Calculus of Variations
Table of Contents
- Variational Calculus
- Deals with the changes in functions and functionals, instead of changes in variables.
1. Variation
- If \(f\) is a function of independent variables, \(\delta f = \varepsilon\eta\) is the small perturbation of \(f\).
- The variation can be formalized with the Fréchet derivative of \(f\).
2. Fundamental Lemma of the Calculus of Variations
2.1. Statement
2.1.1. Single Function
- For a continuous function \(f\) on an open interval \((a,b)\), if \[ \int_{a}^{b} f(x)h(x)\, dx=0 \] for all compactly supported smooth functions \(h\) on \((a,b)\), then \[ f(x)\equiv 0 \] on the interval \((a,b)\).
2.1.2. Multiple Functions
- For a tuple of continuous functions \((f_i)_{i=0}^n\) on an open
interval \((a,b)\), if
- \[ \int_a^b ( f_0(x) \, h(x) + f_1(x) \, h'(x) + \dots + f_n(x) \, h^{(n)}(x) ) \, \mathrm{d}x = 0 \]
- for all compactly supported smooth function \(h\) on \((a,b)\),
- Then there exists the tuple of continuously differentiable
functions \((u_i)_{i=0}^{n-1}\) on \((a,b)\), such that
\[
\begin{align*} f_0 &= u_0',\\ f_i &= u_{i-1} + u_i',\\ f_n &= u_{n-1}. \end{align*}\]
2.1.3. Vector-Valued Function
- For a continuous vector-valued function \(f\colon (a,b) \to \mathbb{R}^d\),
- It is identically the zero vector.
2.1.4. Multivariable Function
- For a continuous multivariable function \(f\colon \Omega\subset \mathbb{R}^d \to \mathbb{R}\) on an open set \(\Omega\).
- \(f\) is identically zero.
2.2. Properties
- The restriction on \(h\) can be loosened up to a function that vanished at the endpoints, \(a\) and \(b\), since the lemma directly implies that.
3. Euler-Lagrange Equation
3.1. Equation
- For a function \(L(q, \dot{q}, t)\),
- \[ \frac{ \partial L }{ \partial q }- \frac{d}{dt}\frac{ \partial L }{ \partial \dot{q} } =0. \]
- This implies that the function \(q\) which satisfies this equation is \[ \operatorname*{arg min}_{q} \left( \int_{a}^{b} L(x,q,q') \, dx \right). \]
3.2. Derivation
3.2.1. Formally
Let \(y(x) = y_0(x) + \varepsilon \eta(x)\). This allows the study the rate of change instead of the infinitesimal variation, which is a standard practice at studying differential calculus.
The local extremum is attained if \[ \frac{d}{d\varepsilon}\int_a^b L(x,y(x),y'(x))\, dx\bigg|_{\varepsilon=0} = 0. \]
\begin{align*} \frac{d}{d\varepsilon}\int_a^b L\, dx\bigg|_{\varepsilon=0} &= \int_a^b \frac{dL}{d\varepsilon}\bigg|_{\varepsilon=0}dx\\ &= \int_a^b \frac{\partial L}{\partial y}\frac{dy}{d\varepsilon} + \frac{\partial L}{\partial y'}\frac{dy'}{d\varepsilon}\bigg|_{\varepsilon=0} dx\\ &= \int_a^b \frac{\partial L}{\partial y}\eta + \frac{\partial L}{\partial y'}\eta' dx\\ &= \int_a^b \frac{\partial L}{\partial y}\eta\,dx + \frac{\partial L}{\partial y'}\eta\bigg|_a^b - \int_a^b \frac{d}{dx}\left(\frac{\partial L}{\partial y'}\right)\eta\,dx\\ &= \int_a^b \left(\frac{\partial L}{\partial y} - \frac{d}{dx}\frac{\partial L}{\partial y'}\right)\eta\,dx \end{align*}3.2.2. Variation Notation
\begin{align*}
0&=\delta \int_{a}^{b} L \, dx \\
&=\int_{a}^{b} \delta L \, dx \\
&=\int_{a}^{b} \frac{ \partial L }{ \partial q } \delta q+\frac{ \partial L }{ \partial q' } \delta q' \, dx \\
&=\int_{a}^{b} \frac{ \partial L }{ \partial q } \delta q +\frac{ \partial L }{ \partial q' } \frac{ d }{ dx } (\delta q) \, dx \\
&=\int_{a}^{b} \frac{ \partial L }{ \partial q } \delta q\, dx + \frac{ \partial L }{ \partial q' } \delta q \,\Bigg|_{a}^{b}-\int_{a}^{b} \frac{ d }{ dx }\left( \frac{ \partial L }{ \partial q' } \right) \delta q \, dx \\
&=\int_{a}^{b} \left( \frac{ \partial L }{ \partial q } - \frac{ d }{ dx } \frac{ \partial L }{ \partial q' } \right) \delta q \, dx
\end{align*}
By the fundamental lemma of the calculus of variations, \[ \frac{ \partial L }{ \partial q }- \frac{d}{dx}\frac{ \partial L }{ \partial q' } =0.\quad\square \]
3.3. Properties
- It implicitly contains \[ \frac{dq}{dx} = q'. \]
- Which together with the Euler-Lagrange equation becomes the
Hanilton's equations via Legendre transformation.
- \[ p := \frac{\partial L}{\partial \dot{q}} \]
- \[ \dot{q} \leadsto \frac{\partial H}{\partial p} \]
- Given a functional of the form:
\[
\int_a^b L(x, f(x), f'(x))\,dx
\]
- \(L\) can be as well be considered as a generalized Lagrangian.1
4. Euler-Poisson Equation
- If \(S\) depends on higher-derivatives of \(y(x)\): \[ S = \int_a^b f(x, y(x), y'(x), \dots, y^{(n)}(x))dx, \]
- and it attains the maximum with \(y(x)\), then \[ \frac{\partial f}{\partial y} - \frac{d}{dx}\left(\frac{\partial f}{\partial y'}\right) + \cdots + (-1)^n\frac{d^n}{dx^n}\left(\frac{\partial f}{\partial y^{(n)}}\right) = 0. \]
- See the fundamental lemma for fundamental lemma for multiple functions
5. Beltrami's Identity
- If \(L\) does not depend on \(x\), the Euler-Lagrange equation
simplifies to
\[
L-y'\frac{\partial L}{\partial y'} = C
\]
where \(C\) is a constant.
- Multiply \(y'\) to the Euler-Lagrange equation, and integrate with respect to \(x\).
- The left hand side is the Legendre transformation of \(L\) with respect to \(y'(x)\).
- This leads to the Hamiltonian mechanics.