An Introduction to the Lie Theory of One-parameter Groups: With Applications to the Solution of Differential Equations |
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An Introduction to the Lie Theory of One-Parameter Groups: With Applications ... Abraham Cohen No preview available - 2023 |
Common terms and phrases
a₁ canonical variables coefficients common solution complete system const contact transformation coördinates determined differential equation invariant differential equation unaltered dx dy dy dx equa family of curves finite transformations function group Uf Hence homogeneous independent solutions infinitesimal transformation integral curves integrating factor invariant under Uf isothermal leave the differential left unaltered linear equation linear partial differential linearly independent lines of curvature method Note obtained one-parameter group ordinary differential equation P₁ parameters partial differential equation path-curves point transformation quadratures r-parameter group radius vector readily seen reduces Remark Riccati equation second order invariant singular solution solution of Aƒ solving takes the form tangent theorem tion two-parameter group type of differential U₁ U₁f U₁ƒ U₂f U₂ƒ Uf=x values whence x₁ y₁ δα аф ду ду ду дх дх ду дх дх эф
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Page 91 - EBSE6, which is $vdtR sin 6, where 6 is the angle between the radius vector and the tangent to the curve...
Page 41 - ... (that is, the slope of the tangent to the curve) at that point (positive if the tangent points upward, and negative if it points downward, moving to the right).
Page 55 - Thus the equation is a homogeneous equation, M and N being here of the first degree. To integrate a homogeneous equation it suffices to assume y = vx. In the transformed equation the variables x and v will then admit of separation. Thus in the...
Page 110 - It will be left as an exercise for the student to show that a similar procedure can be applied to a path which does not enclose the wire.
Page 3 - Lie group theory we shall call the system of demand functions "weakly" self-dual if the demand and inverse demand transformations can be arranged in pairs, the members of which are mutually inverse. That is to say, if, corresponding to a set of values of the essential parameters a...
Page 3 - So called because the effect of any one of them is to stretch the vector going from the origin to the point (x, y) in the ratio -.leaving its direction unaltered.
Page 20 - A single infinity of curves determined by an equation involving * an arbitrary constant is equally determined by a unique differential equation of the first order, of which the equation involving the arbitrary constant is the general solution. lif(x,y)=c and ш(х,у, t}=c...
Page 220 - point" (u, v) to the point (u', v'). The point (o, o) corresponds to the initial state of the given periodic motion and T will carry this point into FIG. 3 itself. T may be expressed as follows by means of power series in u and v : u = au bv v' = cu + dv + • • • where the dots stand for terms of higher degree than the first in u, v, while a, b, c, d may be readily calculated in terms of a.2 An interesting fact is that the determinant ad — be is exactly equal to i, a result which could have...
Page 34 - Rotationen um die £-Axe: #! = x cos t — y sin t, yi = x sin t + y cos t, deren infinitesimale Transformation ist: TJJ. 8f . df ür = — yj~ 4- x ^ . y dx
Page 40 - Find the curves such that the perpendicular distance from the origin to the tangent to a curve at any point is equal to the abscissa of that point.