Abstract
Given two points q o and q 1 in \({\mathbb{R}}^{N}\)and an interval \([{t}_{o},{t}_{1}] \subset \mathbb{R}\), consider the convex set \(\mathcal{K}\)of all smooth curves parameterized with t ∈ [t o , t 1], and of extremities q o and q 1, i.e.,
Given \(q \in \mathcal{K}\)and a vector-valued function φ ∈ C∞o(to, t1), the curve {q + λφ} is still in \(\mathcal{K}\)for all \(\lambda \in \mathbb{R}\).
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Notes
- 1.
A first form of such a principle was introduced by Hamilton in [69]. While motivated by the principle of path of least time of geometrical optics (see §4), Hamilton perceived its general breadth. From his introduction: …A certain quantity which in one physical theory is the “action” and in another “time,” expended by light in going from any first to any second point, is found to be less than if the light had gone in any other than its actual path …The mathematical novelty of my method consists in considering this quantity as a function…and in reducing all researches respecting optical systems of rays to the study of this single function: a reduction which presents mathematical optics under an entirely novel view, and one analogous (as it appears to me) to the aspect under which Descartes presented the application of Algebra to Geometry…. Applications to mechanics are in [72, 73].
- 2.
- 3.
With perhaps improper symbolism we have used the same symbol to denote q(t) and q(s), the instantaneous position of the ray, with different parametric representations.
- 4.
Although optical trajectories are curves in \({\mathbb{R}}^{3}\), wewill regard them as in \({\mathbb{R}}^{N}\), to stress that the principles of geometrical optics aredimension-independent.
- 5.
The property of η → F(η) being homogeneous of degree 1 does not imply any restriction on the rank of the Hessian of F. Give examples of homogeneous functions of degree 1 whose Hessian has rank \(1, 2,\ldots, (N - 1)\).
- 6.
It is not asserted that the front Φ(q o ; t + Δt) is reached at q + Δq, only that such a front is reached in the least time Δt.
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DiBenedetto, E. (2011). Variational Principles. In: Classical Mechanics. Cornerstones. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4648-6_9
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DOI: https://doi.org/10.1007/978-0-8176-4648-6_9
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