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Using symmetry to solve differential equations Martin Jackson Mathematics and Computer Science, University of Puget Sound March 6, 2012 Outline 1 “Magic” coordinates Outline 1 “Magic” coordinates 2 Symmetries of a differential equation Outline 1 “Magic” coordinates 2 Symmetries of a differential equation 3 Using a symmetry to find “magic” coordinates Outline 1 “Magic” coordinates 2 Symmetries of a differential equation 3 Using a symmetry to find “magic” coordinates 4 Finding symmetries of a differential equation Outline 1 “Magic” coordinates 2 Symmetries of a differential equation 3 Using a symmetry to find “magic” coordinates 4 Finding symmetries of a differential equation 5 Topics for another time “Magic” coordinates “Magic” coordinates • start with a differential equation dy = f (x, y ) dx “Magic” coordinates • start with a differential equation dy = f (x, y ) dx 2 1 y 0 -1 -2 0 1 2 x 3 4 dy = f (x, y ) dx “Magic” coordinates dy = f (x, y ) dx ds • goal: find new variables to get = g (r ) dr dy = f (x, y ) dx • start with a differential equation 2 1 y 0 -1 -2 0 1 2 x 3 4 “Magic” coordinates dy = f (x, y ) dx ds • goal: find new variables to get = g (r ) dr ds dy = g (r ) = f (x, y ) dr dx • start with a differential equation y 2 0 1 -1 s 0 -1 -2 -3 -2 -4 0 1 2 x 3 4 -2 -1 0 r 1 2 • Example: dy =y dx dy =y dx • find new coordinates (r , s) to simplify problem • Example: dy =y dx • find new coordinates (r , s) to simplify problem • Example: 2 1 y 0 -1 -2 -2 -1 0 x 1 2 dy =y dx • find new coordinates (r , s) to simplify problem • Example: y 2 2 1 1 s 0 -1 0 -1 -2 -2 -2 -1 0 x 1 2 -2 -1 0 r 1 2 dy =y dx • find new coordinates (r , s) to simplify problem • Example: y 2 2 1 1 s 0 -1 0 -1 -2 -2 -2 -1 0 1 2 -2 x • obvious choice: r = y and s = x -1 0 r 1 2 dy =y dx • find new coordinates (r , s) to simplify problem • Example: y 2 2 1 1 s 0 -1 0 -1 -2 -2 -2 -1 0 1 2 -2 x • obvious choice: r = y and s = x • change coordinates: -1 0 r 1 2 dy =y dx • find new coordinates (r , s) to simplify problem • Example: y 2 2 1 1 s 0 -1 0 -1 -2 -2 -2 -1 0 1 2 -2 x -1 0 r • obvious choice: r = y and s = x • change coordinates: ds dr 1 2 dy =y dx • find new coordinates (r , s) to simplify problem • Example: y 2 2 1 1 s 0 -1 0 -1 -2 -2 -2 -1 0 1 2 -2 x -1 0 r • obvious choice: r = y and s = x • change coordinates: ds dx = dr dy 1 2 dy =y dx • find new coordinates (r , s) to simplify problem • Example: y 2 2 1 1 s 0 -1 0 -1 -2 -2 -2 -1 0 1 2 -2 x -1 0 r • obvious choice: r = y and s = x • change coordinates: ds dx 1 = = dr dy dy /dx 1 2 dy =y dx • find new coordinates (r , s) to simplify problem • Example: y 2 2 1 1 s 0 -1 0 -1 -2 -2 -2 -1 0 1 2 -2 -1 x 0 r • obvious choice: r = y and s = x • change coordinates: ds dx 1 1 = = = dr dy dy /dx y 1 2 dy =y dx • find new coordinates (r , s) to simplify problem • Example: y 2 2 1 1 s 0 -1 0 -1 -2 -2 -2 -1 0 1 2 -2 x -1 0 r • obvious choice: r = y and s = x • change coordinates: ds dx 1 1 1 = = = = dr dy dy /dx y r 1 2 Using “magic” coordinates Using “magic” coordinates • change coordinates: ds 1 dy = y =⇒ = dx dr r Using “magic” coordinates • change coordinates: ds 1 dy = y =⇒ = dx dr r • integrate: Z s= 1 dr = ln r + C r Using “magic” coordinates • change coordinates: ds 1 dy = y =⇒ = dx dr r • integrate: Z s= 1 dr = ln r + C r • change back: x = ln y + C Using “magic” coordinates • change coordinates: ds 1 dy = y =⇒ = dx dr r • integrate: Z s= 1 dr = ln r + C r • change back: x = ln y + C • solve: y = Ce x Another example of “magic” coordinates Another example of “magic” coordinates • Example: dy y y2 = + 3 dx x x Another example of “magic” coordinates • Example: dy y y2 = + 3 dx x x 2 1 y 0 -1 -2 0 1 2 x 3 4 Another example of “magic” coordinates dy y y2 = + 3 dx x x y 1 • a not-so-obvious choice: r = and s = − x x • Example: 2 1 y 0 -1 -2 0 1 2 x 3 4 Another example of “magic” coordinates dy y y2 = + 3 dx x x y 1 • a not-so-obvious choice: r = and s = − x x • change coordinates: • Example: d(− x1 ) ds 1 = = ... = 2 y dr d( x ) r 2 1 y 0 -1 -2 0 1 2 x 3 4 Another example of “magic” coordinates dy y y2 = + 3 dx x x y 1 • a not-so-obvious choice: r = and s = − x x • change coordinates: • Example: d(− x1 ) ds 1 = = ... = 2 y dr d( x ) r y 2 0 1 -1 s 0 -1 -2 -3 -2 -4 0 1 2 x 3 4 -2 -1 0 r 1 2 • change coordinates: dy y y2 1 ds = + 3 =⇒ = 2 dx x x dr r • change coordinates: dy y y2 1 ds = + 3 =⇒ = 2 dx x x dr r • integrate: Z s= 1 1 dr = − + C 2 r r • change coordinates: dy y y2 1 ds = + 3 =⇒ = 2 dx x x dr r • integrate: Z s= 1 1 dr = − + C 2 r r • change back: − 1 x =− +C x y • change coordinates: dy y y2 1 ds = + 3 =⇒ = 2 dx x x dr r • integrate: Z 1 1 dr = − + C 2 r r s= • change back: − 1 x =− +C x y • solve: y= x2 1 + Cx • change coordinates: dy y y2 1 ds = + 3 =⇒ = 2 dx x x dr r • integrate: Z 1 1 dr = − + C 2 r r s= • change back: − 1 x =− +C x y • solve: y= x2 1 + Cx • Question: how do we come up with r = y 1 and s = − ? x x • change coordinates: dy y y2 1 ds = + 3 =⇒ = 2 dx x x dr r • integrate: Z 1 1 dr = − + C 2 r r s= • change back: − 1 x =− +C x y • solve: y= x2 1 + Cx • Question: how do we come up with r = • Answer: symmetry! y 1 and s = − ? x x Transforming the plane Transforming the plane • define a mapping T of the plane to itself Transforming the plane • define a mapping T of the plane to itself • Example: reflection across the y -axis: Transforming the plane • define a mapping T of the plane to itself • Example: reflection across the y -axis: y Hx,yL x Transforming the plane • define a mapping T of the plane to itself • Example: reflection across the y -axis: y ` ` Hx ,y L Hx,yL x Transforming the plane • define a mapping T of the plane to itself • Example: reflection across the y -axis: y ` ` Hx ,y L Hx,yL x • denote this (x̂, ŷ ) = T (x, y ) = (−x, y ) Transformation flows Transformation flows • define a one-parameter family of mappings T that maps the plane to itself for each value of the parameter Transformation flows • define a one-parameter family of mappings T that maps the plane to itself for each value of the parameter • Examples: Demo Transformation flows • define a one-parameter family of mappings T that maps the plane to itself for each value of the parameter • Examples: Demo (x̂, ŷ ) = T (x, y ) = (x + , y ) Transformation flows • define a one-parameter family of mappings T that maps the plane to itself for each value of the parameter • Examples: Demo (x̂, ŷ ) = T (x, y ) = (x + , y ) (x̂, ŷ ) = (x, y + ) Transformation flows • define a one-parameter family of mappings T that maps the plane to itself for each value of the parameter • Examples: Demo (x̂, ŷ ) = T (x, y ) = (x + , y ) (x̂, ŷ ) = (x, y + ) (x̂, ŷ ) = (e x, y ) Transformation flows • define a one-parameter family of mappings T that maps the plane to itself for each value of the parameter • Examples: Demo (x̂, ŷ ) = T (x, y ) = (x + , y ) (x̂, ŷ ) = (x, y + ) (x̂, ŷ ) = (e x, y ) (x̂, ŷ ) = (e x, e y ) Transformation flows • define a one-parameter family of mappings T that maps the plane to itself for each value of the parameter • Examples: Demo (x̂, ŷ ) = T (x, y ) = (x + , y ) (x̂, ŷ ) = (x, y + ) (x̂, ŷ ) = (e x, y ) (x̂, ŷ ) = (e x, e y ) (x̂, ŷ ) = (e x, e − y ) Transformation flows • define a one-parameter family of mappings T that maps the plane to itself for each value of the parameter • Examples: Demo (x̂, ŷ ) = T (x, y ) = (x + , y ) (x̂, ŷ ) = (x, y + ) (x̂, ŷ ) = (e x, y ) (x̂, ŷ ) = (e x, e y ) (x̂, ŷ ) = (e x, e − y ) (x̂, ŷ ) = x y , 1 − x 1 − x Transformation flows • define a one-parameter family of mappings T that maps the plane to itself for each value of the parameter • Examples: Demo (x̂, ŷ ) = T (x, y ) = (x + , y ) (x̂, ŷ ) = (x, y + ) (x̂, ŷ ) = (e x, y ) (x̂, ŷ ) = (e x, e y ) (x̂, ŷ ) = (e x, e − y ) (x̂, ŷ ) = x y (x, y ) , = 1 − x 1 − x 1 − x Transformation flows • define a one-parameter family of mappings T that maps the plane to itself for each value of the parameter • Examples: Demo (x̂, ŷ ) = T (x, y ) = (x + , y ) (x̂, ŷ ) = (x, y + ) (x̂, ŷ ) = (e x, y ) (x̂, ŷ ) = (e x, e y ) (x̂, ŷ ) = (e x, e − y ) (x̂, ŷ ) = x y (x, y ) , = 1 − x 1 − x 1 − x (x̂, ŷ ) = (x cos − y sin , x sin + y cos ) Lie group structure • each of these transformation flows has certain algebraic properties: T0 = Id T ◦ Tδ = T+δ T−1 = T− Lie group structure • each of these transformation flows has certain algebraic properties: T0 = Id T ◦ Tδ = T+δ T−1 = T− • so each of these is a group under composition Lie group structure • each of these transformation flows has certain algebraic properties: T0 = Id T ◦ Tδ = T+δ T−1 = T− • so each of these is a group under composition • each flow is also “nice” as a function of Lie group structure • each of these transformation flows has certain algebraic properties: T0 = Id T ◦ Tδ = T+δ T−1 = T− • so each of these is a group under composition • each flow is also “nice” as a function of • so each is a one-parameter group of transformations that is “nice” as a function of Lie group structure • each of these transformation flows has certain algebraic properties: T0 = Id T ◦ Tδ = T+δ T−1 = T− • so each of these is a group under composition • each flow is also “nice” as a function of • so each is a one-parameter group of transformations that is “nice” as a function of • these are called one-parameter Lie groups Lie group structure • each of these transformation flows has certain algebraic properties: T0 = Id T ◦ Tδ = T+δ T−1 = T− • so each of these is a group under composition • each flow is also “nice” as a function of • so each is a one-parameter group of transformations that is “nice” as a function of • these are called one-parameter Lie groups • now ready to define symmetries of geometric objects Symmetries of a curve • look at the effect of a transformation flow on a geometric object Symmetries of a curve • look at the effect of a transformation flow on a geometric object • Example: What happens to the unit circle under each of the previous transformation flows? Demo Symmetries of a curve • look at the effect of a transformation flow on a geometric object • Example: What happens to the unit circle under each of the previous transformation flows? Demo • under most of them, the circle is not mapped to itself Symmetries of a curve • look at the effect of a transformation flow on a geometric object • Example: What happens to the unit circle under each of the previous transformation flows? Demo • under most of them, the circle is not mapped to itself • the circle is mapped to itself for rotation through any angle Symmetries of a curve • look at the effect of a transformation flow on a geometric object • Example: What happens to the unit circle under each of the previous transformation flows? Demo • under most of them, the circle is not mapped to itself • the circle is mapped to itself for rotation through any angle • the circle has a symmetry for each angle so the circle has a one-parameter symmetry flow (in this case, a one-parameter Lie symmetry) Symmetries of a curve • look at the effect of a transformation flow on a geometric object • Example: What happens to the unit circle under each of the previous transformation flows? Demo • under most of them, the circle is not mapped to itself • the circle is mapped to itself for rotation through any angle • the circle has a symmetry for each angle so the circle has a one-parameter symmetry flow (in this case, a one-parameter Lie symmetry) • in general, a geometric object in the plane has a symmetry flow (or a Lie symmetry) if there is a “nice” transformation flow of the plane that maps that object to itself First-order differential equations as geometric objects First-order differential equations as geometric objects • geometric view of a first-order ODE as a slope field. First-order differential equations as geometric objects • geometric view of a first-order ODE as a slope field. • Examples: First-order differential equations as geometric objects • geometric view of a first-order ODE as a slope field. • Examples: dy =y dx 2 1 y 0 -1 -2 -2 -1 0 x 1 2 First-order differential equations as geometric objects • geometric view of a first-order ODE as a slope field. • Examples: y y2 dy = + 3 dx x x dy =y dx y 2 2 1 1 0 y -1 0 -1 -2 -2 -2 -1 0 x 1 2 0 1 2 x 3 4 First-order differential equations as geometric objects • geometric view of a first-order ODE as a slope field. • Examples: y y2 dy = + 3 dx x x dy =y dx y 2 2 1 1 0 y -1 0 -1 -2 -2 -2 -1 0 x 1 2 0 1 2 3 4 x • to understand how a slope field transforms, first look at how slopes transform Effect of a transformation on slopes Effect of a transformation on slopes • Example: translation in x Effect of a transformation on slopes • Example: translation in x • each point is mapped by (x̂, ŷ ) = (x + , y ) Effect of a transformation on slopes • Example: translation in x • • each point is mapped by (x̂, ŷ ) = (x + , y ) each tangent line segment at (x, y ) is mapped to a tangent line segment at (x̂, ŷ ) Effect of a transformation on slopes • Example: translation in x • • each point is mapped by (x̂, ŷ ) = (x + , y ) each tangent line segment at (x, y ) is mapped to a tangent line segment at (x̂, ŷ ) y m Hx,yL x Effect of a transformation on slopes • Example: translation in x • • each point is mapped by (x̂, ŷ ) = (x + , y ) each tangent line segment at (x, y ) is mapped to a tangent line segment at (x̂, ŷ ) y ` m m Hx,yL ` ` Hx ,y L x Effect of a transformation on slopes • Example: translation in x • • each point is mapped by (x̂, ŷ ) = (x + , y ) each tangent line segment at (x, y ) is mapped to a tangent line segment at (x̂, ŷ ) y ` m m Hx,yL ` ` Hx ,y L x • each transformed tangent line segment has slope m̂ that is the same as the original slope m so (x̂, ŷ , m̂) = T (x, y , m) = (x + , y , m) • Example: scaling in x: (x̂, ŷ ) = T (x, y ) = (e x, y ) • Example: scaling in x: (x̂, ŷ ) = T (x, y ) = (e x, y ) • tangent line segments will be scaled in the x-direction • Example: scaling in x: (x̂, ŷ ) = T (x, y ) = (e x, y ) • tangent line segments will be scaled in the x-direction y m Hx,yL x • Example: scaling in x: (x̂, ŷ ) = T (x, y ) = (e x, y ) • tangent line segments will be scaled in the x-direction y ` m m Hx,yL ` ` Hx ,y L x • Example: scaling in x: (x̂, ŷ ) = T (x, y ) = (e x, y ) • tangent line segments will be scaled in the x-direction y ` m m Hx,yL ` ` Hx ,y L x • scaling in x will scale run and have no effect on rise so • Example: scaling in x: (x̂, ŷ ) = T (x, y ) = (e x, y ) • tangent line segments will be scaled in the x-direction y ` m m Hx,yL ` ` Hx ,y L x • scaling in x will scale run and have no effect on rise so m̂ = ˆ rise run ˆ • Example: scaling in x: (x̂, ŷ ) = T (x, y ) = (e x, y ) • tangent line segments will be scaled in the x-direction y ` m m Hx,yL ` ` Hx ,y L x • scaling in x will scale run and have no effect on rise so m̂ = ˆ rise rise = run ˆ e run • Example: scaling in x: (x̂, ŷ ) = T (x, y ) = (e x, y ) • tangent line segments will be scaled in the x-direction y ` m m Hx,yL ` ` Hx ,y L x • scaling in x will scale run and have no effect on rise so m̂ = ˆ rise rise rise = = e − run ˆ e run run • Example: scaling in x: (x̂, ŷ ) = T (x, y ) = (e x, y ) • tangent line segments will be scaled in the x-direction y ` m m Hx,yL ` ` Hx ,y L x • scaling in x will scale run and have no effect on rise so m̂ = ˆ rise rise rise = = e − = e − m run ˆ e run run • Example: scaling in x: (x̂, ŷ ) = T (x, y ) = (e x, y ) • tangent line segments will be scaled in the x-direction y ` m m Hx,yL ` ` Hx ,y L x • scaling in x will scale run and have no effect on rise so m̂ = • so ˆ rise rise rise = = e − = e − m run ˆ e run run (x̂, ŷ , m̂) = T (x, y , m) = (e x, y , e − m) • Example: scaling in x: (x̂, ŷ ) = T (x, y ) = (e x, y ) • tangent line segments will be scaled in the x-direction y ` m m Hx,yL ` ` Hx ,y L x • scaling in x will scale run and have no effect on rise so m̂ = • so ˆ rise rise rise = = e − = e − m run ˆ e run run (x̂, ŷ , m̂) = T (x, y , m) = (e x, y , e − m) Demo Symmetries of a first-order differential equation Symmetries of a first-order differential equation • start with a differential equation dy = f (x, y ). dx Symmetries of a first-order differential equation dy = f (x, y ). dx • look at the effect of a transformation T on a slope field • start with a differential equation Symmetries of a first-order differential equation dy = f (x, y ). dx • look at the effect of a transformation T on a slope field dy • Example: = y under reflection across the x-axis dx • start with a differential equation Symmetries of a first-order differential equation dy = f (x, y ). dx • look at the effect of a transformation T on a slope field dy • Example: = y under reflection across the x-axis dx • start with a differential equation 2 1 y 0 -1 -2 -2 -1 0 x 1 2 Symmetries of a first-order differential equation dy = f (x, y ). dx • look at the effect of a transformation T on a slope field dy • Example: = y under reflection across the x-axis dx • start with a differential equation y 2 2 1 1 ` y 0 0 -1 -1 -2 -2 -2 -1 0 x 1 2 -2 -1 0 ` x 1 2 Symmetries of a first-order differential equation dy = f (x, y ). dx • look at the effect of a transformation T on a slope field dy • Example: = y under reflection across the x-axis dx • start with a differential equation y 2 2 1 1 ` y 0 0 -1 -1 -2 -2 -2 -1 0 x 1 2 -2 -1 0 ` x 1 2 • T is a symmetry of the differential equation if the slope field maps to itself (so each solution is mapped to a solution) • our interest is in symmetry under a transformation flow • our interest is in symmetry under a transformation flow • Example: explore dy = y under various transformation flows dx Demo • our interest is in symmetry under a transformation flow • Example: explore • dy = y under various transformation flows dx Demo translation in x: • our interest is in symmetry under a transformation flow • Example: explore • • dy = y under various transformation flows dx Demo translation in x: scaling in x: • our interest is in symmetry under a transformation flow • Example: explore dy = y under various transformation flows dx Demo translation in x: scaling in x: • translation in y : • • • our interest is in symmetry under a transformation flow • Example: explore dy = y under various transformation flows dx Demo translation in x: scaling in x: • translation in y : • scaling in y : • • • our interest is in symmetry under a transformation flow • Example: explore dy = y under various transformation flows dx Demo translation in x: a symmetry flow scaling in x: • translation in y : • scaling in y : • • • our interest is in symmetry under a transformation flow • Example: explore dy = y under various transformation flows dx Demo translation in x: a symmetry flow scaling in x: not a symmetry flow • translation in y : • scaling in y : • • • our interest is in symmetry under a transformation flow • Example: explore dy = y under various transformation flows dx Demo translation in x: a symmetry flow scaling in x: not a symmetry flow • translation in y : not a symmetry flow • scaling in y : • • • our interest is in symmetry under a transformation flow • Example: explore dy = y under various transformation flows dx Demo translation in x: a symmetry flow scaling in x: not a symmetry flow • translation in y : not a symmetry flow • scaling in y : a symmetry flow • • • our interest is in symmetry under a transformation flow • Example: explore dy = y under various transformation flows dx Demo translation in x: a symmetry flow scaling in x: not a symmetry flow • translation in y : not a symmetry flow • scaling in y : a symmetry flow • • • Example: explore flows y y2 dy = + 3 under various transformation dx x x Demo • our interest is in symmetry under a transformation flow • Example: explore dy = y under various transformation flows dx Demo translation in x: a symmetry flow scaling in x: not a symmetry flow • translation in y : not a symmetry flow • scaling in y : a symmetry flow • • • Example: explore flows • scaling in y : y y2 dy = + 3 under various transformation dx x x Demo • our interest is in symmetry under a transformation flow • Example: explore dy = y under various transformation flows dx Demo translation in x: a symmetry flow scaling in x: not a symmetry flow • translation in y : not a symmetry flow • scaling in y : a symmetry flow • • • Example: explore flows y y2 dy = + 3 under various transformation dx x x Demo • scaling in y : • projective transformation T (x, y ) = (x, y ) : 1 − x • our interest is in symmetry under a transformation flow • Example: explore dy = y under various transformation flows dx Demo translation in x: a symmetry flow scaling in x: not a symmetry flow • translation in y : not a symmetry flow • scaling in y : a symmetry flow • • • Example: explore flows y y2 dy = + 3 under various transformation dx x x Demo • scaling in y : not a symmetry flow • projective transformation T (x, y ) = (x, y ) : 1 − x • our interest is in symmetry under a transformation flow • Example: explore dy = y under various transformation flows dx Demo translation in x: a symmetry flow scaling in x: not a symmetry flow • translation in y : not a symmetry flow • scaling in y : a symmetry flow • • • Example: explore flows y y2 dy = + 3 under various transformation dx x x Demo • scaling in y : not a symmetry flow • projective transformation T (x, y ) = (x, y ) a symmetry : flow 1 − x • our interest is in symmetry under a transformation flow • Example: explore dy = y under various transformation flows dx Demo translation in x: a symmetry flow scaling in x: not a symmetry flow • translation in y : not a symmetry flow • scaling in y : a symmetry flow • • • Example: explore flows • y y2 dy = + 3 under various transformation dx x x Demo scaling in y : not a symmetry flow (x, y ) a symmetry : flow 1 − x • before working with symmetries of a differential equation, look at a convenient way to picture a transformation flow • projective transformation T (x, y ) = Visualizing a transformation flow Visualizing a transformation flow • have looked at how a grid of points moves Visualizing a transformation flow • have looked at how a grid of points moves • can also look at paths traced out by points Visualizing a transformation flow • have looked at how a grid of points moves • can also look at paths traced out by points • Example: (x̂, ŷ ) = (e x, e − y ) Visualizing a transformation flow • have looked at how a grid of points moves • can also look at paths traced out by points • Example: (x̂, ŷ ) = (e x, e − y ) 2 1 y 0 -1 -2 -2 -1 0 x 1 2 Visualizing a transformation flow • have looked at how a grid of points moves • can also look at paths traced out by points • Example: (x̂, ŷ ) = (e x, e − y ) 2 2 1 y 1 y 0 0 -1 -1 -2 -2 -2 -1 0 x 1 2 -2 -1 0 x 1 2 Visualizing a transformation flow • have looked at how a grid of points moves • can also look at paths traced out by points • Example: (x̂, ŷ ) = (e x, e − y ) 2 2 1 y 1 y 0 0 -1 -1 -2 -2 -2 -1 0 x 1 2 -2 -1 ~ = (ξ, η) • denote the tangent vector field X 0 x 1 2 Computing the tangent vector field Computing the tangent vector field • the tangent vector field is given by ~ = X d x̂ d ŷ , d d ! =0 Computing the tangent vector field • the tangent vector field is given by d x̂ d ŷ , d d ~ = X • Example: (x̂, ŷ ) = (e x, e − y) ! =0 Computing the tangent vector field • the tangent vector field is given by d x̂ d ŷ , d d ~ = X • Example: (x̂, ŷ ) = (e x, e ~ = X d x̂ d ŷ , d d ! =0 − y) ! =0 Computing the tangent vector field • the tangent vector field is given by d x̂ d ŷ , d d ~ = X • Example: (x̂, ŷ ) = (e x, e ~ = X d x̂ d ŷ , d d − ! =0 = e x, −e − y =0 y) ! =0 Computing the tangent vector field • the tangent vector field is given by d x̂ d ŷ , d d ~ = X • Example: (x̂, ŷ ) = (e x, e ~ = X d x̂ d ŷ , d d ! =0 = e x, −e − y = (x, −y ) − =0 y) ! =0 Computing the tangent vector field • the tangent vector field is given by d x̂ d ŷ , d d ~ = X • Example: (x̂, ŷ ) = (e x, e − ! =0 y) 2 ~ = X ! d x̂ d ŷ , d d 1 =0 = e x, −e − y y 0 =0 -1 = (x, −y ) -2 -2 -1 0 x 1 2 • Example: (x̂, ŷ ) = x y , 1 − x 1 − x • Example: (x̂, ŷ ) = ~ = X d x̂ d ŷ , d d ! =0 x y , 1 − x 1 − x • Example: (x̂, ŷ ) = ~ = X d x̂ d ŷ , d d ! =0 x y , 1 − x 1 − x = xy x2 , 2 (1 − x) (1 − x)2 ! =0 • Example: (x̂, ŷ ) = ~ = X d x̂ d ŷ , d d ! =0 x y , 1 − x 1 − x = xy x2 , 2 (1 − x) (1 − x)2 ! =0 = x 2 , xy • Example: (x̂, ŷ ) = ~ = X d x̂ d ŷ , d d ! x y , 1 − x 1 − x = =0 xy x2 , 2 (1 − x) (1 − x)2 ! =0 2 1 y 0 -1 -2 -2 -1 0 x 1 2 = x 2 , xy • a few more examples of tangent vector fields: • a few more examples of tangent vector fields: Scaling in x (x̂, ŷ ) = (e x, y ) ~ = (x, 0) X 2 1 y 0 -1 -2 -2 -1 0 x 1 2 • a few more examples of tangent vector fields: Rotation (x̂, ŷ ) = (x cos − y sin , x sin + y cos ) ~ = (−y , x) X Scaling in x (x̂, ŷ ) = (e x, y ) ~ = (x, 0) X y 2 2 1 1 y 0 0 -1 -1 -2 -2 -2 -1 0 x 1 2 -2 -1 0 x 1 2 Using a symmetry to solve the differential equation Using a symmetry to solve the differential equation • find coordinates (r , s) in which the symmetry field is vertical and uniform Using a symmetry to solve the differential equation • find coordinates (r , s) in which the symmetry field is vertical and uniform ~ = (x 2 , xy ) X 2 1 y 0 -1 -2 -2 -1 0 x 1 2 Using a symmetry to solve the differential equation • find coordinates (r , s) in which the symmetry field is vertical and uniform ~ = (x 2 , xy ) X y ~ = (0, 1) X 2 2 1 1 0 s -1 0 -1 -2 -2 -2 -1 0 x 1 2 -2 -1 0 r 1 2 Using a symmetry to solve the differential equation • find coordinates (r , s) in which the symmetry field is vertical and uniform ~ = (x 2 , xy ) X y ~ = (0, 1) X 2 2 1 1 0 s -1 0 -1 -2 -2 -2 -1 0 x 1 2 -2 -1 0 r 1 2 • in the new coordinates, differential equation reduces to an antiderivative problem since symmetry maps solutions to solutions by translation in the dependent variable ~ = (1, 0) • Example: translation in x: X ~ = (1, 0) • Example: translation in x: X ~ = (1, 0) X 2 1 y 0 -1 -2 -2 -1 0 x 1 2 ~ = (1, 0) • Example: translation in x: X • from the geometry, can see that choosing r = y , s = x works ~ = (1, 0) X 2 1 y 0 -1 -2 -2 -1 0 x 1 2 ~ = (1, 0) • Example: translation in x: X • from the geometry, can see that choosing r = y , s = x works ~ = (0, 1) X ~ = (1, 0) X y 2 2 1 1 s 0 -1 0 -1 -2 -2 -2 -1 0 x 1 2 -2 -1 0 r 1 2 ~ = (1, 0) • Example: translation in x: X • from the geometry, can see that choosing r = y , s = x works ~ = (0, 1) X ~ = (1, 0) X y 2 2 1 1 s 0 -1 0 -1 -2 -2 -2 -1 0 1 2 -2 x • this is a symmetry flow for dy =y dx -1 0 r 1 2 ~ = (1, 0) • Example: translation in x: X • from the geometry, can see that choosing r = y , s = x works ~ = (0, 1) X ~ = (1, 0) X y 2 2 1 1 s 0 -1 0 -1 -2 -2 -2 -1 0 1 2 -2 x • this is a symmetry flow for -1 0 r 1 2 dy =y dx • in these “magic coordinates”, this ODE becomes ds 1 = dr r ~ = (0, y ) • Example: scaling in y : X ~ = (0, y ) • Example: scaling in y : X ~ = (0, y ) X 2 1 y 0 -1 -2 -2 -1 0 x 1 2 ~ = (0, y ) • Example: scaling in y : X • from the geometry, see that r = x and s is some function of y ; a choice that works is s = ln y ~ = (0, y ) X 2 1 y 0 -1 -2 -2 -1 0 x 1 2 ~ = (0, y ) • Example: scaling in y : X • from the geometry, see that r = x and s is some function of y ; a choice that works is s = ln y ~ = (0, 1) X ~ = (0, y ) X y 2 2 1 1 0 s -1 0 -1 -2 -2 -2 -1 0 x 1 2 -2 -1 0 r 1 2 ~ = (0, y ) • Example: scaling in y : X • from the geometry, see that r = x and s is some function of y ; a choice that works is s = ln y ~ = (0, 1) X ~ = (0, y ) X y 2 2 1 1 0 s -1 0 -1 -2 -2 -2 -1 0 1 2 -2 x -1 0 r 1 2 dy = y so can now transform dx the differential equation to the new coordinates • this is also a symmetry flow for Finding “magic” coordinates • choose r (x, y ) so that r = constant curves are tangent to ~ = (ξ, η) X Finding “magic” coordinates • choose r (x, y ) so that r = constant curves are tangent to ~ = (ξ, η) X ~ = (x 2 , xy ) • Example: X Finding “magic” coordinates • choose r (x, y ) so that r = constant curves are tangent to ~ = (ξ, η) X ~ = (x 2 , xy ) • Example: X 2 1 y 0 -1 -2 0 1 2 x 3 4 Finding “magic” coordinates • choose r (x, y ) so that r = constant curves are tangent to ~ = (ξ, η) X ~ = (x 2 , xy ) • Example: X 2 1 y 0 -1 -2 0 1 2 x 3 4 Finding “magic” coordinates • choose r (x, y ) so that r = constant curves are tangent to ~ = (ξ, η) X ~ = (x 2 , xy ) • Example: X 2 1 y 0 -1 -2 0 1 2 x 3 4 • equivalent to choosing r (x, y ) so that the derivative in the ~ is 0: direction of X Finding “magic” coordinates • choose r (x, y ) so that r = constant curves are tangent to ~ = (ξ, η) X ~ = (x 2 , xy ) • Example: X 2 1 y 0 -1 -2 0 1 2 x 3 4 • equivalent to choosing r (x, y ) so that the derivative in the ~ is 0: direction of X ~ · ∇r ~ = ξ ∂r + η ∂r = 0 X ∂x ∂y • choose s(x, y ) so that s = constant curves are nowhere tangent to r = constant curves and the derivative in the ~ is uniform: direction of X ~ · ∇s ~ = ξ ∂s + η ∂s = 1 X ∂x ∂y • choose s(x, y ) so that s = constant curves are nowhere tangent to r = constant curves and the derivative in the ~ is uniform: direction of X ~ · ∇s ~ = ξ ∂s + η ∂s = 1 X ∂x ∂y ~ = (x 2 , xy ) • Example: X 2 1 y 0 -1 -2 0 1 2 x 3 4 • choose s(x, y ) so that s = constant curves are nowhere tangent to r = constant curves and the derivative in the ~ is uniform: direction of X ~ · ∇s ~ = ξ ∂s + η ∂s = 1 X ∂x ∂y ~ = (x 2 , xy ) • Example: X 2 1 y 0 -1 -2 0 1 2 x 3 4 • choose s(x, y ) so that s = constant curves are nowhere tangent to r = constant curves and the derivative in the ~ is uniform: direction of X ~ · ∇s ~ = ξ ∂s + η ∂s = 1 X ∂x ∂y ~ = (x 2 , xy ) • Example: X 2 1 y 0 -1 -2 0 1 2 x 3 4 • choose s(x, y ) so that s = constant curves are nowhere tangent to r = constant curves and the derivative in the ~ is uniform: direction of X ~ · ∇s ~ = ξ ∂s + η ∂s = 1 X ∂x ∂y ~ = (x 2 , xy ) • Example: X 2 1 y 0 -1 -2 0 1 2 x 3 4 • looks intimidating but not so bad in practice since need only a specific solution rather than the general solution ~ = (x 2 , xy ) • Example: X ~ = (x 2 , xy ) • Example: X • must find a solution to ∂r ∂r + xy =0 ∂x ∂y ∂s ∂s x2 + xy =1 ∂x ∂y x2 ~ = (x 2 , xy ) • Example: X • must find a solution to ∂r ∂r + xy =0 ∂x ∂y ∂s ∂s x2 + xy =1 ∂x ∂y x2 • top equation:r is constant along curves given by ~ = (x 2 , xy ) • Example: X • must find a solution to ∂r ∂r + xy =0 ∂x ∂y ∂s ∂s x2 + xy =1 ∂x ∂y x2 • top equation:r is constant along curves given by dx dy = 2 x xy ~ = (x 2 , xy ) • Example: X • must find a solution to ∂r ∂r + xy =0 ∂x ∂y ∂s ∂s x2 + xy =1 ∂x ∂y x2 • top equation:r is constant along curves given by dx dy dx dy = =⇒ = 2 x xy x y ~ = (x 2 , xy ) • Example: X • must find a solution to ∂r ∂r + xy =0 ∂x ∂y ∂s ∂s x2 + xy =1 ∂x ∂y x2 • top equation:r is constant along curves given by dx dy dx dy y = =⇒ = =⇒ = constant 2 x xy x y x ~ = (x 2 , xy ) • Example: X • must find a solution to ∂r ∂r + xy =0 ∂x ∂y ∂s ∂s x2 + xy =1 ∂x ∂y x2 • top equation:r is constant along curves given by dx dy dx dy y y = =⇒ = =⇒ = constant so r = works 2 x xy x y x x ~ = (x 2 , xy ) • Example: X • must find a solution to ∂r ∂r + xy =0 ∂x ∂y ∂s ∂s x2 + xy =1 ∂x ∂y x2 • top equation:r is constant along curves given by dx dy dx dy y y = =⇒ = =⇒ = constant so r = works 2 x xy x y x x • solve bottom equation by looking for s depending only on x so x2 ∂s + xy · 0 = 1 ∂x ~ = (x 2 , xy ) • Example: X • must find a solution to ∂r ∂r + xy =0 ∂x ∂y ∂s ∂s x2 + xy =1 ∂x ∂y x2 • top equation:r is constant along curves given by dx dy dx dy y y = =⇒ = =⇒ = constant so r = works 2 x xy x y x x • solve bottom equation by looking for s depending only on x so x2 ∂s ∂s 1 + xy · 0 = 1 =⇒ = 2 ∂x ∂x x ~ = (x 2 , xy ) • Example: X • must find a solution to ∂r ∂r + xy =0 ∂x ∂y ∂s ∂s x2 + xy =1 ∂x ∂y x2 • top equation:r is constant along curves given by dx dy dx dy y y = =⇒ = =⇒ = constant so r = works 2 x xy x y x x • solve bottom equation by looking for s depending only on x so x2 ∂s ∂s 1 1 + xy · 0 = 1 =⇒ = 2 =⇒ s = − ∂x ∂x x x ~ = (x 2 , xy ) • Example: X • must find a solution to ∂r ∂r + xy =0 ∂x ∂y ∂s ∂s x2 + xy =1 ∂x ∂y x2 • top equation:r is constant along curves given by dx dy dx dy y y = =⇒ = =⇒ = constant so r = works 2 x xy x y x x • solve bottom equation by looking for s depending only on x so x2 ∂s ∂s 1 1 + xy · 0 = 1 =⇒ = 2 =⇒ s = − ∂x ∂x x x • these are the “magic” coordinates we used at the start Review the general plan • start with a differential equation in the form meaning slopes can vary in both x and y dy = f (x, y ) dx Review the general plan • start with a differential equation in the form meaning slopes can vary in both x and y dy = f (x, y ) dx • find a symmetry flow for the differential equation with tangent ~ = (ξ, η) vector field X Review the general plan • start with a differential equation in the form meaning slopes can vary in both x and y dy = f (x, y ) dx • find a symmetry flow for the differential equation with tangent ~ = (ξ, η) vector field X • for that symmetry flow, find new coordinates r and s so that ~ = (0, 1) the tangent vector field is X Review the general plan • start with a differential equation in the form meaning slopes can vary in both x and y dy = f (x, y ) dx • find a symmetry flow for the differential equation with tangent ~ = (ξ, η) vector field X • for that symmetry flow, find new coordinates r and s so that ~ = (0, 1) the tangent vector field is X • in the new coordinates, slopes can vary only in r and not in s since translation in s maps slope field to slope field Review the general plan • start with a differential equation in the form meaning slopes can vary in both x and y dy = f (x, y ) dx • find a symmetry flow for the differential equation with tangent ~ = (ξ, η) vector field X • for that symmetry flow, find new coordinates r and s so that ~ = (0, 1) the tangent vector field is X • in the new coordinates, slopes can vary only in r and not in s since translation in s maps slope field to slope field • thus, in the new coordinates, the differential equation has the form ds = g (r ) dr Review the general plan • start with a differential equation in the form meaning slopes can vary in both x and y dy = f (x, y ) dx • find a symmetry flow for the differential equation with tangent ~ = (ξ, η) vector field X • for that symmetry flow, find new coordinates r and s so that ~ = (0, 1) the tangent vector field is X • in the new coordinates, slopes can vary only in r and not in s since translation in s maps slope field to slope field • thus, in the new coordinates, the differential equation has the ds = g (r ) dr • integrate! form Review the general plan • start with a differential equation in the form meaning slopes can vary in both x and y dy = f (x, y ) dx • find a symmetry flow for the differential equation with tangent ~ = (ξ, η) vector field X • for that symmetry flow, find new coordinates r and s so that ~ = (0, 1) the tangent vector field is X • in the new coordinates, slopes can vary only in r and not in s since translation in s maps slope field to slope field • thus, in the new coordinates, the differential equation has the ds = g (r ) dr • integrate! form • change back to the original coordinates Review the general plan • start with a differential equation in the form meaning slopes can vary in both x and y dy = f (x, y ) dx • find a symmetry flow for the differential equation with tangent ~ = (ξ, η) vector field X Wait a minute, how do we do that!? • for that symmetry flow, find new coordinates r and s so that ~ = (0, 1) the tangent vector field is X • in the new coordinates, slopes can vary only in r and not in s since translation in s maps slope field to slope field • thus, in the new coordinates, the differential equation has the ds = g (r ) dr • integrate! form • change back to the original coordinates Finding the symmetries of a differential equation Finding the symmetries of a differential equation • defining condition: (x̂, ŷ ) = T (x, y ) is a symmetry flow if dy d ŷ = f (x, y ) =⇒ = f (x̂, ŷ ) dx d x̂ Finding the symmetries of a differential equation • defining condition: (x̂, ŷ ) = T (x, y ) is a symmetry flow if dy d ŷ = f (x, y ) =⇒ = f (x̂, ŷ ) dx d x̂ ~ = (ξ, η) by • strategy: determine the tangent vector field X linearizing the defining condition Finding the symmetries of a differential equation • defining condition: (x̂, ŷ ) = T (x, y ) is a symmetry flow if dy d ŷ = f (x, y ) =⇒ = f (x̂, ŷ ) dx d x̂ ~ = (ξ, η) by • strategy: determine the tangent vector field X linearizing the defining condition • start with (x̂, ŷ ) = (x, y ) + (ξ, η) + higher-order terms to be ignored Finding the symmetries of a differential equation • defining condition: (x̂, ŷ ) = T (x, y ) is a symmetry flow if dy d ŷ = f (x, y ) =⇒ = f (x̂, ŷ ) dx d x̂ ~ = (ξ, η) by • strategy: determine the tangent vector field X linearizing the defining condition • start with (x̂, ŷ ) = (x, y ) + (ξ, η) + higher-order terms to be ignored • substitute into defining condition: d(y + η) = f (x + ξ, y + η) d(x + ξ) Finding the symmetries of a differential equation • defining condition: (x̂, ŷ ) = T (x, y ) is a symmetry flow if dy d ŷ = f (x, y ) =⇒ = f (x̂, ŷ ) dx d x̂ ~ = (ξ, η) by • strategy: determine the tangent vector field X linearizing the defining condition • start with (x̂, ŷ ) = (x, y ) + (ξ, η) + higher-order terms to be ignored • substitute into defining condition: d(y + η) = f (x + ξ, y + η) d(x + ξ) • after the dust settles: ∂η ∂η ∂ξ ∂f ∂f ∂ξ +f − = ξ+ η −f2 ∂x ∂y ∂x ∂y ∂x ∂y Finding the symmetries of a differential equation • defining condition: (x̂, ŷ ) = T (x, y ) is a symmetry flow if dy d ŷ = f (x, y ) =⇒ = f (x̂, ŷ ) dx d x̂ ~ = (ξ, η) by • strategy: determine the tangent vector field X linearizing the defining condition • start with (x̂, ŷ ) = (x, y ) + (ξ, η) + higher-order terms to be ignored • substitute into defining condition: d(y + η) = f (x + ξ, y + η) d(x + ξ) • after the dust settles: ∂η ∂η ∂ξ ∂f ∂f ∂ξ +f − = ξ+ η −f2 ∂x ∂y ∂x ∂y ∂x ∂y • first-order linear PDE for ξ(x, y ) and η(x, y ) ∂η ∂η ∂ξ ∂ξ ∂f ∂f +f − −f2 = ξ+ η ∂x ∂y ∂x ∂y ∂x ∂y ∂η ∂η ∂ξ ∂ξ ∂f ∂f +f − −f2 = ξ+ η ∂x ∂y ∂x ∂y ∂x ∂y • Example: dy = y so f (x, y ) = y dx ∂η ∂η ∂ξ ∂ξ ∂f ∂f +f − −f2 = ξ+ η ∂x ∂y ∂x ∂y ∂x ∂y dy = y so f (x, y ) = y dx • try ξ = ax + by + c and η = αx + βy + γ • Example: ∂η ∂η ∂ξ ∂ξ ∂f ∂f +f − −f2 = ξ+ η ∂x ∂y ∂x ∂y ∂x ∂y dy = y so f (x, y ) = y dx • try ξ = ax + by + c and η = αx + βy + γ • substitute: • Example: α + (β − a)y − by 2 = αx + βy + γ ∂η ∂η ∂ξ ∂ξ ∂f ∂f +f − −f2 = ξ+ η ∂x ∂y ∂x ∂y ∂x ∂y dy = y so f (x, y ) = y dx • try ξ = ax + by + c and η = αx + βy + γ • substitute: • Example: α + (β − a)y − by 2 = αx + βy + γ • match coefficients: ∂η ∂η ∂ξ ∂ξ ∂f ∂f +f − −f2 = ξ+ η ∂x ∂y ∂x ∂y ∂x ∂y dy = y so f (x, y ) = y dx • try ξ = ax + by + c and η = αx + βy + γ • substitute: • Example: α + (β − a)y − by 2 = αx + βy + γ • match coefficients: 1: α=γ x: 0=α y: β−a=β 2 y : b=0 ∂η ∂η ∂ξ ∂ξ ∂f ∂f +f − −f2 = ξ+ η ∂x ∂y ∂x ∂y ∂x ∂y dy = y so f (x, y ) = y dx • try ξ = ax + by + c and η = αx + βy + γ • substitute: • Example: α + (β − a)y − by 2 = αx + βy + γ • match coefficients: 1: α=γ x: 0=α y: 2 y : β−a=β b=0 ξ=c =⇒ η = βy ∂η ∂η ∂ξ ∂ξ ∂f ∂f +f − −f2 = ξ+ η ∂x ∂y ∂x ∂y ∂x ∂y dy = y so f (x, y ) = y dx • try ξ = ax + by + c and η = αx + βy + γ • substitute: • Example: α + (β − a)y − by 2 = αx + βy + γ • match coefficients: 1: α=γ x: 0=α y: 2 y : • β−a=β ξ=c =⇒ η = βy b=0 ~ = (1, 0) and X ~ = (0, y ) are symmetry vector fields so X Topics for another time • Classifying first-order ODEs by symmetry Topics for another time • Classifying first-order ODEs by symmetry • Symmetries of higher-order ODEs Topics for another time • Classifying first-order ODEs by symmetry • Symmetries of higher-order ODEs • Symmetries of PDEs Topics for another time • Classifying first-order ODEs by symmetry • Symmetries of higher-order ODEs • Symmetries of PDEs • Finding invariant solutions (example: fundamental solution for the heat equation) Topics for another time • Classifying first-order ODEs by symmetry • Symmetries of higher-order ODEs • Symmetries of PDEs • Finding invariant solutions (example: fundamental solution for the heat equation) • Algebraic structure of symmetries: Lie groups and Lie algebras Topics for another time • Classifying first-order ODEs by symmetry • Symmetries of higher-order ODEs • Symmetries of PDEs • Finding invariant solutions (example: fundamental solution for the heat equation) • Algebraic structure of symmetries: Lie groups and Lie algebras • Variational symmetries Topics for another time • Classifying first-order ODEs by symmetry • Symmetries of higher-order ODEs • Symmetries of PDEs • Finding invariant solutions (example: fundamental solution for the heat equation) • Algebraic structure of symmetries: Lie groups and Lie algebras • Variational symmetries • Nonclassical symmetries A few references A few references Peter Hydon, Symmetry Methods for Differential Equations: A Beginner’s Guide, Cambridge, 2000. A few references Peter Hydon, Symmetry Methods for Differential Equations: A Beginner’s Guide, Cambridge, 2000. Peter Olver, Applications of Lie Groups to Differential Equations 2nd ed., Springer, 1993.