System of first order ODEs
Particular case of system of ODEs. They are also called dynamical systems.
Consider the equation
$$ \dot{\boldsymbol{x}}=F(\boldsymbol{x}), \quad \boldsymbol{x} \in \mathbb{R}^{n} $$with $F: \mathbb{R}^n\mapsto \mathbb{R}^n$.
A solution for this equation with initial condition $\varphi\left(t_{0}\right)=x_{0}$ is any
$$ \varphi : I \rightarrow \mathbb{R}^{n} $$such that
$$ \left.\frac{d \varphi}{d t}\right|_{t=\tau}=F( \varphi(\tau)) $$for every $\tau\in I$, and $\varphi\left(t_{0}\right)=x_{0}$.
This problem has always a solution, which is smooth and unique (if we consider the maximality of the domain): this is the existence and uniqueness theorem for ODEs. The statement and proof can be read at [Arnold 1991] page 93 (chapter 2, S7, 2) or [Lee 2013] page 314. The hypothesis can be weaker (the function $F$ only needs to be a Lipschitz continuous function), and I think this is what states the Picard-Lindelöf theorem.
Every system can be identified with a vector field in $\mathbb{R}^n$: if $F=(F_1,F_2,\ldots)$ then correspond to the vector field
$$ X=\sum_i F_i \partial x_i $$So, in addition, the existence and uniqueness theorem establishes the 1-1 relation between vector fields and local one-parameter groups:
On the other hand, a formal power series solution is obtained with the Lie series.
The system, since it has an associated vector field, it is also associated to a linear homogeneous PDE of which is its characteristic equation. See discussion at the end of that link about method of characteristics.
Particularly important is the linear case, which can be split in system of first order linear ODEs- real case and system of first order linear ODEs- complex case.
Examples
Important examples appears at Hamiltonian mechanics.
________________________________________
________________________________________
________________________________________
Author of the notes: Antonio J. Pan-Collantes
INDEX: