Lie Symmetry Methods for Solving Differential Equations (II)

求解微分方程的李对称方法（二） 苏剑林 · 2013-11-26 · Mathematics integral differential equation symmetry Lie group

Because of carelessness during a system reinstall, I lost the Word document for Lie Symmetry Methods for Solving Differential Equations, and worse still, the USB drive that held a copy of it was lost as well~ Nothing to be done — I had to retype the whole article from scratch. Fortunately, I had once printed it out on paper, so I still had the old hard copy to work from. I am now releasing Lie Symmetry Methods for Solving Differential Equations (II), hoping it can offer a bit of a resource to anyone interested in Lie symmetry methods.

Compared with Part (I), in Part (II) I have retyped everything using CTeX — and indeed, $\LaTeX$ really is the best of the software options for writing math papers. Even though it's pure code editing, that actually suits my preference for a clean, simple style. Content-wise, Part (II) adds material on first-order systems of ordinary differential equations, and also revises some details from Part (I). By the end of this article, the discussion gives a reasonably complete account of the ideas behind Lie symmetry integration for first-order systems of ODEs, and the content has grown to 13 pages. In the upcoming Part (III), I'll cover Lie algebra; and if there ends up being a Part (IV), it will discuss computer implementations of Lie symmetry methods. I hope you'll enjoy this series. And I'm even more looking forward to hearing your thoughts after reading (including catching my mistakes) ^_^more

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 Symmetries of the equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.5 Solving differential equations using symmetries. . . . . . . . . . . . . . . . . 5
2.5.1 The canonical coordinate method. . . . . . . . . . . . . . . . . . . . . . . 6
2.5.2 The integrating factor method. . . . . . . . . . . . . . . . . . . . . . . . 6
2.6 Computing symmetries. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.7 Equations with a given symmetry. . . . . . . . . . . . . . . . . . . . . . . . 8
2.7.1 The direct integration method. . . . . . . . . . . . . . . . . . . . . . . . 8
2.7.2 The canonical coordinate method. . . . . . . . . . . . . . . . . . . . . . . 8
2.8 Ordinary differential equations with known symmetries. . . . . . . . . . . . 8
2.9 Worked examples for this chapter. . . . . . . . . . . . . . . . . . . . . . . . 9
3 First-Order Systems of Ordinary Differential Equations 10
3.1 Infinitesimal transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 Canonical coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3 First prolongation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.4 Symmetries of the system. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.5 Reducing the order by one using symmetry: the canonical coordinate method. . . . . . . . . . . . . . 12
3.6 Computing symmetries. . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.7 Systems with a given symmetry. . . . . . . . . . . . . . . . . . . . . . . . 13
3.8 Worked examples for this chapter. . . . . . . . . . . . . . . . . . . . . . . . 13

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Lie Symmetry Methods for Solving Differential Equations.pdf