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Some Mechanical Problems in a Geometric Setting

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Classical Mechanics

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Abstract

The Maupertuis variational principle is the oldest least-action principle of classical mechanics. Its precise formulation was given by Euler and Lagrange; for its history, see Yourgrau and Mandelstam (Variational Principles in Dynamics and Quantum Theory. Pitman/W.B. Sanders, London/Philadelphia, 1968). However, the traditional formulation (as a variational problem subject to the constraint that only the motions with fixed total energy are considered), remained problematic, as emphasized by V. Arnold (double citation): “In his Lectures on Dynamics (1842–1843), C. Jacobi commented: “In almost all textbooks, even the best, this Principle is presented in such a way that it is impossible to understand”. I do not choose to break with tradition” (Arnold, Mathematical Methods of Classical Mechanics, 2nd edn. Springer, New York, 1989).

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Notes

  1. 1.

    Equation (6.7) can also be taken as a starting point for obtaining \(\tilde{L}\).

  2. 2.

    More exactly, the notions of a geodesic and a minimal length line coincide only in the Riemann space with a Riemann connection. Here we do not distinguish these notions. They are discussed in more detail in the following sections.

  3. 3.

    Detailed discussion of the coordinate formulation of Riemann geometry for non-mathematicians can be found, for example, in [15]. For the coordinate-free formulation, see [35].

  4. 4.

    Note that this definition does not mention coordinates, representing an example of the coordinate-free definition of differential geometry.

  5. 5.

    The line was called the trajectory in Sects. 6.1 and 6.2.

  6. 6.

    Accordingly, any vector proportional to ξ is called a tangent vector to the line determined by the curve.

  7. 7.

    We consider only torsion-free affine connections.

  8. 8.

    Parallel transport of the covariantly constant field along any line takes it into itself, see below.

  9. 9.

    Components ξ a(q b) at the point q b = q b(τ) are defined as ξ a(q b) ≡ ξ a(τ).

  10. 10.

    Let us point out that Eq. (6.79) itself cannot be rewritten in terms of D b .

  11. 11.

    Note that they do not depend on α or on the length of ξ a.

  12. 12.

    N ab are known as the coefficients of second quadratic form of the surface.

  13. 13.

    For the case of the Riemann connection, dynamical parametrization is precisely the natural parametrization, see page 216.

  14. 14.

    In the flat limit the sequence y (1) μ, x μ, y (2) μ of events can be associated with emission, reflection and absorbtion of a photon with the propagation law ds = 0. Then the middle point (6.148) should be considered simultaneous with x 0.

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Authors and Affiliations

  1. Universidade Federal de Juiz de Fora Dept. of Mathematics, Juiz de Fora, Minas Gerais, Brazil

    Alexei Deriglazov

  2. Laboratory of Mathematical Physics, Tomsk Polytechnic University, Tomsk, Russian Federation

    Alexei Deriglazov

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Deriglazov, A. (2017). Some Mechanical Problems in a Geometric Setting. In: Classical Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-319-44147-4_6

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