2.7: Calculus of Variations with Many Variables
- Page ID
- 29529
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)We’ve found the equations defining the curve \(y(x)\) along which the integral
\[J[y]=\int_{x_{1}}^{x_{2}} f\left(y, y^{\prime}\right) d x\]
has a stationary value, and we’ve seen how it works in some two-dimensional curve examples.
But most dynamical systems are parameterized by more than one variable, so we need to know how to go from a curve in \((x,y)\) to one in a space \(\left(x, y_{1}, y_{2}, \ldots y_{n}\right)\), and we need to minimize (say)
\[J\left[y_{1}, y_{2}, \ldots y_{n}\right]=\int_{x_{1}}^{x_{2}} f\left(y_{1}, y_{2}, \ldots y_{n}, y_{1}^{\prime}, y_{2}^{\prime}, \ldots y_{n}^{\prime}\right) d x\]
In fact, the generalization is straightforward: the path deviation simply becomes a vector,
\[\delta \vec{y}(x)=\left(\delta y_{1}(x), \delta y_{2}(x), \ldots, \delta y_{n}(x)\right)\]
Then under any infinitesimal variation \[\delta \mathbf{y}(x) \text { (writing also } \left.\mathbf{y}=\left(y_{1}, \ldots y_{n}\right)\right)\]
\[\delta J[\vec{y}]=\int_{x_{1}}^{x_{2}} \sum_{i=1}^{n}\left[\dfrac{\partial f\left(\vec{y}, \vec{y}^{\prime}\right)}{\partial y_{i}} \delta y_{i}(x)+\dfrac{\partial f\left(\vec{y}, \vec{y}^{\prime}\right)}{\partial y_{i}^{\prime}} \delta y_{i}^{\prime}(x)\right] d x=0\]
Just as before, we take the variation zero at the endpoints, and integrate by parts to get now n separate equations for the stationary path:
\[\dfrac{\partial f\left(\vec{y}, \vec{y}^{\prime}\right)}{\partial y_{i}}-\dfrac{d}{d x}\left(\dfrac{\partial f\left(\vec{y}, \vec{y}^{\prime}\right)}{\partial y_{i}^{\prime}}\right)=0, \quad i=1, \ldots, n\]