26.2. Examples

26.2.1. Shortest path

Shortest path between two given points \(A\), \(B\). The length of a curve \(\gamma_{AB}\) connecting the two points reads

\[L_{\gamma_{AB}} = \int_{\gamma_{AB}} ds \ ,\]

with \(s\) the arc-length parameter. The value of \(s\) at the extremes of integration is not independent on the curve \(\gamma_{AB}\), and thus on the result. In order to write the integrals w.r.t. a parameter with given values at the extreme points, a change of parameter is required. Let’s define a parameter \(\ell\), so that the curve in space can be represented as \(\gamma: \, \mathbf{r}(t)\), \(t \in [t_A, t_B]\), with \(t_A\), \(t_B\) given, and \(\mathbf{r}(t_A) = \mathbf{r}_A\), \(\mathbf{r}(t_B) = \mathbf{r}_B\) given. The elementary length \(ds\) becomes

\[| d \mathbf{r} | = \underbrace{| \mathbf{r}'(s)|}_{=1, \, \text{by def of $s$}} ds = | \mathbf{r}'(\ell) | \, d \ell \ .\]

The integral thus becomes

\[L_{\gamma_{AB}} = L[\mathbf{r}(\ell)] = \int_{\ell_A}^{\ell_B} |\mathbf{r}'(\ell)| \, d \ell \ ,\]

where the dependence on the curve \(\mathbf{r}(\ell)\) is made explicit, and the values of the parameter \(\ell\) at the extreme is given. Variation of the functional reads

\[\begin{split}\begin{aligned} \delta L & = \delta \int_{\ell_A}^{\ell_B} |\mathbf{r}'(\ell)| \, d \ell = \\ & = \int_{\ell_A}^{\ell_B} \delta |\mathbf{r}'(\ell)| \, d \ell = \\ & = \int_{\ell_A}^{\ell_B} \delta \mathbf{r}' \cdot \dfrac{\mathbf{r}'}{|\mathbf{r}'|} d \ell = \\ & = \underbrace{ \left.\delta \mathbf{r} \cdot \hat{\mathbf{t}} \right|_{\ell_A}^{\ell_B}}_{= 0} - \int_{\ell_A}^{\ell_B} \delta \mathbf{r} \cdot \dfrac{d \hat{\mathbf{t}}}{d \ell} d \ell \ , \end{aligned}\end{split}\]

being \(\hat{\mathbf{t}}(\ell) = \frac{\mathbf{r}'(\ell)}{|\mathbf{r}'(\ell)|}\), the unit-length tangent vector to the curve. Stationariety of \(L\), for any possible variation \(\delta \mathbf{r}(\ell)\) implies

\[\dfrac{d \hat{\mathbf{t}}}{d \ell} = \mathbf{0} \ ,\]

Thus, the solution reads \(\mathbf{r}(\ell) = \alpha \hat{\mathbf{t}} \, \ell + \mathbf{r}_0\), and prescribing the boundary conditions

\[\mathbf{r}(\ell) = \mathbf{r}_A + \frac{\ell-\ell_A}{\ell_B-\ell_A} \left( \mathbf{r}_B-\mathbf{r}_A \right) \ .\]

Details about the integration

26.2.2. Fermat principle

In geometrical optics, Fermat principles asserts that a light ray between two points represent the path with the shortest travelling time connecting them. Let \(\gamma\) a curve in space. The travelling time of a light ray with speed \(c\) reads

\[T = \int_{\gamma_{AB}} dt = \int_{\gamma_{AB}} \dfrac{1}{c} d s \ ,\]

begin \(ds = c dt\). Let \(\ell\) be a parameter, so that \(\gamma: \, \mathbf{r}(\ell)\) is a parametrization of the curve, with the extreme points independent from the result. The speed of light may depend on the position in space, \(c(\mathbf{r})\), and it can be written as a function of the speed of light in vacuum and the refractive index \(n(\mathbf{r})\) as \(c(\mathbf{r}) = \frac{c_0}{n(\mathbf{r})}\). Making the dependence on \(\mathbf{r}(\ell)\), \(\mathbf{r}'(\ell)\) explicit in the functional,

\[T[\mathbf{r}(\ell)] = \int_{\ell_A}^{\ell_B} n(\mathbf{r}(\ell)) \, |\mathbf{r}'(\ell)| \, d \ell \ .\]

Fermat principle reads

\[\begin{split}\begin{aligned} 0 & = \delta T = \\ & = \delta \int_{\ell_A}^{\ell_B} n(\mathbf{r}(\ell)) \, |\mathbf{r}'(\ell)| \, d \ell = \\ & = \int_{\ell_A}^{\ell_B} \left\{ \delta \mathbf{r} \cdot \nabla_{\mathbf{r}} n |\mathbf{r}|' + \delta \mathbf{r}' \cdot \frac{\mathbf{r}'}{|\mathbf{r}'|} n(\mathbf{r}) \right\} \, d \ell = \\ & = \int_{\ell_A}^{\ell_B} \delta \mathbf{r} \cdot \left\{ \nabla_{\mathbf{r}} n \, |\mathbf{r}|' - \dfrac{d}{d \ell} \left( \frac{\mathbf{r}'}{|\mathbf{r}'|} n(\mathbf{r}) \right) \right\} \, d \ell + \underbrace{ \left. \delta \mathbf{r} \cdot \frac{\mathbf{r}'}{|\mathbf{r}'|} n(\mathbf{r}) \right|_{\ell_A}^{\ell_B}}_{=0} \ , \end{aligned}\end{split}\]

From the arbitrarieness of \(\delta \mathbf{r}\),

\[\begin{split}\begin{aligned} 0 & = \dfrac{d}{d \ell} \left( n(\mathbf{r}(\ell)) \dfrac{\mathbf{r}'(\ell)}{|\mathbf{r}'(\ell)|} \right) - |\mathbf{r}'(\ell)| \, \nabla n(\mathbf{r}(\ell)) = \\ & = \dfrac{\mathbf{r}'}{|\mathbf{r}|} \mathbf{r}' \cdot \nabla n - n \dfrac{d}{d \ell} \left( \dfrac{\mathbf{r}'}{|\mathbf{r}'|} \right) - |\mathbf{r}'| \nabla n = \\ & = | \mathbf{r}' | \left[ \mathbb{I} - \dfrac{\mathbf{r}' \cdot \mathbf{r}'}{|\mathbf{r}'|^2} \right] \cdot \nabla n - n \dfrac{d}{d\ell} \left( \dfrac{\mathbf{r}'}{|\mathbf{r}'|} \right) \ . \end{aligned}\end{split}\]

and thus

\[\mathbb{P}_{\perp \hat{\mathbf{t}}} \cdot \nabla n = \dfrac{n}{|\mathbf{r}'|} \dfrac{d \hat{\mathbf{t}}}{d \ell} \ ,\]

or, re-introducting the “physical” variable \(s\) (the arc-length is the parameter with physical, geometrical, non arbitrary meaning; the equations should be invariant from the parametrization, so we should be happy of the following result, written in invariant form),

(26.1)\[\mathbb{P}_{\perp \hat{\mathbf{t}}} \cdot \nabla n = n \dfrac{d \hat{\mathbf{t}}}{ds} \ .\]

being \(\mathbb{P}_{\perp \hat{\mathbf{t}}}\) the orthogonal projector in the direction perpendicular to the unit tangent vector \(\hat{\mathbf{t}}\). Using the results of geometry of curves, the derivative \(\hat{\mathbf{t}}'(s) = \kappa(s) \hat{\mathbf{n}}(s)\), being \(\hat{\mathbf{n}}\) the unit normal vector pointing towards the local center of curvature (center of the osculator circle, tangent with the same second order derivative), and \(\kappa(s) = \frac{1}{R(s)}\) is the local curvature, and \(R(s)\) the radius of curvature (the radius of the osculator circle). Thus

\[\kappa \hat{\mathbf{n}} = \left[ \mathbb{I} - \hat{\mathbf{t}} \otimes \hat{\mathbf{t}} \right] \cdot \frac{\nabla n}{n}\]

Projecting this equation on the local Frenet basis \(\{ \hat{\mathbf{t}}, \hat{\mathbf{n}}, \hat{\mathbf{b}} \}\),

\[\begin{split}\left\{ \begin{aligned} t: & \, 0 = \frac{1}{n} \hat{\mathbf{t}} \cdot \nabla n \\ n: & \, \kappa = \frac{1}{n} \hat{\mathbf{n}} \cdot \nabla n \\ b: & \, 0 = \frac{1}{n} \hat{\mathbf{b}} \cdot \nabla n \\ \end{aligned} \right.\end{split}\]

26.2.3. Lagrange equations in classical mechanics

26.2.4. Lagrange equations in special relativity

26.2.4.1. Using tensor formalism

Equations of motion and physical principles must be written using physical properties, and thus be independent from an arbitrary parametrization.

Strong formulation. Let the equation of motion of a particle be

\[m \dfrac{d \mathbf{U}}{d \tau} = \mathbf{K} \ ,\]

with \(m\) the rest mass, \(\tau\) the proper time, \(\mathbf{U} = \frac{d \mathbf{X}}{d \tau}\) the 4-velocity, and \(\mathbf{K}\) the 4-force.

Weak formulation. Multiplying by an arbitrary test 4-vector \(\mathbf{W}(\tau)\) and integrating over an arbitrary interval \(\tau \in [ \tau_0, \tau_1]\),

\[\begin{aligned} 0 & = \int_{\tau_0}^{\tau_1} \mathbf{W} \cdot \left\{ m \dfrac{d \mathbf{U}}{d \tau} - \mathbf{K} \right\} d \tau \ . \end{aligned}\]

The parametrization of the trajectory is changed from \(\tau\) to an arbitrary parameter \(\lambda\), so that \(\mathbf{X}_{0,1} = \mathbf{X}(\tau_{0,1}) = \mathbf{X}(\tau(\lambda_{0,1}))\) are prescribed for given values \(\lambda_{0,1}\). As the invariant \(d \tau\) is defined through

\[c^2 d \tau^2 = d s^2 = d \mathbf{X} \cdot d \mathbf{X} = \mathbf{X}'(\lambda) \cdot \mathbf{X}'(\lambda) \, d \lambda^2 \ ,\]

the relation between the differentials and the rule of derivation of composite functions read

\[d \tau = \dfrac{\sqrt{\mathbf{X}' \cdot \mathbf{X}'}}{c} \, d \lambda \qquad , \qquad \dfrac{d}{d \tau} = \dfrac{d \lambda}{d \tau} \dfrac{d}{d \lambda} = \dfrac{c}{\sqrt{\mathbf{X}' \cdot \mathbf{X}'}} \dfrac{d}{d \lambda} \ .\]

The velocity vector becomes

\[\mathbf{U} = \dfrac{d \mathbf{X}}{d \tau} = \dfrac{d \lambda}{d \tau} \dfrac{d \mathbf{X}}{d \lambda} = \dfrac{c}{\sqrt{\mathbf{X}' \cdot \mathbf{X}'}} \mathbf{X}' \ . \]

The integral thus becomes

\[\begin{split}\begin{aligned} 0 & = \int_{\tau_0}^{\tau_1} \mathbf{W} \cdot \left\{ m \dfrac{d \mathbf{U}}{d \tau} - \mathbf{K} \right\} d \tau = \\ & = \int_{\lambda_0}^{\lambda_1} \mathbf{W} \cdot \left\{ m \dfrac{c}{\sqrt{\mathbf{X}' \cdot \mathbf{X}'}} \left( \dfrac{c}{\sqrt{\mathbf{X}' \cdot \mathbf{X}'}} \mathbf{X}' \right)' - \mathbf{K} \right\} \dfrac{\sqrt{\mathbf{X}' \cdot \mathbf{X}'}}{c} d \lambda = \\ & = \int_{\lambda_0}^{\lambda_1} \mathbf{W} \cdot \left( \dfrac{mc}{\sqrt{\mathbf{X}' \cdot \mathbf{X}'}} \mathbf{X}' \right)' \, d \lambda - \int_{\lambda_0}^{\lambda_1} \mathbf{W} \cdot \mathbf{K} \dfrac{\sqrt{\mathbf{X}' \cdot \mathbf{X}'}}{c} d \lambda = \\ \end{aligned}\end{split}\]

Lagrange mechanics immediately follows choosing the test function \(\mathbf{W} = \delta \mathbf{X}\).

• Free particle. For a free particle, \(\mathbf{K} = \mathbf{0}\).

  \[\begin{split}\begin{aligned} 0 & = \int_{\lambda_0}^{\lambda_1} \delta \mathbf{X} \cdot \left( \dfrac{mc}{\sqrt{\mathbf{X}' \cdot \mathbf{X}'}} \mathbf{X}' \right)' \, d \lambda = \\ & = \underbrace{\left. \left[ \delta \mathbf{X} \cdot \dfrac{mc}{\sqrt{\mathbf{X}' \cdot \mathbf{X}'}} \mathbf{X}' \right) \right|_{\lambda_0}^{\lambda_1}}_{= 0} - \int_{\lambda_0}^{\lambda_1} \delta \mathbf{X}' \cdot \dfrac{mc}{\sqrt{\mathbf{X}' \cdot \mathbf{X}'}} \mathbf{X}' \, d \lambda = \\ & = - \delta \int_{\lambda_0}^{\lambda_1} mc \, \sqrt{\mathbf{X}' \cdot \mathbf{X}'} \, d \lambda = \\ & = - \delta \int_{\tau_0}^{\tau_1} mc^2 \, d \tau = \\ & = - \delta \int_{s_0}^{s_1} mc \, d s = \\ & = \delta S \ , \end{aligned}\end{split}\]

  where the change of the independent parameters is made after the variation is put outside the integral \(\int_{\lambda_0}^{\lambda_1}\), with given extreme values.

• Particle subjecd to Lorentz force,

  \[\mathbf{K} = q \mathbf{F}(\mathbf{X}) \cdot \mathbf{U} = q \mathbf{F}(\mathbf{X}) \frac{c}{\sqrt{\mathbf{X}' \cdot \mathbf{X}'}} \mathbf{X}' \ .\]

  The second integral becomes

  \[\begin{split}\begin{aligned} - \int_{\lambda_0}^{\lambda_1} \delta \mathbf{X} \cdot \mathbf{K} \dfrac{\sqrt{\mathbf{X}' \cdot \mathbf{X}'}}{c} d \lambda & = - q \int_{\lambda_0}^{\lambda_1} \delta \mathbf{X} \cdot \mathbf{F} \cdot \mathbf{U} \dfrac{\sqrt{\mathbf{X}' \cdot \mathbf{X}'}}{c} d \lambda = \\ & = - q \int_{\lambda_0}^{\lambda_1} \delta \mathbf{X} \cdot \mathbf{F} \cdot \mathbf{X}' \, d \lambda = \\ & = - q \int_{\lambda_0}^{\lambda_1} \delta \mathbf{X} \cdot \left[ \nabla \mathbf{A} - \nabla^T \mathbf{A} \right] \cdot \mathbf{X}' \, d \lambda = && \text{(see details, below)} \\ & = - \delta \int_{\lambda_0}^{\lambda_1} q \mathbf{A}(\mathbf{X}) \cdot \mathbf{X}' \, d \lambda = \\ & = - \delta \int_{\tau_0}^{\tau_1} q \mathbf{A}(\mathbf{X}) \cdot \mathbf{U} \, d \tau \end{aligned}\end{split}\]

  The variational principle thus reads

  \[\begin{split}\begin{aligned} 0 & = \delta S = \\ & = \delta \int_{\tau_0}^{\tau_1} \left\{ - m c^2 - q \mathbf{A}(\mathbf{X}(\tau)) \cdot \mathbf{U}(\tau) \right\} \, d \tau = \\ & = \delta \int_{\lambda_0}^{\lambda_1} \left\{ - m c \sqrt{\mathbf{X}'(\lambda) \cdot \mathbf{X}'(\lambda)} - q \mathbf{A}\left(\mathbf{X}(\lambda)\right) \cdot \mathbf{X}'(\lambda) \right\} \, d \lambda \ . \end{aligned}\end{split}\]
  EM field force - details

  \[\begin{split}\begin{aligned} \delta \int_{\lambda_0}^{\lambda_1} \mathbf{A}(\mathbf{X}) \cdot \mathbf{X}' \, d \lambda & = \int_{\lambda_0}^{\lambda_1} \delta \mathbf{X} \cdot \nabla \mathbf{A}(\mathbf{X}) \cdot \mathbf{X}' \, d \lambda + \int_{\lambda_0}^{\lambda_1} \mathbf{A}(\mathbf{X}) \cdot \delta \mathbf{X}' \, d \lambda \\ & = \int_{\lambda_0}^{\lambda_1} \delta \mathbf{X} \cdot \nabla \mathbf{A}(\mathbf{X}) \cdot \mathbf{X}' \, d \lambda + \left[ \mathbf{A}(\mathbf{X}) \cdot \delta \mathbf{X} \right]_{\lambda_0}^{\lambda_1} - \int_{\lambda_0}^{\lambda_1} \dfrac{d}{d \lambda} \mathbf{A}(\mathbf{X}(\lambda)) \cdot \delta \mathbf{X} = \\ & = \int_{\lambda_0}^{\lambda_1} \delta \mathbf{X} \cdot \nabla \mathbf{A}(\mathbf{X}) \cdot \mathbf{X}' \, d \lambda - \int_{\lambda_0}^{\lambda_1} \mathbf{X}' \cdot \mathbf{A}(\mathbf{X}(\lambda)) \cdot \delta \mathbf{X} = \\ & = \int_{\lambda_0}^{\lambda_1} \delta \mathbf{X} \cdot \left[ \nabla \mathbf{A}(\mathbf{X}) - \nabla^T \mathbf{A}(\mathbf{X}) \right] \cdot \mathbf{X}' \, d \lambda \end{aligned}\end{split}\]
  From the variational principle to the equations of motion

  \[\begin{split}\begin{aligned} 0 & = \delta \int_{\lambda_0}^{\lambda_1} \mathcal{L}\left( \mathbf{X}(\lambda), \mathbf{X}'(\lambda), \lambda \right) \, d \lambda = \\ & = \int_{\lambda_0}^{\lambda_1} \left\{ \delta \mathbf{X}'(\lambda) \cdot \nabla_{\mathbf{X}'} \mathcal{L} + \delta \mathbf{X}(\lambda) \cdot \nabla_{\mathbf{X}} \mathcal{L} \right\} \, d \lambda = \\ & = \underbrace{\left.\left[ \delta \mathbf{X} \cdot \nabla_{\mathbf{X}'} \mathcal{L} \right]\right|_{\lambda_0}^{\lambda_1}}_{=0} - \int_{\lambda_0}^{\lambda_1} \delta \mathbf{X} \cdot \left\{ \dfrac{d}{d\lambda} \left( \nabla_{\mathbf{X}'} \mathcal{L} \right) - \nabla_{\mathbf{X}} \mathcal{L} \right\} \, d \lambda \ , \end{aligned}\end{split}\]

  and, since \(\delta \mathbf{X}\) must be arbitary, Lagrange equations follow

  \[\dfrac{d}{d\lambda} \left( \nabla_{\mathbf{X}'} \mathcal{L} \right) - \nabla_{\mathbf{X}} \mathcal{L} = \mathbf{0} \ .\]

  If the Lagrangian funcion is

  \[\mathcal{L}(\mathbf{X}, \mathbf{X}', \lambda) = - m c \sqrt{\mathbf{X}'(\lambda) \cdot \mathbf{X}'(\lambda)} - q \mathbf{A}\left(\mathbf{X}(\lambda)\right) \cdot \mathbf{X}'(\lambda) \ ,\]

  its derivatives are

  \[\begin{split}\begin{aligned} \nabla_{\mathbf{X}'} \mathcal{L} & = - \dfrac{m c}{\sqrt{\mathbf{X}' \cdot \mathbf{X}'}} \mathbf{X}' - q \mathbf{A}(\mathbf{X}) = - m \mathbf{U} - q \mathbf{A} \\ \nabla_{\mathbf{X} } \mathcal{L} & = - q \nabla \mathbf{A} \cdot \mathbf{X}' = - q \nabla \mathbf{A} \cdot \mathbf{U} \dfrac{\sqrt{\mathbf{X}' \cdot \mathbf{X}'}}{c} \\ \end{aligned}\end{split}\]

  and

  \[\begin{split}\begin{aligned} \dfrac{d}{d\lambda} \nabla_{\mathbf{X}'} \mathcal{L} & = \dfrac{d \tau}{d\lambda} \dfrac{d}{d \tau} \left( - m \mathbf{U} - q \mathbf{A} \right) = \\ & = \dfrac{\sqrt{\mathbf{X}' \cdot \mathbf{X}'}}{c} \dfrac{d }{d \tau} \left(- m \mathbf{U} - q \mathbf{A}(\mathbf{X}) \right) = \\ & = \dfrac{\sqrt{\mathbf{X}' \cdot \mathbf{X}'}}{c} \left(- m \dfrac{d }{d \tau}\mathbf{U} - q \mathbf{U} \cdot \nabla \mathbf{A}(\mathbf{X}) \right) \ . \end{aligned}\end{split}\]

  Putting together all the pieces of the Lagrange equations, and dividing by \(\frac{\sqrt{\mathbf{X}' \cdot \mathbf{X}'}}{c}\) - different from zero, if the 3-velocity of the particle is \(|\mathbf{v}| < c\) -,

  \[0 = - m \dfrac{d \mathbf{U}}{d \tau} - q \mathbf{U} \cdot \nabla \mathbf{A} + q \nabla \mathbf{A} \cdot \mathbf{U} \ ,\]

  and thus

  \[m \dfrac{d \mathbf{U}}{d \tau} = q \left[ \nabla \mathbf{A} - \nabla^T \mathbf{A} \right] \cdot \mathbf{U} \ .\]

26.2.4.2. Using coordinates

With \(\mathbf{X}\left( q^{\mu}(\lambda) \right)\),…