Abstract
The Lagrange equations arising from a Lagrange function are second order differential equations. With this formalism, it is possible to realize constraints (such as occur in applications when objects are affixed to an axle or connected by rods) by simply restricting the Lagrange function.
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Notes
- 1.
Image: courtesy of Zentrum Mathematik (Technical University of Munich, Germany).
- 2.
This also applies in special relativity, in each inertial frame, and one can convert between different inertial frames.
- 3.
As the statement of Exercise 8.5 (a) indicates, this condition is by no means necessary.
- 4.
When we apply the operator \(\mathrm {D} I(\gamma )\) to a vector h, we write \(\gamma \) as an index.
- 5.
We assume here for simplicity of the discussion that L does not explicitly depend on the time.
- 6.
The length functional \(\mathcal{L}\) is not to be confused with the Lagrangian L !
- 7.
According to the Theorema Egregium by Gauss it can be written in terms of the metric g, its first and second derivatives. So it is an intrinsic quantity.
- 8.
The system is integrable in the sense of the definition on page 330.
We can also understand the fact that \(p_\varphi \) is constant by noting that \(p_\varphi \) is the 3-component of the angular momentum. It is conserved because the surface of revolution is invariant under rotation about the 3-axis: see Noether’s theorem on page 351.
- 9.
The discussion in [Ar2], §45 for the planar double pendulum is problematic because the metric on the configuration space becomes degenerate in this case.
- 10.
Named after the Dutch mathematician Willebrord van Roijen Snell (1580–1626). The law was first found by the Arabic mathematician Abu Sa‘d Ibn Sahl (ca. 940–1000), in the form \(n_1\sin \alpha _1 = n_2\sin \alpha _2\), with the refractive indices \(n_1, n_2\) in the two media.
- 11.
In practice, one frequently corrects for the colors blue and red (see figure) and therefore uses quantities called Abbe numbers of the different kinds of glass, rather than their refractive indices.
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Knauf, A. (2018). Variational Principles. In: Mathematical Physics: Classical Mechanics. UNITEXT(), vol 109. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55774-7_8
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DOI: https://doi.org/10.1007/978-3-662-55774-7_8
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