Consider the differential equation
This equation can be solved in integration (quadrature) and every solution is obtained by shifting one particular solution.(Constants of integration)
Is it possible to change coordinates in the first equation and reduce to the second case where we can solve equation explicitly.
One way to view the reason for this solvability is the symmetry- is independent of , so the vector field is invariant under vertical shifts.
So given one solution, all the solutions are obtained by shifting it along axis.
Can we always change coordinates to solve the first equation? Assume that we have a symmetry to the vector field, does that help you to solve the equation?
Assume a symmetry group acts continuously on our system:
Let be the the infinitesimal generator of a one-parameter subgroup of these symmetries.
Lie (1874): Assume is a 1-parameter group leaving the system invariant. There exists a function such that
and is the solution to
Proof: If is a solution then,
Thus we see that
So is the integrating factor for our problem.
Examples:
1.
The vertical translations leave the equation invariant.
Hence
works and we just get
2.
Rotations leave this equation invariant: Infact, we see that the equation says that the vector field is rotationally symmetric, but the angle between the vectors and the radial direction changes with the radius (at a fixed radius this angle is contant)
is the generator for rotations.
3.
The symmetry is the transformation:
The generator is .
4.
The symmetry is the transformation:
The generator is .
5. Riccati equation:
The symmetry is the transformation: , with generator
6.
The symmetry is the transformation: with generator
7.
The symmetry is the transformation: with generator
Differential Galois Theory: If there is solvable -dimensional stability group for
Then the solution can be found by repeated integration.