Abstract
We can infer from the examples given in Chap. 1 that linear dynamical systems are interesting on their own.
La filosofia è scritta in questo grandissimo libro che continuamente ci sta aperto innanzi agli occhi (io dico l’universo), ma non si può intendere, se prima non s’impara a intender la lingua, e conoscer i caratteri ne’quali è scritto. Egli è scritto in lingua matematica, e i caratteri son triangoli, cerchi ed altre figure geometriche, senza i quali mezzi è impossibile a intenderne umanamente parola; senza questi è un aggirarsi vanamente per un oscuro laberinto.
Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth.
Galileo Galilei Il Saggiatore
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Notes
- 1.
‘Differentiable’ here could be understood simply as the statement that the maps \(t\mapsto \phi _{t}e_{i}\), with \(\left\{ e_{i}\right\} \) a basis in \(E\), are differentiable.
- 2.
The derivative of \(\phi _{t}\) could be easily understood as thinking of \(\phi _{t}\) as a curve of matrices, once we have selected any basis on \(E\), then the space of \(n\times n\) matrices can be identified with \(\mathbb {R}^{n^{2}}\).
- 3.
Notice that the operator-valued power series \(\sum _{k=0}^{\infty }\frac{t^{k}}{k!}A^{k}\) is convergent because it can be bounded by the numerical series \(\sum _{k=0}^{\infty }\frac{t^{k}}{k!}||A||^{k}\) with \(||\cdot ||\) any norm in the finite-dimensional linear space of linear maps on \(E\).
- 4.
In Quantum Mechanics a similar decomposition of the total space can be achieved by using a compact group of symmetries for \(A\). The irreducible invariant subspaces of our group will be finite-dimensional and the restriction of the Hamiltonian operator \(A\) to each invariant subspace gives raise to a finite-dimensional problem. Motions in central potentials are often studied in this way by using the rotation group and spherical harmonics. The radial part is then studied as a one-dimensional problem.
- 5.
Also-called a ‘product integral’ (see, e.g.: [DF79]).
- 6.
See however below, Remark 2.5.
- 7.
To be quite honest, the class of functions that extend more naturally the algebra of polynomials is the algebra of real analytic functions. However in this book we will restrict our attention to the algebra of smooth functions.
- 8.
These fields are not independent. In fact, denoting them collectively as: \(\mathbf {X}=\left( X_{\left( 1\right) },X_{\left( 2\right) },X_{\left( 3\right) }\right) \), with: \(\mathbf {x}=\left( x^{1},x^{2},x^{3}\right) \), it is obvious that: \(\mathbf {x}\cdot \mathbf {X} =0\).
- 9.
See however below, Appendix C, Sect. C.2 for a similar discussion in a more general context.
- 10.
However this idea will also work in a more abstract setting in the sense that it is possible to show that there is a one-to-one correspondence between equivalence classes of curves possessing a contact of order \(1\) at \(x\) and linear first-order differential operators \(v\) acting locally on functions at \(x\).
- 11.
Also-called the Euler differential operator.
- 12.
The leaves are essentially the equipotential surfaces of two opposite electric charges (a dipole) located at \(\left( -1,0,0\right) \) and \(\left( 1,0,0\right) \) respectively.
- 13.
This is an application of a deep result in the theory of Lie groups, also-called Hilbert’s fifth problem that shows that any finite-dimensional locally compact topological group without “small subgroup” is a Lie group [MZ55].
- 14.
The map \(\exp :\mathfrak {gl}(n) \rightarrow GL(n)\) is differentiable with differential the identity at \(I\), hence by the inverse theorem there is local inverse of \(\exp \) which is differentiable, that is the map \(\ln \) we are using.
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Cariñena, J.F., Ibort, A., Marmo, G., Morandi, G. (2015). The Language of Geometry and Dynamical Systems: The Linearity Paradigm. In: Geometry from Dynamics, Classical and Quantum. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-9220-2_2
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