Definition A.4.1.
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.
Subsection A.4.1 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.4.2 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*}
Remark A.4.3.
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.4.4.
The requirement that a function must be stationary gives rise to the Euler-Lagrange equation.
Variation.
Let \(\eta(x)\) be an admissible variation function and \(\varepsilon\) be a small parameter.
Definition A.4.5 Perturbation.
In calculus of variations, a perturbation is a small change
\begin{equation*}
\delta y = \epsilon \eta(x) \, .
\end{equation*}
Definition A.4.6 Variation.
A variation is an infinitesimal perturbation of an admissible path \(y(x)\text{.}\) A typical variation is written as
\begin{equation*}
y(x) \rightarrow y(x) + \underbrace{\varepsilon \eta(x)}_{\delta y} \, .
\end{equation*}
Remark A.4.7.
Variations determine how a functional changes in response to small changes in its input.
Definition A.4.8 First Variation.
The first variation is the first order change \(\delta J\) in the functional with respect to parameter \(\varepsilon\) after a perturbation \(\delta y\) is applied to \(y(x)\text{.}\)
Remark A.4.9.
First variation in calculus of variations is analogous to the first derivative in original calculus.
Paths.
Definition A.4.10 Path.
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.4.11 Admissible Path.
An admissible path is a path satisfying the boundary conditions along with any additional constraints for a particular variational problem.
Definition A.4.12 Stationary Point.
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*}
