Solving First Order Ordinary Differential Equations Using Lie Symmetries

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Solving First Order Ordinary Differential
Equations Using Lie Symmetries

• Katie Thompson

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Overview

• Discrete symmetries vs. continuous symmetries.
• What is a Lie symmetry?
• Using Lie symmetries to solve first order
  ordinary differential equations.
• Examining standard methods taught in a first
  course in ordinary differential equations.

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Transformations that are Symmetries

• We look at diffeomorphisms on the plane, i.e.,
  maps ?R2? R2.
• ?(x,y)(f(x,y),g(x,y)), which are invertible and
  infinitely differentiable.
• The set of diffeomorphisms ? forms a group
  under composition.

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Symmetries of Differential Equations

• A symmetry of a differential equation is a
  transformation that sends solutions to solutions.
• The identity map is a trivial example of a
  symmetry.
• Let y(x)?(y). Then a translation in x sends
  solutions to solutions.

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Discrete vs. Continuous Symmetries
y

• A rigid square has a finite, discrete set of
  symmetries.
• We interest ourselves in infinite sets of
  continuous symmetries.
• Consider the unit circle x2y21.

?
x
Any rotation by ?? R about the center is a
symmetry. We denote the set of symmetries
??(x,y)(xcos?-ysin?,xsin?xcos?).
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One-Parameter Lie Group Properties

• The set of diffeomorphisms ?? for
• ?? R forms a one-parameter Lie group if the
  following hold
• ? 0 is the trivial symmetry.
• ???????.
• Notice that the above imply that ?-???-1.

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Example of a One-Parameter Lie Group
y

• Again, consider the unit circle x2y21.

?
x
Any rotation by ?? R about the center is a
symmetry. We denote the set of symmetries
??(x,y)(xcos?-ysin?,xsin?xcos?).
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Symmetry Condition

• For the first-order ODE y(x)?(x,y), a symmetry
  group preserves the equation, i.e.

• This is equivalent to the condition

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Example
Scaling is a symmetry of this ODE.
Verify the symmetry condition
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Goal

• We want to use the Lie symmetries to separate
  variables and find the general solution for a
  first-order ordinary differential equation.
• From the symmetries, we find canonical
  coordinates. The canonical coordinates are the
  key to separating variables.

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Infinitesimal Generators

• Let p(x,y) be a point in R2 and let ?? be a
  one-parameter Lie group.
• Then the orbit of p under the group action is the
  set, Op??(p)?? R.
• Then we define the infinitesimal generator of ??
  at the point p as the tangent vector to Op at ?0.

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Canonical Coordinates

• We want to calculate canonical coordinates,
  (r,s)(r(x,y),s(x,y)).
• These canonical coordinates are built so that the
  Lie symmetries become translations (r,s?).
• Then the ODE reduces to

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Canonical Coordinates

• We know that for the translation, (r,s?), the
  tangent vector at (r,s) is given by (0,1).
• We use the chain rule to get

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Canonical Coordinates

• We can find r by solving

• Solve r(x,y) for y.
• We find s by calculating

• Then evaluate the integral at rr(x,y).

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Solve ODEs Using Canonical Coordinates

• To reduce the ODE to separation of variables,
  rewrite in terms of canonical coordinates.

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First-Order ODEs of Homogeneous Type

• These are differential equations of the form

• Make the substitution z(x)y(x)/x.
• Use this substitution to solve the ODE.
• This method comes from the symmetry

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Canonical Coordinates

• Then canonical coordinates are given by

• We rewrite in terms of canonical coordinates to
  get

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Example of 1st Order ODE of Homogeneous Type
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Riccati Equation
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Solutions to Riccati Equations

• Apply the transformation (ye-2?, xe?) to the
  solution when c is nonzero.

• When c0, the transformation sends the curve to
  itself.

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Beyond 1st Order ODEs

• Higher Order ODEs Symmetries are used to reduce
  the order of the differential equation.
• Partial Differential Equations
• Methods to find symmetric solutions are well
  understood.
• General methods to find non-symmetric families of
  solutions are not known, and is an open area of
  research.

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References

• Edwards, C. Henry and Penney, David E.
  Differential Equations Computing and Modeling.
  Upper Saddle River, NJ Prentice Hall 2000.
• Hydon, Peter E. Symmetry Methods for Differential
  Equations A Beginners Guide. Cambridge, UK
  Cambridge University Press, 2000.