Abstract
It is common in textbooks on classical mechanics to discuss canonical transformations on the basis of the integral form of the canonicity conditions and a theory of integral invariants [1, 12, 14]. We prefer to deduce all the properties of canonical transformations by direct analysis of the canonicity conditions given by Eqs. (2.100) and (2.104). For convenience, we have made the subject matter of the next chapter independent from this one, so the reader can o mit this and continue from the next chapter. It is worth noting that time-independent canonical transformations are an important tool to analyze the structure of a general singular theory.
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Notes
- 1.
Sometimes these are called contact transformations .
- 2.
note that this is not possible in the time-independent case: if H(z) depends essentially on all the variables, then the same is true for \(\tilde H=H(z(z'))\), see Eq. (3.6)).
References
E. Cartan, Leçons sur les Invariants Intégraux (Hermann, Paris, 1922)
H. Goldstein, Classical Mechanics, 2nd edn. (Addison-Wesley, Reading, MA, 1980)
F.R. Gantmacher, Lectures on Analytical Mechanics (MIR, Moscow, 1970)
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Deriglazov, A. (2010). Canonical Transformations of Two-Dimensional Phase Space. In: Classical Mechanics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14037-2_3
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DOI: https://doi.org/10.1007/978-3-642-14037-2_3
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