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Properties of Canonical Transformations

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Classical Mechanics

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Abstract

As we have seen in Sect. 2.7, the canonical form of Hamiltonian equations is not preserved by general phase-space transformations. Those that leave the form of the equations unaltered were called canonical transformations. In this chapter, we discuss their properties for the case of phase space of an arbitrary dimension.

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Notes

  1. 1.

    Below, we discuss only a free canonical transformation. For an arbitrary canonical transformation, the situation is similar, see [14].

  2. 2.

    The left-hand side of this expression is known as a Lagrange bracket.

  3. 3.

    Since the general solution (4.95) determines the canonical transformation (4.92), the Eq. (4.97) state that we search for the generating function of the evolution.

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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). Properties of Canonical Transformations. In: Classical Mechanics. Springer, Cham. https://doi.org/10.1007/978-3-319-44147-4_4

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