Most elementary numerical schemes found useful for solving classical trajectory problems are canonical transformations. This fact should be made more widely known among teachers of computational physics and Hamiltonian mechanics. From the perspective of advanced mechanics, unlike that of numerical schemes, there are no bewildering number of seemingly arbitrary elementary schemes based on Taylor's expansion. There are only two canonical first and second order algorithms, on the basis of which one can comprehend the structures of higher order symplectic and non-symplectic schemes. This work shows that most elementary algorithms up to the fourth-order can be derived from canonical transformations and Poisson brackets of advanced mechanics.

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This work uses a simple notation to denote Lie operators such as Ĥ or Ŝ. The direct use9 of {·,S} leads to unwieldy expression for powers of Ŝn as {{{·,S},S},S}... The old time typographical notation17,18 of “:S:” seemed strange to modern readers. The use of “LS14,15 or “DS8 is redundant, with the symbol “L” or “D” serving no purpose. Here, we follow a practice already familiar to students from quantum mechanics, by denoting a Lie operator with a “caret” over its defining function, Ŝ. Also, it is essential to define Ĥ={·,H}, and notĤ={H,·}, otherwise one gets a confusing negative sign18 for moving forward in time.

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