Definition A.3.1.
A function space \(\mathcal{F}\) is a collection of functions \(y\in\mathcal{F}\) that share the same domain and codomain, and specified mathematical properties.
Section A.3 Calculus of Variations
Subsection A.3.1 Calculus of Variations Primer
Function Space.
Functional.
A functional is a mapping \(J : \mathcal{F} \to \mathbb{R}\text{,}\) where \(\mathcal{F}\) denotes a function space.
Definition A.3.2. Standard Form of a Functional.
In the calculus of variations, the standard form of a functional \(J[y]\) is typically written as
\begin{equation*}
J[y] = \int_{x_1}^{x_2}{F\big(x,y(x),y_x(x)\big)\,dx} \, .
\end{equation*}
Definition A.3.3.
A functional is said to be stationary at a point along an admissible path \(y=f(x)\) if all first order changes to that path at that point are zero. This is mathematically represented as
\begin{equation*}
\delta J = 0 \, .
\end{equation*}
Remark A.3.4.
A point along an admissible path that gives a stationary functional is not necessarily a minimum or maximum. It may also be a saddle point.
Remark A.3.5.
The requirement that a function must be stationary gives rise to the Euler-Lagrange equation.
Path.
Definition A.3.6.
A path is a function \(f(x_1,x_2,\dots)\) that assigns the value of one or more dependent variables to each value of an independent variable \(y=f(x_1,x_2,\dots)\text{.}\) In the context of calculus of variations, a path is the object over which a functional is optimized.
Definition A.3.7.
An admissible path is a path satisfying the boundary conditions along with any additional constraints for a particular variational problem.
Subsection A.3.2 Calculus of Variations Formulation of the Euler-Lagrange Equation
Definition A.3.8. The Euler-Lagrange Equation.
For a functional \(J[y] := \int_{x_1}^{x_2}{F\big(x,y(x),y_x(x)\big) \, dx}\text{,}\) the Euler-Lagrange Equation is
\begin{equation*}
\frac{\partial F}{\partial x} - \frac{d}{dx} \big( \frac{\partial F}{\partial y_x} \big) = 0
\end{equation*}
Remark A.3.9.
The Euler-Lagrange equation provides the necessary conditions for extremizing a path.
