Abstract
In describing real-world phenomena, we often seek a local or leading order approximation, e.g., by an orthogonal projection onto a finite dimensional linear subspace of vectors or functions.
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Notes
- 1.
This defines the critical angle for a Fata Morgana to occur.
- 2.
In conserving photon energy, E of the incident and refracted photons being the same, this is equivalent to extremizing the action \(S=E\Delta t\), i.e., distance measured in total phase, relevant to covariant formulations of Fermat’s principle.
- 3.
In flat space with Euclidean metric, the triangle inequality holds locally and globally. In curved space with Euclidean signature, it holds locally.
- 4.
For numerical integration over the real line, see, e.g., [4].
- 5.
These are the Euler-Lagrange equations of a free particle in polar coordinates.
References
Demir, D., 2011, A table top demonstration of radiation pressure (Diplomarbeit, University of Vienna).
Feynman, R.P., Leighton, R.B., & Sands, M., Lectures on Physics, Vol. I, 1963, Ch. 26.
Gautschi, W., 2004, Orthogonal Polynomials: Computation and Approximation. Numerical Mathematics and Scientific Computation (Oxford University Press, New York).
Takahasi, H., & Mori, M., 1974, “Double Exponential Formulas for Numerical Integration,” Publ. RIMS, Kyoto University, vol. 9, 721–741.
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van Putten, M.H. (2017). Projections and Minimal Distances. In: Introduction to Methods of Approximation in Physics and Astronomy. Undergraduate Lecture Notes in Physics. Springer, Singapore. https://doi.org/10.1007/978-981-10-2932-5_5
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