10.1. Functionals: definitions

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Chapter 10. Variational methods

10.1. Functionals, extremums and variations

  1. Functionals: definitions
  2. Variations of functionals
  3. Stationary points of functionals
  4. Extremums of functionals

Functionals: definitions

Definition 1. Functional is a map from some space of functions (or subset in the space of functions) $\mathsf{H}$ to $\mathbb{R}$ (or $\mathbb{C}$): \begin{equation} \Phi: \mathsf{H}\ni u\to \Phi[u]\in \mathbb{R}. \label{eq-10.1.1} \end{equation}

Remark 1. Important that we consider a whole function as an argument, not its value at some particular point!

Example 1.

  1. On the space $C(I)$ of continuos functions on the closed interval $I$ consider functional $\Phi[u]=u(a)$ where $a\in I$ (value at the point);
  2. On $C(I)$ consider functionals $\Phi[u]=\max_{x\in I} u(x)$, $\Phi[u]=\min_{x\in I} u(x)$ and $\Phi[u]=\max_{x\in I} |u(x)|$, $\Phi[u]=\min_{x\in I} |u(x)|$;
  3. Consider $\Phi[u]=\int_I f(x) u(x)\,dx$ where $f(x)$ is some fixed function.
  4. On the space $C^1(I)$ of continuos and continuously differentiable functions on the closed interval $I$ consider functional $\Phi[u]=u'(a)$.

Definition 2.

  1. Sum of functionals $\Phi_1+\Phi_2$ is defined as $(\Phi_1+\Phi_2)[u]=\Phi_1[u]+\Phi_2[u]$;
  2. Product of functional by a number: $\lambda \Phi$ is defined as $(\lambda \Phi)[u]=\lambda (\Phi[u])$;
  3. Function of functionals: $F(\Phi_1,\ldots,\Phi_s)$ is defined as $F(\Phi_1,\ldots,\Phi_s)[u]=F (\Phi_[u],\ldots,\Phi_s[u])$.

Definition 3. Functional $\Phi[u]$ is called linear if \begin{gather} \Phi [u+v]= \Phi[u]+\Phi [v],\label{eq-10.1.2}\\ \Phi [\lambda u]= \lambda \Phi[u] \label{eq-10.1.3} \end{gather} for all functions $u$ and scalars $\lambda$.

Remark 2. Linear functionals will be crucial in the definition of distributions later.

Exercise 1. Which functionals of Example 1. are linear?

Variations of functionals: 1-variable

We start from the classical variational problems: a single real valued function $q(t)$ of $t\in [t_0,t_1]$, then consider vector-valued function. This would lead us to ODEs (or their systems), and rightfully belongs to advanced ODE course.

Let us consider functional \begin{equation} S[q]= \int_{I} L(q(t),\dot{q}(t),t)\,dt \label{eq-10.1.4} \end{equation} where in traditions of Lagrangian mechanics we interpret $t\in I=[t_0,t_1]$ as a time, $q(t)$ as a coordinate, and $\dot{q}(t):=q'_t(t)$ as a velocity.

Let us consider $q+\delta q$ where $\delta q $ is a small function. We do not formalize this notion, just $\delta q=\varepsilon \varphi$ with fixed $\varphi$ and $\varepsilon\to 0$ is considered to be small. We call $\delta q$ variation of $q$ and important is that we change a function as a whole object. Let us consider

\begin{multline} \delta S:=S[q+\delta q]-S[q]= \int_I\Bigl(L(q+\delta q,\dot{q} + \delta \dot{q},t) -L(q,\dot{q},t)\Bigr)\,dt\\ \approx \int_I \Bigl(\frac{\partial L}{\partial q}\delta q + \frac{\partial L}{\partial \dot{q}}\delta \dot{q}\Bigr)\,dt\qquad \label{eq-10.1.5} \end{multline} where we calculated the linear part of expression in the parenthesis; if $\delta u=\varepsilon \varphi$ and all functions are sufficiently smooth then $\approx$ would mean equal modulo $o(\varepsilon)$ as $\varepsilon\to 0$.

Definition 4.

  1. Function $L$ we call Lagrangian.
  2. The right-hand expression of (\ref{eq-10.1.5}) which is a linear functional with respect to $\delta q$ we call variation of functional $S$.

Assumption 1. All functions are sufficiently smooth.

Under this assumption, we can integrate the right-hand expression of (\ref{eq-10.1.5}) by parts: \begin{multline} \delta S:= \int_I \Bigl(\frac{\partial L}{\partial q}\delta q + \frac{\partial L}{\partial \dot{q}}\delta \dot{q}\Bigr)\,dt\\ = \int_I\Bigl(\frac{\partial L}{\partial q} - \frac{d}{dt} \frac{\partial L}{\partial \dot{q}} u\Bigr)\delta u \,dt - \Bigl( \frac{\partial L}{\partial \dot{q}} \Bigr)\delta q \Bigr|_{t=t_0}^{t=t_1},\qquad \label{eq-10.1.6} \end{multline}

Stationary points of functionals

Definition 5. If $\delta S=0$ for all admissible variations $\delta q$ we call $q$ a stationary point or extremal of functional $S$.

Remark 3.

  1. We consider $q$ as a point in the functional space.
  2. In this definition we did not specify which variations are admissible. Let us consider as admissible all variations which are $0$ at both ends of $I$: \begin{equation} \delta q (t_0)=\delta q(t_1)=0. \label{eq-10.1.7} \end{equation} We will consider different admissible variations later.

In this framework \begin{equation} \delta S= \int_I\Bigl(\frac{\partial L}{\partial q} - \frac{d}{dt} \frac{\partial L}{\partial \dot{q}} \Bigr)\delta q \,dt . \label{eq-10.1.8} \end{equation}

Lemma 1. Let $f$ be a continuos function in $I$. If $\int_I f(t)\varphi(t)\,dt=0$ for all $\varphi$ such that $\varphi(t_0)=\varphi(t_1)=0$ then $f=0$ in $I$.

Proof. Indeed, let us assume that $f(\bar{t})> 0$ at some point $\bar{t}\in I$ (case $f(\bar{t})< 0$ is analyzed in the same way). Then $f(t)>0$ in some vicinity $\mathcal{V}$ of $\bar{t}$. Consider function $\varphi(x)$ which is $0$ outside of $\mathcal{V}$, $\varphi\ge 0$ in $\mathcal{V}$ and $\varphi(\bar{t})>0$. Then $f(t)\varphi(t)$ has the same properties and $\int_I f(t)\varphi(t)\, dt>0$. Contradiction!

As a corollary we arrive to

Theorem 1. Let us consider a functional (\ref{eq-10.1.4}) and consider as admissible all $\delta u$ satisfying (\ref{eq-10.1.7}). Then $u$ is a stationary point of $\Phi$ if and only if it satisfies Euler-Lagrange equation \begin{equation} \frac{\delta S}{\delta q}:= \frac{\partial L}{\partial q} - \frac{d}{dt} \left(\frac{\partial L}{\partial \dot{q}}\right) =0. \label{eq-10.1.9} \end{equation}

Remark 4.

  1. Equation (\ref{eq-10.1.9}) is the 2nd order ODE.
  2. If $L_{q}=0$ then it is integrates to \begin{equation} \frac{\partial L}{\partial \dot{q}}=C. \label{eq-10.1.10} \end{equation}
  3. The following equality holds: \begin{equation} \frac{d}{dt} \left(\frac{\partial L}{\partial \dot{q}}\dot{q}-L\right)=-\frac{\partial L}{\partial t}. \label{eq-10.1.11} \end{equation} The proof will be provided for vector-valued $\mathbf{q}(t)$.
  4. In particular, if $\frac{\partial L}{\partial t}=0$ ($L$ does not depend explicitely on $t$), then \begin{equation} \frac{d}{dt} \left(\frac{\partial L}{\partial \dot{q}}\dot{q}-L\right)=0\implies \frac{\partial L}{\partial \dot{q}}\dot{q}-L=C. \label{eq-10.1.12} \end{equation}

Extremums of functionals

Definition 6. If $S[q]\ge S[q+\delta q]$ for all small admissible variations $\delta q$ we call $u$ a local maximum of functional $S$. If $S[q]\le S[q+\delta q]$ for all small admissible variations $\delta q$ we call $q$ a local minimum of functional $S$.

Here again we do not specify what is small admissible variation.

Theorem 2. If $q$ is a local extremum (that means either local minimum or maximum) of $S$ and variation exits, then $q$ is a stationary point.

Proof. Consider case of minimum. Let $\delta q =\varepsilon \varphi$. Then $S[q+\delta q]- S [q]=\varepsilon (\delta S)(\varphi) +o(\varepsilon)$. If $\pm \delta S > 0$ then choosing $\mp \varepsilon < 0$ we make $\varepsilon (\delta S)(\varphi)\le -2\sigma \varepsilon$ with some $\sigma>0$. Meanwhile for sufficiently small $\varepsilon$ $o(\varepsilon)$ is much smaller and $S [q+\delta q]- S [q]\le -2\sigma \varepsilon<0$ and $q$ is not a local minimum.

Remark 5. We consider neither sufficient conditions of extremums nor second variations (similar to second differentials). In some cases they will be obvious.

$\Leftarrow$  $\Uparrow$  $\downarrow$  $\Rightarrow$