Introduction to Symmetry Analysis

Description

Symmetry analysis based on Lie group theory is the most important method for solving nonlinear problems aside from numerical computation. The method can be used to find the symmetries of almost any system of differential equations and the knowledge of these symmetries can be used to reduce the complexity of physical problems governed by the equations. This is a broad, self-contained, introduction to the basics of symmetry analysis for first and second year graduate students in science, engineering and applied mathematics. Mathematica-based software for finding the Lie point symmetries and Lie-Bäcklund symmetries of differential equations is included on a CD along with more than forty sample notebooks illustrating applications ranging from simple, low order, ordinary differential equations to complex systems of partial differential equations. MathReader 4.0 is included to let the user read the sample notebooks and follow the procedure used to find symmetries.

PDF p. 148 gives a good definition of symplectic manifolds , in the context of classical mechanics, Hamilton's equations, and Poisson brackets. (OED: "συμπλεκτικός twining or plaiting together, copulative")

The conserved elements of a Hamiltonian system … define a vector space. The rules of algebra in this space are given by the skew-symmetry of the composition operator (Poisson bracket) (4.43), the additive properties in (4.44), and the Jacobi identity (4.47). A vector space with these special properties is called a symplectic space, and the solution of the Hamiltonian system is said to lie on a symplectic manifold. This odd word comes from the greek symplektikos meaning “twining together,” from syn (together) and plekein (to twine). It is an apt description of the solution trajectories of a periodically forced Hamiltonian system, which can be visualized as a family of spiraling curves on a torus in a three-dimensional phase space where the third dimension is the phase angle of the forcing function.

PDF pp. 501-2 show how Legendre transform s are a type of contact transform.

PDF pp. 520-4, §14.4.3 is on symmetries of Kepler's problem in the context of the generalized contact transfroms, Lie-Bäcklund transfroms (cf. Symmetries of Integro-Differential Equations PDF p. 61 Example 1.5.6); cf. "Example 15.2 (A particle moving under the influence of a spherically symmetric inverse-square body force)." (PDF pp. 551-7), where Cantwell (PDF p. 524) "use[s] these groups to generate invariants of the motion for the system (14.132)."

PDF pp. 91-3 2.5: "Buckingham’s Π Theorem – The Dimensional-Analysis Algorithm"

Author is friends with Notre Dame profs, and he visited there in fall semester 1998.

ZIP is SymmetryAnalysisSoftwareCompressed.zip

This textbook is designed mainly for first- and second-year graduate students in science, engineering and applied mathematics, although the material is presented in a form that should be understandable to an upper-level undergraduate with a background in differential equations. The main goal is to teach methods of symmetry analysis and to instill in the student a sense of confidence in dealing with complex problems. The central theme is that anytime one is confronted with a physical problem and a set of equations to solve, the first step is to study the problem using dimensional analysis and the second is to use the methods of symmetry analysis to work out the Lie groups (symmetries) of the governing equations. This may or may not produce a simplification, but it will almost always bring clarity to the problem. It is the author's firm belief that symmetry analysis should be as familiar to the student as Fourier analysis.
Most of the theory, with a large number of relatively short worked examples, is developed in the first part of the book, while the second part contains a number of fully worked problems where the role of symmetry analysis as a part of the complete solution is illustrated. The emphasis is on applications, and the exercises provided at the end of each chapter are designed to help the reader practice the material.
In more detail, the book is organized as follows. The exposition starts with a historical preface [pp. xxi-xli]. Chapters 1–3 and 5–10 cover a standard course: introduction to symmetry, dimensional analysis, introduction to systems of ODEs and first-order PDEs, state-space analysis, introduction to one-parameter Lie groups, symmetry analysis of first order ODEs, introduction to differential functions, symmetry analysis of higher-order ODEs and PDEs, introduction to laminar boundary layers. Chapters 4 and 11–16 contain more advanced material: classical dynamics, incompressible flow, compressible flow, similarity rules for turbulent shear flows, Lie-Bäcklund transformations, variational symmetries and conservation laws, Bäcklund transformations and nonlocal groups. To make the book more complete some background material of advanced nature is presented in Appendices 1–3.
The first exercises involving the identification of Lie symmetries should be worked by hand so that the reader has a chance to practice the Lie algorithm, but it becomes quickly apparent that the calculations are huge even in fairly simple cases. This is one of the main reasons why Lie theory was never adopted in the mainstream curricula in science and engineering. Fortunately, we now live in an era when powerful symbol manipulation software packages are widely available. A special Mathematica®-based package was developed by the author of the book and it is included on CD along with more than sixty sample notebooks that are carefully coordinated with the examples and exercises in the text. Details of the package are described in Appendix 4. Reviewed by Victor V. Zharinov

After reading

• Michael J. Crowe, “Duhem and History and Philosophy of Mathematics,” Synthese 83, no. 3 (June 1, 1990): 431–47—

in which the Crowe, an emeritus prof. in the history of mathematics from Notre Dame (cf. his book on the development of vector notation), convincingly argues against Duhem's idealistic views of the historical development of mathematics—and the historical introduction of Cantwell's book, I've become interested in the development of non-Euclidean geometry, which seems to related to my question on whether mathematical suppositions of physical theories are uniquely determined (i.e., whether mathematics applies to physics in one way or multiple ways or whether mathematical abstraction is unique). Duhem's attacks on German science and the esprit de géométrie would seem to make him opposed to Riemannian geometry, with which he was familiar (he knew of Beltrami's work, corresponded with geometers like Engel and Klein, and knew Darboux et al. directly). (Interestingly, Riemann worked in Weber's physics lab!) I also learned that Gauss, who allegedly read Kant's Critique 5×, wanted to apply non-Euclidean geometries to physics in part to oppose Kant's notion of space. Lastly, the history of non-Euclidean does not begin with Lobatschewsky (1793-1856) but with the Jesuit Gerolamo Saccheri (1667-1733), whose work Euclid Vindicated from Every Blemish has recently been translated into English.
Here's the view of Assis 2014 p. 310, which follows Fr. O'Rahilly's:

We agree with O’Rahilly as regards the several problems and confusions which Einstein’s theories brought to physics. [O’R65, Volume 2, Chapter XIII, Section 5, pp. 662-71.]
In our view, several theoretic concepts of modern physics have the same role as the epicycles in the old ptolemaic theory: Length contraction, time dilation, Lorentz’s invariance, Lorentz’s transformations, covariant laws, invariant laws, Minkowski’s metric, Minkowski’s spacetime, four-dimensional space-time, energy-momentum tensor, Riemannian geometry applied to physics, virtual photon, Schwarzschild’s line element, tensorial algebras in four-dimensional spaces, quadrivectors, metric tensor g μν , Christoffel’s symbols, string theory, super strings, curvature of space, dark matter, dark energy, wormholes, etc. The relational mechanics presented in this book is totally against Einstein’s theories and eliminates all these epicycles.

cf. also Peirce on non-Euclidean geometry: CP 8.91 (Peirce outlines several astronomical methods, some of which modern cosmology today uses to assess the geometry of the universe.)
Studying the history of math has really shown me how shaky mathematics is, which helps address my question of whether sacra divina est "dignior aliis scientiis" (I q. 1 a. 5), even "dignior " than mathematics.

I do not think Einstein himself, with the advent of his GR, was actually in the business of pursuing the “geometrization” of physics (along the lines, say of a Descartes). He disputed the occasional accusation, holding to a strong commitment to physics which it would probably be unfair to deny him (yet “the several problems and confusions which Einstein’s theories brought to physics,” dixit Pr. Assis, are undeniable). However, my contention is that it is certainly fair to say that his mathematical description of GR introduces a mathematical and physical incongruity, in the way it geometrizes the gravitational field (due to a specific conception and use of the metric involving Riemannian geometry) to induce a physical internal curvature of spacetime.

S. Crothers, as you well know, has convincingly shown in many of his papers that the equations of Einsteinian GR are predicated upon a universe devoid of matter, since Einstein himself set his energy-momentum tensor equal to… zero. Therefore, GR’s geometrized “gravity” spacetime inevitably seems to rest upon the massive contraction of a void physical “cause.”

Furthermore, the real connection with Riemann only comes with the introduction of his (and Gauss’s) adjusted metrical continuum and “ n -dimensioned manifoldness,” specifically preparing Einstein’s pseudo-Cartesian metric field g μν. Einstein mathematically conceived the Riemann-based metric on a four-dimensional differentiable manifold called “spacetime,” and on that basis, went on to conceive the latter’s “curvature.”

The concept of connection was independently developed by Christoffel and did not actually include curvature. The latter emerged with Levi-Civita around the time Einstein put together (on the basis of Poincaré’s relativity) the framework for an extension of SR into GR, and later with Cartan recasting Newtonian gravitational theory by means of metric and curvature tensors.

Another, I believe also decisive contention against Einstein’s relativity theory implied in what I stated above in connection to S. Crothers’ own criticism, can be summed up as follows: Einstein’s RT is ultimately an attempt at a fancied cosmological monism (which would probably have seduced a Spinoza), whether ontologically or only descriptively geometrical (I leave the question open, while I tend to believe that Einstein himself did not see physics as geometry). A monism (pretending to be physical) because, in the geometry of “spacetime,” the individuality—and therefore the causal (electric) interactions—of (charged) celestial bodies vanishes away in the single encompassing moving structure of SR physics inclusive of gravity. Hence, there is in effect no more cosmological field of real application for the crucial idea of physical causality in such a generalized monistic view of blurred space and time (never mind that causality is intrinsic to the interlaced workings of physical reality and their properly scientific apprehension in a physical theory).

About Riemann, while he did devote himself to investigating the “Hypotheses which Lie at the Bases of Geometry” to expand on such concept as the “metric properties of space,” he often refrained from directly and unduly jumping to conclusions about physics. I too had come across the fact that he worked for some time in Weber’s lab. And Weber remained familiar with his subsequent research and contributions.

Sacra divina est multum “dignior” quam mathematica... St. Thomas says it best though: “[…] ista scientia [sacra divina]… omnes alias transcendit tam speculativas quam practicas.” It so does, one might add, notably by its object, extent, effect, inner light, consistency and ultimate (supernatural) comprehensibility. But I would still suggest that they relate in a certain and crucial epistemological (and pedagogical) sense—in particular by way of some analogies in which we might say that articles of the Creed are to sound doctrines as axioms are to theorems.