Due to carelessness during a system reinstallation, the author lost the Word document for “Lie Symmetry Methods for Solving Differential Equations.” Even more unfortunately, the USB drive containing the document was also lost. Consequently, I had no choice but to re-enter this article from scratch. Fortunately, I had previously printed a hard copy, so I had an old draft for reference. I am now releasing “Lie Symmetry Methods for Solving Differential Equations (II),” hoping to provide some resources for friends interested in Lie symmetry methods.
Compared to Part (I), Part (II) has been entirely re-entered using
CTex. As expected, LaTeX is indeed the leader among software for writing
mathematical papers. Although it involves pure code editing, it
perfectly fits my pursuit of a concise and clear style. In terms of
content, Part (II) adds material on systems of first-order ordinary
differential equations and modifies some details from Part (I). With the
completion of this article, the approach to Lie symmetry integration for
systems of first-order ordinary differential equations is preliminarily
and relatively completely described, with the content expanding to 13
pages. In the upcoming Part (III), content on Lie algebras will be
provided; if there is a Part (IV), it will discuss the computer
implementation of Lie symmetry methods. I hope everyone enjoys this
series of articles. I look forward to your feedback (including error
corrections) ^_^
Table of Contents
1 Introduction to Lie Symmetry Methods 2
2 First-order Ordinary Differential Equations 2
2.1 Infinitesimal Transformations . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Canonical Coordinates . . . . . . . . . . . . . . . . . . . . . . . 3
2.3 First Prolongation . . . . . . . . . . . . . . . . . . . . . . . 4
2.4 Symmetry of Equations . . . . . . . . . . . . . . . . . . . . . . 5
2.5 Solving Differential Equations Using Symmetry . . . . . . . . . . . . . . . . . 5
.1 Canonical Coordinate Method . . . . . . . . . . . . . . . . . . 6
.2 Integrating Factor Method . . . . . . . . . . . . . . . . . . 6
2.6 Calculating Symmetries . . . . . . . . . . . . . . . . . . . . . . 7
2.7 Equations with Given Symmetries . . . . . . . . . . . . . . . . . . 8
.1 Direct Integration Method . . . . . . . . . . . . . . . . . . 8
.2 Canonical Coordinate Method . . . . . . . . . . . . . . . . . . 8
2.8 ODEs with Known Symmetries . . . . . . . . . . . . . . . . . . 8
2.9 Examples in this Chapter . . . . . . . . . . . . . . . . . . . . . . 9
3 Systems of First-order Ordinary Differential Equations 10
3.1 Infinitesimal Transformations . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Canonical Coordinates . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 First Prolongation . . . . . . . . . . . . . . . . . . . . . . . 11
3.4 Symmetry of Systems of Equations . . . . . . . . . . . . . . . . . . . . 11
3.5 Reducing Order by One Using Symmetry: Canonical Coordinate Method . . . . . . . . . . . . 12
3.6 Calculating Symmetries . . . . . . . . . . . . . . . . . . . . . . 12
3.7 Equations with Given Symmetries . . . . . . . . . . . . . . . . . . 13
3.8 Examples in this Chapter . . . . . . . . . . . . . . . . . . . . . . 13
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Symmetry Methods for Solving Differential Equations.pdf
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