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Symmetry Methods for Differential Equations Background: Many ingenious techniques for obtaining exact solutions of differential equations are based on exploiting symmetries of differential equations. It is quite convenient to find symmetries of a given differential equation (even the unfamiliar ones) and to use them systematically to obtain exact solutions…
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Symmetry Methods for Differential Equations
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Download file to see previous pages Lie’s methodology is based on this philosophy. The main challenge was to find the group, which leaves the solutions of a differential equation invariant, meaning which group maps solutions into solutions. This factor was considered the trivial constant, which can be added to any indefinite integral. The additive constant represents an element in a translation group. In the simplest first order ODE, one independent variable x and one dependent variable y can be represented by: dy/dx = g(x).General formulations of constraint equation ( dy/dx=p) and a surface equation( F(x,y,p)=0) are used to write down the solutions by quadratures. Lie’s methodology provides an algorithm, for determining, whether an ODE possesses symmetry and if so, the kind of symmetry. Transformations to a set of canonical variables like R,S,T is algorithmic. A canonical variable R(x,y) signifies the new variable like x, while S(x,y) is the new variable like y and T (x,y,p) forms the new constraint between S and R( similar to dy/dx). Under these new coordinate system the surface and constraint equations are designated by F(R,-T)= 0 and dS/dR =f (R, -T) respectively. The system is reduced to quadratures and integration follows. Chapter 1: Concept of Symmetry and Transformations Concept of Symmetry Symmetry of geometrical objects or a physical system refers to the property of being “unchanged” under certain transformations. Hence, symmetry of a physical system or geometric object is an intrinsic or observed feature of the system that remains preserved under a specified change. The transformations can be continuous (for example, rotations of a circle) or may be discontinuous (for example, rotations of a regular polygon). An object is said to bear a rotational symmetry if the object is turned around at its centre point by certain number of degrees and the object still looks the same. Thus it matches itself a number of times while it is being rotated. For example a flower with 5 petals will have symmetry of order 5, as it will match itself 5 times. Invariance is example of one such symmetry under arbitrary differentiable co-ordinate transformations. Invariance is specified algebraically that leaves some quantity unchanged. For example, humidity may be constant throughout a room, since humidity is independent of position within the room; it is invariant under a shift in the measurer’s position. Thus when a sphere is rotated about its center, it will appear exactly the same as it was before rotation. So the sphere exhibits a spherical symmetry. This means a rotation about any axis of the sphere preserves the shape of the sphere. The concept can be illustrated with the example of an electrical wire. The electric field of a wire exhibits cylindrical symmetry. The strength of an electric field at a specified distance (?) from the charged wire with infinite length will bear the same magnitude at each point on the surface of the cylinder (electrical field) with its axis being the wire having a radius (?). If the wire is rotated on its own axis, it will not change its position or the charge density and hence the electric field will be preserved. Hence the field strength at a rotated position is the same. When some configuration of charges (non-stationary) produces an ...Download file to see next pagesRead More
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