26.3. Extra

26.3.1. Parametrization of the integral with the arc-length

Parametrization of the curve \(\gamma_{AB}: \, \mathbf{r}(s)\), with a physical parameter \(s \in [s_A, s_B]\) whose values at extreme points may depend on the solution itself: i.e. even though \(\mathbf{r}_A\), \(\mathbf{r}_B\) are prescribed, the values \(s_A\), \(s_B\), s.t. \(\mathbf{r}_A = \mathbf{r}(s_A)\), \(\mathbf{r}_B = \mathbf{r}(s_B)\), are not. The functional reads

\[S_{\gamma_{AB}} = S[\mathbf{r}(s)] = \int_{\mathbf{r} \in \gamma_{AB}} \mathcal{L}(\mathbf{r}, s) \, ds = \int_{s_A}^{s_B} \mathcal{L}(\mathbf{r}(s), s) \, ds \]

Method 1.

\[\begin{split}\begin{aligned} \Delta S_{\gamma_{AB}} & = S[\mathbf{r}(s_\delta)+\varepsilon \delta\mathbf{r}(s_\delta)] - S[\mathbf{r}(s)+\varepsilon \delta\mathbf{r}(s)] = \\ & = \int_{\mathbf{r} \in \mathbf{r}(s_\delta)+\varepsilon \delta \mathbf{r}(s_\delta)} \mathcal{L}(\mathbf{r}, s_{\delta}) \, ds_{\delta} - \dots = \\ \end{aligned}\end{split}\]

Here \(d s_{\delta}\) is the arc-length of the elementary vector of the curve \(\gamma_\delta\), \(d \mathbf{r}_{\delta} = d \mathbf{r} + \varepsilon d \delta \mathbf{r}\), i.e.

\[\begin{split}\begin{aligned} d s_{\delta}^2 & = \left( d \mathbf{r} + \varepsilon d \delta \mathbf{r} \right) \cdot \left( d \mathbf{r} + \varepsilon d \delta \mathbf{r} \right) = \\ & = | d \mathbf{r} |^2 + 2 \varepsilon d \mathbf{r} \cdot d \delta \mathbf{r} + o(\varepsilon) = \\ & \simeq | d \mathbf{r} |^2 \left( 1 + 2 \varepsilon \dfrac{d \delta \mathbf{r} \cdot d \mathbf{r}}{|d \mathbf{r}|^2} \right) + o(\varepsilon) \ , \end{aligned}\end{split}\]

so that first-order approximation reads

\[\begin{split}\begin{aligned} d s_{\delta} & = | d \mathbf{r} | \left( 1 + \varepsilon \dfrac{d \delta \mathbf{r} \cdot d \mathbf{r}}{|d \mathbf{r}|^2} \right) = \\ & = ds + \varepsilon \hat{\mathbf{t}} \cdot d \delta \mathbf{r} \ . \end{aligned}\end{split}\]

The first integral in the difference thus becomes

\[\begin{split}\begin{aligned} & \int_{s_{\delta} = s_{\delta,A}}^{s_{\delta,B}} \mathcal{L}(\mathbf{r}_\delta(s_\delta), s_\delta ) \left( ds + \varepsilon \hat{\mathbf{t}} \cdot d \delta \mathbf{r} \right) \\ & \qquad = \int_{s_{\delta} = s_{\delta,A}}^{s_{\delta,B}} \left[ \mathcal{L}(\mathbf{r}(s_\delta), s_\delta ) + \varepsilon \delta \mathbf{r} \cdot \nabla \mathcal{L}(\mathbf{r}(s_\delta), s_\delta) + o(\varepsilon) \right] \left( \dfrac{ds}{d s_\delta} + \varepsilon \hat{\mathbf{t}} \cdot \frac{d \delta \mathbf{r}}{d s_\delta} \right) d s_{\delta} = \\ & \qquad = \int_{s = s_{A}}^{s_{B}} \mathcal{L}(\mathbf{r}(s), s) \, ds + \varepsilon \int_{s = s_{A}}^{s_{B}} \delta \mathbf{r} \cdot \nabla \mathcal{L}(\mathbf{r}(s), s) \, ds + \\ & \qquad + \varepsilon \left.\left[ \delta \mathbf{r} \cdot \hat{\mathbf{t}} \mathcal{L} \right]\right|_{A}^{B} - \varepsilon \int_{s_\delta = s_{\delta,A}}^{s_{\delta,B}} \delta \mathbf{r} \cdot \dfrac{d}{ds}\left( \hat{\mathbf{t}} \mathcal{L} \right) d s_\delta + o(\varepsilon) = \\ & \qquad = \int_{s = s_{A}}^{s_{B}} \mathcal{L}(\mathbf{r}(s), s) \, ds + \varepsilon \int_{s = s_{A}}^{s_{B}} \delta \mathbf{r} \cdot \left\{ \nabla \mathcal{L}(\mathbf{r}(s), s) - \dfrac{d}{ds_\delta}\left( \hat{\mathbf{t}} \mathcal{L} \right) \right\} \, ds + \\ & \qquad + \varepsilon \left.\left[ \delta \mathbf{r} \cdot \hat{\mathbf{t}} \mathcal{L} \right]\right|_{A}^{B} + o(\varepsilon) \ . \end{aligned}\end{split}\]

Thus, the variation becomes

\[\begin{split}\begin{aligned} \delta_{\mathbf{r}} S & = \lim_{\varepsilon \rightarrow 0} \dfrac{1}{\varepsilon} \Delta_{\mathbf{r}} S = \\ & = \int_{s = s_{A}}^{s_{B}} \delta \mathbf{r} \cdot \left\{ \nabla \mathcal{L}(\mathbf{r}(s), s) - \dfrac{d}{ds}\left( \hat{\mathbf{t}} \mathcal{L} \right) \right\} \, ds + \left.\left[ \delta \mathbf{r} \cdot \hat{\mathbf{t}} \mathcal{L} \right]\right|_{A}^{B} \end{aligned}\end{split}\]

having assumed a function \(s(s_\delta)\) associating every point on curve \(\gamma_{AB}\) to a value of the parameter \(s_{\delta}\), and having used the rules for changing independent variables in the integrals, \(\frac{ds}{ds_{\delta}} ds_{\delta} = d_s\), and for the derivative of compsite functions \(\frac{d}{ds_\delta} = \frac{d s}{d s_\delta} \frac{d}{d s}\).

Fermat principle

Following Fermat principle, the differential equations of the trajectory of a light ray come from the condition

\[\delta \int_{\gamma} \frac{1}{c} \, ds = \frac{1}{c_0} \delta \int_{\gamma} n \, ds\ .\]

Here the integrand is \(\mathcal{L}(\mathbf{r}(s),s) = n(\mathbf{r}(s),s)\), and following the general formulation of the variational problem with prescribed ends \(\delta \mathbf{r}|_{A,B} = \mathbf{0}\) gives

\[\begin{split}\begin{aligned} \mathbf{0} & = \nabla \mathcal{L} - \dfrac{d}{ds} \left( \hat{\mathbf{t}} \mathcal{L} \right) = \\ & = \nabla \mathcal{L} - \kappa \hat{\mathbf{n}} \mathcal{L} - \hat{\mathbf{t}} \hat{\mathbf{t}} \cdot \mathcal{L} = \\ & = \mathbb{P}_{\perp \hat{\mathbf{t}}} \cdot \nabla \mathcal{L} - \kappa \hat{\mathbf{n}} \mathcal{L} \ , \end{aligned}\end{split}\]

i.e. the very same equations as (26.1), having recalled that the derivative of the tangent unit vector to a line w.r.t. the arc-length is proportional to the unit normal vector, through the curvature, \(\frac{d \hat{\mathbf{t}}}{d s}= \kappa \hat{\mathbf{n}}\).

Method 2.

26.3.2. Integral with non prescirbed ends