Abstract
In many respects, mechanics carries geometrical structures. This could be felt very clearly at various places in the first four chapters. The most important examples are the structures of the space–time continua that support the dynamics of nonrelativistic and relativistic mechanics, respectively. The formulation of Lagrangian mechanics over the space of generalized coordinates and their time derivatives, as well as of Hamilton–Jacobi canonical mechanics over the phase space, reveals strong geometrical features of these manifolds.
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Notes
- 1.
In what follows we shall often call \(V_p\), i.e. the restriction of the vector field to \(T_pM\), a representative of the vector field.
- 2.
The precise statement is this: the real vector space of \(\mathbb R\)-linear derivations on \(\mathcal{F} (M)\) is isomorphic to the real vector space \(\mathcal{V} (M)\).
- 3.
Below we shall also use the notation TF, instead of \(\mathrm{d}F\), a notation which is customary in the mathematical literature.
- 4.
Using well-known techniques of linear algebra one can show that at each point \(p \in M\) one can find a basis such that \(g_{ik}\) is diagonal, i.e. \(g = \sum ^n_{i=1} \varepsilon _i \mathrm{d}x^i \otimes \mathrm{d} x^i\), with \(\varepsilon _i = \pm 1\). If all \(\varepsilon _i\) are equal to \(+1\), the metric is said to be Riemannian . In all other cases it is said to be semi-Riemannian .
- 5.
For this reason V.I. Arnol’d (1978) calls the Lie derivative the fisherman’s derivative. The fisherman sees only the river in front of him. He sees all kinds of objects floating by on the river and takes their differential along the lines of the river’s flow.
- 6.
This prescription is called the inner product : \(i_X \alpha (Y_1 , \ldots , Y_k) {\;\mathop {=}\limits ^\mathrm{def}\;}\alpha (X, Y_1, \ldots , Y_k)\) is said to be the inner product of X with \(\alpha \). The indentity (5.94) then reads \(L_X \alpha = i_X (d \alpha ) + \mathrm{d} (i_X \alpha )\).
- 7.
A mapping \(\varPhi : M \rightarrow N\) is said to be regular in the point \(p \in M\) if the corresponding differential, or tangent, mapping from \(T_pM\) to \(T \varPhi _{(p)}N\) is surjective.
- 8.
As is easy to guess, this is true if \(\det (\partial ^2 H/ \partial p_k \partial p_i)\) vanishes nowhere.
- 9.
The global properties of symplectic manifolds are the subject of an important research field of mathematics. The present state of the art is described in the book by Hofer and Zehnder (1994). This book should be readily accessible for the mathematically minded reader.
- 10.
The condition (5.128) expresses the fact that the Levi–Civita connection has vanishing torsion.
- 11.
V.I. Arnol’d : Ann. Inst. Fourier 16, 319 (1966).
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Scheck, F. (2018). Geometric Aspects of Mechanics. In: Mechanics. Graduate Texts in Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55490-6_5
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DOI: https://doi.org/10.1007/978-3-662-55490-6_5
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