One-dimensional separable variable ODE
where , initial value undecided
Example
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.
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[exponential-of-vector-field] Question
let open in
The vector field is an analytic function
If you know matrix Lie groups, then you should know that Lie algebras can be mapped to Lie groups via
This also holds for analytic functions; in the sense of analytic topological convergence, should generate a local analytic diffeomorphism. The value of at should be
polynomial like
Or adding
Such that it corresponds to the ODE . We know that ODE theory can also give local diffeomorphisms through vector fields
Example
Comparing the results of pure vector fields to the results of ODE integral curves, you will find the results are the same. Take the case of constant coefficient linear or one-dimensional separable ODEs as an example
compare , expect with
Example [harmonic-oscillator]
Harmonic oscillator first-orderized
Trigonometric case takes
Thus
Or written in the form of complex exponentials
Hyperbolic case takes , similarly
The characteristic polynomial equation of the harmonic oscillator equation is or . We are interested in the trigonometric case or , whose prototype is or . This gives a motivation for complex numbers
In the case where the harmonic oscillator is a real-valued function, in the complex exponential representation of the solution, to keep the result in , when , the coefficients in front of should be complex conjugates of each other
- ,
compare , expect with
โฆ
Or
[vector-field-as-ฮด-diffeomorphism] Near the local analytic homeomorphism , the vector field serves as the coordinate of the local analytic homeomorphism group . This is similar to geodesic-coordinate
ODE, itโs also a one-parameter homomorphism embedding
Usually denoted as
For proof techniques, see wiki:Cauchy-Kovalevskaya_theorem, where the convergence radius of the power series is estimated using a special upper bound control method, similar to what was done in inverse-analytic
, ==>
[integral-curve] Picard iteration (wiki) representation of ODE solutions or integral curves e.g.
A time-varying vector field ODE is a special kind of vector field on
If it is a time-varying linear ODE then (alias Dyson series)
The solution to a constant coefficient ODE can be written in analytic form, by converting the ODE into a first-order constant coefficient linear ODE regarding , and then writing matrix in Jordan normal form
[Lie-bracket] Lie bracket
The conjugate-action of
Suppose generate . The first-order derivative is , while the second-order derivative mixing is , which can also be understood as first then , so that a โlinear representation of the Lie groupโ is obtained midway
Note that after swapping the order of , is a different mapping
for ,
[Lie-derivative] Lie derivative alias drag derivative
let generate a one-parameter diffeomorphism through
let
Jacobi identity or
The Lie derivative can also be defined for tensor fields โฆ
[first-order-PDE-integrable-condition] alias [Frobenius-theorem] generalizes first-order ODE integral curves to first-order PDE system integral surfaces; in this case, the linear space spanned by the vector fields needs to form a Lie subalgebra, or use the more general concept of involutive/integrable subbundles. Solutions to the PDE can come from successive ODE integral curves along coordinate directions, and the result doesnโt depend on the choice of path. In the case of first-order linear PDE systems, the integrability condition becomes the symmetry of second-order partial derivatives under coordinates