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Poisson Brackets Environment

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Elementary Symplectic Topology and Mechanics

Part of the book series: Lecture Notes of the Unione Matematica Italiana ((UMILN,volume 16))

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Abstract

The set of real valued1 square matrices \(A = (A_{\mathit{ij}}),i,j = 1,\ldots,m\), is a popular non-commutative algebra: we are concerned with sum A + B, and products AB and λ A, for \(\lambda \in \mathbb{R}\).

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Notes

  1. 1.

    Representation of endomorphisms in some finite dimensional linear space.

  2. 2.

    The right arrow ⇒ of (3.3) follows from (3.4) since tA and sB are commuting, the left arrow ⇐ is caught by computing \(\frac{d} {\mathit{dt}}\big\vert _{t=0} \frac{d} {\mathit{ds}}\big\vert _{s=0}\) on both hand sides of e tA e sB = e sB e tA. Here, we just recall that \(\frac{d} {\mathit{dt}}e^{\mathit{tA}} = \mathit{Ae}^{\mathit{tA}}\).

  3. 3.

    See Chap. 1, dedicated to Differential Geometry.

References

  1. R. Abraham, J.E. Marsden, Foundations of Mechanics. Advanced Book Program, 2nd edn. (Benjamin/Cummings Publishing, Reading, 1978), xxii+m–xvi+806pp

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Cardin, F. (2015). Poisson Brackets Environment. In: Elementary Symplectic Topology and Mechanics. Lecture Notes of the Unione Matematica Italiana, vol 16. Springer, Cham. https://doi.org/10.1007/978-3-319-11026-4_3

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