Solving Differential Equations with Diffeomorphisms

Change of Variables §

• (see here. is independent of the coordinate system used for both and provided that both are connected to the Cartesian coordinate system by a diffeomorphic change of variables.

  Thus, diffeomorphisms can be used to transform an ODE to one we know how to solve through a change in variables.

• The vector field on the - axis with a unique component is denoted . This allows us to compute a diffeomorphism on a vector field

  • In general, let be a fixed coordinate system in . Then the basis vector fields are denoted
    The vector field with components is denoted
  • Notice that the basis vector fields represent a change of coordinates matrix. If we have a given vector field, we can then obtain the vector fields in this coordinate system using matrix vector multiplication (which produces the sum above).
  • Another way to write this is with the Jacobian Matrix . Suppose we have functions and basis vector fields with
    Define
    Then the change of variables is obtained as

• (Arnold 5.3.1) Let be the image of the vector field in under a diffeomorphism of a domain onto a domain (i.e., ).

  The differential equation

  Are equivalent in the sense that if is a solution of the first, then is a solution of the second equations and conversely.

  Thus, the change of variables maps the first equation onto the second. Similarly, the change of variables maps the second onto the first.

  • Proof: Verify that the proposed solutions are solutions
    With , .

• To solve the differential equation

  It suffices to construct a diffeomorphism that maps the direction field of the above to an equation we know how to solve.

• Let be a one parameter diffeomorphism group and be another onto diffeomorphism.

  The image of the flow under the action of the diffeomorphism is the flow where . Alternatively, this can be written as

  For any .

  The orbits of are mapped to the orbits of . We say that and are equivalent.

  • Clearly the image is also a one-parameter diffeomorphism group.
  • (Arnold 5.5.1) The diffeomorphism takes the vector field into the field ; Conversely, if a diffeomorphism takes into , then it takes to .

Integration §

• (Arnold 6.2.1.) Suppose a one-parameter group of symmetries of a direction field on the plane is known. Then the equation defined by this direction field can be integrated explicitly in a neighborhood of each nonstationary point of the symmetry group.

  A point is called nonstationary for a transformation group if not all transformations of the group leave it fixed

• (Arnold 6.2.2) In a neighborhood of every nonstationary point of action of a one parameter group of diffeomorphisms on the plane, one can choose coordinates such that the given one-parameter diffeomorphism group can be written in the form of a group of translations

  For sufficiently small .

  That is, indexes the orbits of a group and is simply the time of the motion.

  • Proof: This comes from the properties of group composition. In particular, consider the rectification below. Using a diffeomorphic mapping , map the -plane to the figure below such that . The result then follows

Rectification of a one-parameter diffeomorphism group. Image taken from Arnold

• An equation is homogeneous if the direction field defining it on the plane is homogeneous; that is it is invariant with respect to the one parameter group of dilations.

  • If is an integral curve, then the curve homothetic to it is also an integral curve.
  • Intuition: An equation is homogeneous if, when all variables are scaled by some constant, the equation remains the same (after cancellation).

• (Arnold 6.3.1) A homogeneous equation

  Can be reduced to an equation with separable variables by the substitution in the domain . This performs a change of variables to coordinates

• The function is homogeneous of degree if it satisfies

  • (Arnold 6.3.2) Euler’s Theorem. A function is homogeneous of degree if and only if it satisfies

    Another way to say this: is an eigenvector of the differentiation operator along the phase velocity field of dilations with eigenvector .

    • Proof: In the forward direction, calculate at .

      In the reverse direction, integrate the differential equation with separable variables defined by the Euler relation. That is, using .

  • The homogeneous function of degree is a common eigenvector of all linear operators with eigenvalues .

  • A necessary and sufficient condition for the direction field of the differential equation to be homogeneous is

• The -process also called inflating is defined by mapping

  • This allows us to turn any algebraic curve with a singularity at the origin into a curve having no singularities except ordinary self-intersections.

• A group of quasi-homogeneous dilations of the plane is a one-parameter group of linear transformations

  We refer to as weights, we denote , and .

• An equation is called quasi-homogeneous with weights if in the direction field that defines it in, the plane is invariant with respect to the group of quasi-homogeneous dilations

  • (Arnold 6.4.1) A quasi-homogeneous equation with weights and can be reduced to an equation with separable variables by passing the coordinates
    in the domain

• A function is called quasi-homogeneous of degree if it satisfies the identity

  That is is a common eigenvector of the operators , where is a quasi-homogeneous dilation with, with eigenvalues .

  • (Arnold 6.4.2) A necessary and sufficient condition for the direction field of the equation
    to be a quasi-homogeneous is that the right-hand side be quasi-homogeneous and its quasi-homogeneous degree be
  • (Arnold 6.5.1) Under a quasi-homogeneous dilation the graph of the function transforms into the graph of the function for which
    Thus, transforms like .

• A quasi-homogeneous vector field is defined by the condition

Links §

• Arnold