Solve Lagrange Multiplier Mystery: ∂Σ{Ni}/∂Nj = ∂N/∂Nj=0

In summary, lagrange multiplier is a method for optimizing a function subject to constraints. The constraints mean that the variables are not independent from each other, but the way they are differentiated assumes they are, leading to a contradiction.
  • #1
kidsasd987
143
4
Hi, I have a question about lagrange multiplier

Let's say we are given with the following constraints

Σ{Ni}=N and Σ{NiEi}=total energy. N and total energy are constants by definition.
if we take the derivative with respect to Nj,

∂Σ{Ni}/∂Nj=∂N/∂Nj
where i=j, ∂Σ{Ni}/∂Nj=1 and ∂N/∂Nj = 0 because N is constant.

On slide 14, it says ∂N/∂Nj = 0 while ∂Σ{Ni}/∂Nj=1 with the preceding constraint Σ{Ni}=N.
Then, we can conclude

∂Σ{Ni}/∂Nj= ∂N/∂Nj=0.

This is quite ambiguous to me.
if we assume we have a constraint x1+x2+x3..+xn=const.

partial of this constraint with respect to xj will be 1=0 therefore it is contradiction.
how shoud I interpret this?
 

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  • #2
A constraint means that your variables are not independent from each other. So when I have ## x_1+x_2+\dots+x_n=const ##, it means when I decrease one of the ##x##s, another increases to maintain the constant. But the way you differentiate it, you assume they're independent variables which is a wrong assumption.
 
  • #3
ShayanJ said:
A constraint means that your variables are not independent from each other. So when I have ## x_1+x_2+\dots+x_n=const ##, it means when I decrease one of the ##x##s, another increases to maintain the constant. But the way you differentiate it, you assume they're independent variables which is a wrong assumption.

hmm I get it thanks .
 

Related to Solve Lagrange Multiplier Mystery: ∂Σ{Ni}/∂Nj = ∂N/∂Nj=0

1. What is a Lagrange multiplier?

A Lagrange multiplier is a method used in calculus to find the maximum or minimum value of a function subject to one or more constraints. It involves using a multiplier, also known as a lambda value, to find the optimal solution.

2. How is a Lagrange multiplier used to solve equations?

A Lagrange multiplier is used to solve equations by incorporating the constraints into the original objective function. This creates a new equation, which can then be solved using the method of partial derivatives.

3. What does the equation ∂Σ{Ni}/∂Nj = ∂N/∂Nj=0 represent?

This equation represents the condition for a Lagrange multiplier to be used. It states that the partial derivative of the objective function with respect to the variable Nj must be equal to 0 in order for a Lagrange multiplier to be applicable.

4. How does the Lagrange multiplier method work?

The Lagrange multiplier method works by creating a new equation, known as the Lagrangian, which includes the constraints and the objective function. This is then solved using the method of partial derivatives to find the optimal solution.

5. What are the applications of the Lagrange multiplier method?

The Lagrange multiplier method has various applications in mathematics and science, including optimization problems in economics, physics, and engineering. It is also used in statistics to find the maximum likelihood estimators and in machine learning for training algorithms.

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