or minimizing, and in your post you did not say which you were doing.

The goal was to find an optimal value subject to the constraint... Does it matter then, and how do you know which one to pick?

Also, both partial derivatives would be correct, right?
 
  • #4
939 said:
The goal was to find an optimal value subject to the constraint... Does it matter then, and how do you know which one to pick?

Also, both partial derivatives would be correct, right?

What is optimal? Maximum? Minimum?

I don't know what you mean when you ask if both partials are correct; you did not give formulas for the partials, so I have no way to know if they are correct or not.
 
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  • #5
Ray Vickson said:
What is optimal? Maximum? Minimum?

I don't know what you mean when you ask if both partials are correct; you did not give formulas for the partials, so I have no way to know if they are correct or not.

Sorry.

I mean if you were NOT asked to find a maximum or minimum, and were ONLY asked to find the lagrangian and the three partial derivatives - would these two answers be correct regardless of how you solved the constraint function?
 
  • #6
939 said:
Sorry.

I mean if you were NOT asked to find a maximum or minimum, and were ONLY asked to find the lagrangian and the three partial derivatives - would these two answers be correct regardless of how you solved the constraint function?

No: you need to solve the equations correctly! I think maybe you meant to say "regardless of which form of Lagrangian you use". Then the answer would be yes, if you make no errors during solving.

However: you don't really need to ask; you can just go ahead and do it both ways to see what you get. In fact, that would be faster than submitting a question and waiting for an answer!
 
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1. What is a Lagrangian?

A Lagrangian is a mathematical function that describes the dynamics of a physical system. It is used to determine the equations of motion for a system and is an important tool in classical mechanics.

2. How do you find the Lagrangian for a system?

To find the Lagrangian for a system, you need to identify the system's degrees of freedom and the corresponding generalized coordinates. Then, you can use the Lagrangian equation L = T - V, where T is the kinetic energy and V is the potential energy of the system, to determine the Lagrangian.

3. What is the importance of finding the Lagrangian?

Finding the Lagrangian is important because it allows us to describe the behavior of a physical system in terms of its generalized coordinates, rather than in terms of forces. This makes it easier to analyze and solve complex systems.

4. Can the Lagrangian be used in all types of physical systems?

Yes, the Lagrangian can be used in all types of physical systems, including classical mechanics, quantum mechanics, and field theory. It is a fundamental tool in theoretical physics and has many applications in various areas of science and engineering.

5. How do you check your work when finding the Lagrangian?

To check your work when finding the Lagrangian, you can use the Euler-Lagrange equations, which are a set of differential equations that describe the dynamics of a system in terms of its Lagrangian. If your Lagrangian is correct, it should satisfy these equations and give you the correct equations of motion for the system.

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