First integral and characteristic equation

Consider the curve
\[
\left\{ \begin{array} { l } { {x} = cos(x) } \\ {{y}}=sin(x) \end{array} \right.
\]
that is solution of the system
\[
\left\{ \begin{array} { l } { \dot{x} = -y } \\ {\dot{y}}=x \end{array} \right.
\]

It is obvious that the function \(f(x,y)=x^2+y^2\) is constant in any solution. We call it a first integral of the system.


Definition:
We say \(f\) is first integral for a ODE system \(\dot{x}=X(x)\) if \(X(f)=0\).


The function \(f\) is invariant by the 1-parameter group created by \(X\). Or equivalently, the solutions \(\gamma(t)\) of the system are contained in the hypersurfaces \(\{f=cte\}\).


On the other hand, a function \(\Phi\) is functionally dependent on \(h_1, \ldots,h_{n-1}\) if \(\Phi=G\left(h_{1}, \cdots, h_{n-1}\right)\). And \(h_1, \ldots,h_{n-1}\) are called functionally independent if the differentials are linearly independent in a open set, that is, the Jacobian of \(H(x)=(h_1(x), \ldots,h_{n-1}(x))\) has maximal rank (keep an eye:     \(x\in \mathbb{R}^N\)).


Theorem:
Given the open set \(U\subseteq \mathbb{R}^n\), \(X \in \mathfrak{X}(U)\) and \(x_0\in U\) such that \(X(x_0)\neq 0\), we can find an open subset \(x_0\in V\subseteq U\) such that the system \( \dot{x}=X(x)\) admits \(n-1\) first integrals which are functionally independent and such that any other first integral depends functionally on them.

The proof is based on the tubular flow theorem. See Arnold Ordinary Differential Equations page 127.

This \(n-1\) first integrals define a curve in \(V\subseteq \mathbb{R}^n\) that is precisely a solution curve \(\gamma(t)\).


Finally, given a first order homogeneous linear PDE
\[
a_{1} \partial u / \partial x_{1}+\cdots+a_{n} \partial u / \partial x_{n}=0
\]
where \(a_k=a_k(x_1,\ldots,x_n)\), we look for solutions
\[
u: \mathbb{R}^n \rightarrow \mathbb{R}
\]

If we define the vector field \(A=(a_1,a_2,\ldots,a_n)\), the ODE
\[
\dot{x}=A(x)
\]
is called the characteristic system or characteristic equation, and their integral curves are called the characteristics of the original PDE.

Of course, a function \(F\) is a solution of the PDE if and only if is a first integral of the characteristic ODE.
 

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