Lagrange multipliers in the calculus of variations

[entripon]

I'm looking for a derivation of the method of Lagrange multipliers as used in the calculus of variations for extremizing a functional subject to constraints. More specifically, I'm trying to understand the relationship between the "method of Lagrange multipliers" from standard calculus and the "Lagrange multipliers" introduced when solving for the generalized constraint forces of a system in Lagrangian dynamics.

The method as applied to standard functions has a nice, intuitive explanation: at the extremum point, the gradient of the function to be extremized must be equal to some multiple of the gradient of the constraint function. Is there a similar explanation in the case of functionals (specifically, as applied to the action functional)? I don't have access to a real text on the calculus of variations, and every online resource I've found (http://www.mpri.lsu.edu/textbook/Chapter8-b.htm" , for example) simply states the method without providing a derivation or explanation.

 

[Valerie Park]

The method of Lagrange multipliers as applied to functionals is essentially the same as when it is applied to standard functions. The only difference is that instead of dealing with a single function, we are dealing with an integral of a function over some region. At the extremum point, the gradient of the functional to be extremized must be equal to some multiple of the gradient of the constraint function. This can be written in terms of the integrand of the functional as:$$ \int \frac{\partial f}{\partial x} \cdot \lambda \cdot \frac{\partial g}{\partial x}\;dx = 0 $$where $f$ is the integrand of the functional, $g$ is the constraint, and $\lambda$ is a Lagrange multiplier. This implies that at the extremum point, the integrand of the functional must be proportional to the constraint:$$ \frac{\partial f}{\partial x} = \lambda \cdot \frac{\partial g}{\partial x} $$Using this relationship, the extremum point can be found by solving the equation above for $x$, which gives the desired extremum point. In the case of Lagrangian dynamics, the method of Lagrange multipliers is used to find the generalized constraint forces. Essentially, the same equation as above applies, but now $g$ is a constraint equation rather than a function. In this case, the Lagrange multipliers represent the magnitudes of the constraint forces.

 

[charng]

I understand your curiosity about the derivation and explanation of the method of Lagrange multipliers in the calculus of variations. This method, also known as the Lagrange multiplier method, is a powerful tool for finding the extremum of a functional subject to constraints. It is closely related to the method of Lagrange multipliers in standard calculus, but has some key differences that make it applicable to functionals.

To understand the relationship between the two methods, it is important to first understand the concept of a functional. A functional is a function that takes in another function as its input and returns a scalar value as its output. In other words, it maps a function to a number. The action functional, which is commonly used in the calculus of variations, takes in a path (a function of time) and returns the action (a scalar value) along that path.

Now, in order to extremize the action functional subject to constraints, we need to find the path that minimizes or maximizes the action while satisfying those constraints. This is where the method of Lagrange multipliers comes in. It allows us to introduce constraints into the calculus of variations by adding them as additional terms in the action functional.

The key idea behind the method of Lagrange multipliers is that at the extremum point, the variations (changes) in the action functional and the constraints must be equal. In other words, the constrained path must have the same variations as the unconstrained path. This is similar to the idea in standard calculus where the gradient of the function to be extremized is equal to the gradient of the constraint function at the extremum point.

To incorporate this idea into the calculus of variations, we introduce Lagrange multipliers as additional variables that act as multipliers for the variations in the constraints. These multipliers are then used to create a new functional, called the augmented functional, which includes the original action functional and the constraint functions multiplied by their respective Lagrange multipliers.

By finding the stationary points of this augmented functional (where the variations are equal), we can solve for the values of the Lagrange multipliers and the extremal path that satisfies the constraints.

In summary, the method of Lagrange multipliers in the calculus of variations is a way to incorporate constraints into the extremization of functionals by introducing Lagrange multipliers as multipliers for the variations in the constraints. This method is similar to the Lagrange multiplier method in standard calculus, but is adapted for function