2.7: Calculus of Variations with Many Variables

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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 \[\delta \mathbf{y}(x) \text { (writing also } \left.\mathbf{y}=\left(y_{1}, \ldots y_{n}\right)\right)\]

\[\delta J[\vec{y}]=\int_{x_{1}}^{x_{2}} \sum_{i=1}^{n}\left[\frac{\partial f\left(\vec{y}, \vec{y}^{\prime}\right)}{\partial y_{i}} \delta y_{i}(x)+\frac{\partial f\left(\vec{y}, \vec{y}^{\prime}\right)}{\partial y_{i}^{\prime}} \delta y_{i}^{\prime}(x)\right] d x=0\]

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:

\[\frac{\partial f\left(\vec{y}, \vec{y}^{\prime}\right)}{\partial y_{i}}-\frac{d}{d x}\left(\frac{\partial f\left(\vec{y}, \vec{y}^{\prime}\right)}{\partial y_{i}^{\prime}}\right)=0, \quad i=1, \ldots, n\]