Applications of Lie Groups to Differential Equations - download pdf or read online

By Peter J. Olver

ISBN-10: 1468402749

ISBN-13: 9781468402742

This e-book is dedicated to explaining a variety of functions of con­tinuous symmetry teams to bodily very important structures of differential equations. Emphasis is put on major functions of group-theoretic equipment, prepared in order that the utilized reader can with no trouble study the fundamental computational ideas required for real actual difficulties. the 1st bankruptcy collects jointly (but doesn't end up) these features of Lie crew conception that are of significance to differential equations. functions lined within the physique of the publication comprise calculation of symmetry teams of differential equations, integration of normal differential equations, together with distinct concepts for Euler-Lagrange equations or Hamiltonian structures, differential invariants and development of equations with pre­scribed symmetry teams, group-invariant options of partial differential equations, dimensional research, and the connections among conservation legislation and symmetry teams. Generalizations of the fundamental symmetry crew suggestion, and purposes to conservation legislation, integrability stipulations, thoroughly integrable platforms and soliton equations, and bi-Hamiltonian structures are lined intimately. The exposition within reason self-contained, and supplemented by way of various examples of direct actual value, selected from classical mechanics, fluid mechanics, elasticity and different utilized components.

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Additional resources for Applications of Lie Groups to Differential Equations (Graduate Texts in Mathematics, Volume 107)

Example text

The proof of this theorem follows easily from the implicit function theorem using arguments similar to the above case of surfaces in 1R3. 8 says that we can choose local coordinates x = (Xl, ... , xm) on M near each XoE N such that F(x) = (x\ ... , xn). Thus, in terms of these coordinates, N = {Xl = ... = xn = O}, and so the x's provide the flat local coordinates for N near Xo. Moreover, the latter m - n components (x n+\ ... ,xm) then provide local coordinates on N itself. In particular, this proves that N is a regular submanifold.

Thus ifv is a vector field whose expression in the x-coordinates is v= and y = m. 16) in the y-coordinates. The next result illustrates our earlier remarks that by suitably choosing local coordinates, we can often simplify the expressions for objects on manifolds, in this case vector fields. 29. Suppose v is a vector field not vanishing at a point vi"" #- O. Then there is a local coordinate chart y = (yl, ... , ym) at that in terms of these coordinates, v = a/a y I. Xo Xo EM: such PROOF. First linearly change coordinates so that Xo = 0 and vi"" = a/ axl.

Are assumed to be connected. This will avoid constantly restating the connectedness condition in the statement of our results. A manifold M is simply-connected if every closed curve C c: M can be continuously deformed to a point. This is equivalent to the existence of a continuous map H:[O,I]x[O,l]~M such that H{t,O)=xo for all 0,;;; t,;;; 1, while H{t, 1), 0,;;; t,;;;1 parametrizes C. For example, ~m is simply-connected, while ~\{O} is not, as there is no way to continuously contract the unit circle to a point without passing through the origin.

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Applications of Lie Groups to Differential Equations (Graduate Texts in Mathematics, Volume 107) by Peter J. Olver

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