Lagrange Equations

Lagrange Multipliers

The method of Lagrange multipliers is a strategy to find the local maxima and minima of a function subject to one or more constraints. It introduces an auxiliary variable, the Lagrange multiplier, to convert a constrained optimization problem into a system of equations.

The Lagrange Multiplier Method

For a function $f(x,y,z)$ subject to the constraint $g(x,y,z)=0$, the Lagrange multiplier method involves solving the following system:

$$\nabla f(x,y,z)=\lambda \nabla g(x,y,z)$$

Where:

• $\nabla f(x,y,z)$ is the gradient of the function we want to optimize.
• $\nabla g(x,y,z)$ is the gradient of the constraint.
• $\lambda$ is the Lagrange multiplier.

In essence, the gradients of $f$ and $g$ must be parallel, reflecting that the rate of change of $f$ in the direction of the constraint is zero.

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Example: Finding Extrema

Find the extrema of the function $f(x,y)=x^2+y^2$ subject to the constraint $g(x,y)=x+y-1=0$.

• Step 1: Set up the Lagrange system:

  $$\nabla f=\lambda \nabla g$$

  This gives:

  $$(2x,2y)=\lambda (1,1)$$

• Step 2: Solve for $\lambda$:

  $$2x=\lambda \text{and}2y=\lambda$$

  So, $x=y$.

• Step 3: Use the constraint $x+y=1$ to find $x$ and $y$:

  $$x+x=1⟹x=\frac{1}{2},y=\frac{1}{2}$$

• Result: The extrema occur at $(\frac{1}{2},\frac{1}{2})$, with $f(x,y)=\frac{1}{2}$.

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Geometric Interpretation

Lagrange multipliers provide a way of finding points where the function’s gradient and the constraint’s gradient are aligned, meaning that any further movement within the constraint will not increase or decrease the function’s value.

Lagrange Equations for Multiple Constraints

For problems with multiple constraints, say $g_1(x,y,z)=0$ and $g_2(x,y,z)=0$, the system of Lagrange equations becomes:

$$\nabla f(x,y,z)={\lambda }_1\nabla g_1(x,y,z)+{\lambda }_2\nabla g_2(x,y,z)$$

Where:

• ${\lambda }_1$ and ${\lambda }_2$ are the Lagrange multipliers associated with each constraint.
• This leads to a system of equations that can be solved to find the critical points of $f$ subject to the given constraints.

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Key Takeaways

• Lagrange multipliers are useful for finding extrema of functions under constraints.
• The method works by finding where the gradients of the function and the constraint(s) are parallel.
• It generalizes to multiple constraints by introducing additional multipliers.

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This structure can be expanded with more examples or exercises on using Lagrange multipliers in optimization problems.