Abstract
A classical approach to the dynamics of Hamiltonian systems (or dynamical systems in general) is based on the notion of a phase space (Chaps. 2 and 3). It turns out that the phase space of a Hamiltonian system possesses certain geometric properties [1].
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- 1.
Henri Poincaré (1854–1912), a great French physicist and mathematician being i.a. a pioneer of chaos theory.
- 2.
Bernhard Riemann (1826–1866), influential German mathematician who made essential contributions to analysis and differential geometry.
- 3.
Paul Finsler (1894–1970), German and Swiss mathematician.
- 4.
We will often use both terms in an interchangeable way.
- 5.
We emphasize that it is not the only way to obtain the metric tensor.
- 6.
Tullio Levi-Civita (1873–1941), Italian mathematician of Jewish origin who investigated celestial mechanics, the three-body problem, and hydrodynamics.
- 7.
Obviously, we can obtain geodesic equations from formula (10.1.5).
- 8.
Elwin Bruno Christoffel (1829–1900), German mathematician and physicist working mainly at the University of Strasbourg.
- 9.
Since, by definition, a Riemannian space has the structure of a differentiable manifold, in general there is no single global coordinate system but many so-called local coordinate systems.
- 10.
Often this quantity is called the absolute derivative of a tensor [8].
- 11.
A similar situation occurs in the case of calculation of Lyapunov exponents.
- 12.
Obviously we apply here the Einstein summation convention, that is, we carry out the summation with respect to repeating indices.
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Awrejcewicz, J. (2012). Geometric Dynamics. In: Classical Mechanics. Advances in Mechanics and Mathematics, vol 29. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3740-6_11
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DOI: https://doi.org/10.1007/978-1-4614-3740-6_11
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