Abstract
This chapter contains some of the most advanced topics in analytical mechanics. Due to the introductory character of the book, the theorems requiring a deep knowledge of geometry and algebra are partially proved or only stated.
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Notes
- 1.
A curve γ1 = x 1(s), \(s \in \left [a,b\right ]\), on a manifold V n is homotopic to a curve γ = x(s), s ∈ [a, b], of V n if there exists a continuous map** F : [a, b] ×[0, 1] → V n such that F(0, s) = x(s) and F(s, 1) = x 1(s), in other words, if γ1 reduces to γ by a continuous transformation.
- 2.
That is, except for the points of Tn(I) belonging to a subset with a Lebesgue measure equal to zero.
- 3.
An example of Cantor’s set can be obtained as follows. Start with the unit interval [0, 1] and choose 0 < k < 1. Remove the middle interval of length k ∕ 2. The length of the two remaining intervals is 1 − k ∕ 2. From each of them remove the middle interval of length k ∕ 8. The four remaining intervals will have a total length of 1 − k ∕ 2 − k ∕ 4. From each of them remove the middle interval of length k ∕ 32. The remaining length will be 1 − k ∕ 2 − k ∕ 4 − k ∕ 8, and so on. After infinitely many steps, the remaining set will have the length
$$1 - k/2 - k/4 - k/8 + \cdots = 1 - k > 0.$$
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Romano, A. (2012). Completely Integrable Systems. In: Classical Mechanics with Mathematica®. Modeling and Simulation in Science, Engineering and Technology. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8352-8_20
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