Abstract
In Chapter 1 we introduced Fermat’s principle in an elementary way, without any critical analysis of its consequences.
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Notes
- 1.
This result can directly be obtained starting from (10.28) or resorting to Euler’s theorem about the homogeneous functions. In fact this theorem states that if \(f(x_1,\ldots ,x_n)\) is a homogeneous function of degree \(\alpha \), that is a function for which
$$ f(\mu x_1,\ldots ,\mu x_n)=\mu ^{\alpha }f(x_1,\ldots ,x_n), $$then, the following identity holds
$$ \alpha f=\frac{\partial f}{\partial x_i}x_i $$(summation over i). Since L in (10.28) is homogeneous of the first order, we obtain (10.32).
- 2.
See Exercise 1 at the end of Chapter 1.
- 3.
Note that at least one of the \(\dot{x}_i\) is different from zero. Consequently, the homogeneous system (10.47) must have solutions different from zero. Then, necessarily
$$ \det \left( \frac{\partial ^{2}L}{\partial \dot{x}_{i}\partial \dot{x}_{h}}\right) =0. $$In this case the Lagrangian function (10.2) is said to be degenerate.
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Romano, A., Cavaliere, R. (2016). Fermat’s Principle and Wave Fronts. In: Geometric Optics. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-43732-3_10
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DOI: https://doi.org/10.1007/978-3-319-43732-3_10
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