Namespaces
Variants
Actions

Difference between revisions of "First integral"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
Line 1: Line 1:
 +
{{MSC|34|70H}}
 +
[[Category:Ordinary differential equations]]
 +
{{TEX|done}}
 +
 
''of an ordinary differential equation''
 
''of an ordinary differential equation''
  
A non-constant continuously-differentiable function whose derivative vanishes identically on the solutions of that equation. For a scalar equation
+
Consider a system of ordinary differential equations of first order in the unknowns $x: \mathbb R \supset I \to \mathbb R^n$:
 
+
\begin{equation}\label{e:ODE}
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040460/f0404601.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
\Phi (t, x(t), \dot{x} (t)) = 0\, .
 
+
\end{equation}
a first integral is a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040460/f0404602.png" /> which occurs in the general solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040460/f0404603.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040460/f0404604.png" /> is an arbitrary constant. Therefore, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040460/f0404605.png" /> satisfies the linear equation
+
A first integral of the system is a (non-constant) continuously-differentiable function $\Psi: \mathbb R \times \mathbb R^n \to \mathbb R$ which is locally constant on any solution of \eqref{e:ODE}, namely such that $\frac{d}{dt} \Psi (t, x(t)) = 0$ for any $x: J \to \mathbb R^n$ solving \eqref{e:ODE}. The domain of definition of $\Psi$ must be suitably adjusted when the domain of definition of $\Phi$ is not the entire space and often one considers $\Psi$ which are defined in a yet smaller domain (i.e. only locally around some particular point).  
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040460/f0404606.png" /></td> </tr></table>
 
 
 
containing first-order partial derivatives. A first integral need not exist throughout the domain of definition of (*), but it always exists in a small region around a point at which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040460/f0404607.png" /> is continuously differentiable. A first integral is not uniquely defined. For example, for the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040460/f0404608.png" />, a first integral is not only <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040460/f0404609.png" /> but also, e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040460/f04046010.png" />.
 
 
 
Knowledge of a first integral for a normal system
 
 
 
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040460/f04046011.png" /></td> </tr></table>
 
 
 
enables one to reduce the order of this system by one, while the search for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040460/f04046012.png" /> functionally-independent first integrals is equivalent to the search for the general solution in implicit form. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040460/f04046013.png" /> are functionally-independent first integrals, then any other first integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040460/f04046014.png" /> can be put in the form
 
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040460/f04046015.png" /></td> </tr></table>
+
'''Example'''
 +
For a scalar equation of the form
 +
\[
 +
\dot{x} (t) = f (t, x(t))
 +
\]
 +
with $f: \mathbb R^2 \to \mathbb R$, it can be easily seen that $F$ is a first integral if and only if $F$ solves the partial
 +
differential equation
 +
\begin{equation}\label{e:first_integral}
 +
\frac{\partial F}{\partial t} + \frac{\partial F}{\partial x} f = 0\, .
 +
\end{equation}
 +
A solution of \eqref{e:first_integral} always exists in a neighborhood of a point where $f$ is [[Lipschitz condition|Lipschitz]].
 +
In fact, if $f$ is Lipschitz on $\mathbb R \times \mathbb R$, then there is a unique global solution of \eqref{e:first_integral} subject to any initial condition of the Cauchy type $f (0, x) = f_0 (x)$ $\forall x$, see [[Transport equation]].
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040460/f04046016.png" /> is some differentiable function.
+
The knowledge of a first integral "reduces" the number of unknowns by $1$. A particularly interesting case is when the system is [[Completely integrable system|completely integrable]], i.e. when there are $n$  functionally independent first integrals $\Psi_1, \ldots , \Psi_n$. This condition is equivalent to the existence of a general formula for solutions of \eqref{e:ODE} in implicit form. The knowledge
 +
of $n$ functionally independent first integrals guarantees also that any other first integral $\Lambda$ must take the form
 +
\[
 +
\Lambda (t, x) = F ( \Psi_1 (t,x), \ldots , \Psi_n (t,x))\, .
 +
\]
 +
First integrals of motions are particularly studied in the theory of [[Hamiltonian system|Hamiltonian systems]]. For phsically relevant cases the first integrals are also called "constants of motions" and some of them correspond to conservation laws for physically relevant quantities. The primary example is the system of equations governing the motion of a particle in a potential field:
 +
\begin{equation}\label{e:potential_field}
 +
\ddot{x} (t) = - \nabla U (x(t))\, .
 +
\end{equation}
 +
Introducing the new variables $v(t) := \dot{x} (t)$ we can turn \eqref{e:potential_field} into a first order system
 +
\begin{equation}\label{e:Hamilton}
 +
\left\{
 +
\begin{array}{ll}
 +
\dot{x} (t) = v (t)\\
 +
\dot{v} (t) = - \nabla U (x(t))\, .
 +
\end{array}\right.
 +
\end{equation}
 +
Well-known first integrals of motion are then the total energy $\Psi (x, v) = \frac{|v|^2}{2} + U (x)$ and the components of the angular momentum $\Psi_{ij} (x,v) = x_j v_i -x_i v_j$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"L.S. Pontryagin,  "Ordinary differential equations" , Addison-Wesley  (1962) (Translated from Russian)</TD></TR></table>
+
{|
 +
|-
 +
|valign="top"|{{Ref|Ar}}|| V.I. Arnold, "Mathematical methods of classical mechanics", Springer Verlag (1989)
 +
|-
 +
|valign="top"|{{Ref|Fi}}|| L.S. Pontryagin,  "Ordinary differential equations" , Addison-Wesley  (1962)  
 +
|-
 +
|}

Revision as of 07:46, 29 June 2014

2010 Mathematics Subject Classification: Primary: 34-XX Secondary: 70H [MSN][ZBL]

of an ordinary differential equation

Consider a system of ordinary differential equations of first order in the unknowns $x: \mathbb R \supset I \to \mathbb R^n$: \begin{equation}\label{e:ODE} \Phi (t, x(t), \dot{x} (t)) = 0\, . \end{equation} A first integral of the system is a (non-constant) continuously-differentiable function $\Psi: \mathbb R \times \mathbb R^n \to \mathbb R$ which is locally constant on any solution of \eqref{e:ODE}, namely such that $\frac{d}{dt} \Psi (t, x(t)) = 0$ for any $x: J \to \mathbb R^n$ solving \eqref{e:ODE}. The domain of definition of $\Psi$ must be suitably adjusted when the domain of definition of $\Phi$ is not the entire space and often one considers $\Psi$ which are defined in a yet smaller domain (i.e. only locally around some particular point).

Example For a scalar equation of the form \[ \dot{x} (t) = f (t, x(t)) \] with $f: \mathbb R^2 \to \mathbb R$, it can be easily seen that $F$ is a first integral if and only if $F$ solves the partial differential equation \begin{equation}\label{e:first_integral} \frac{\partial F}{\partial t} + \frac{\partial F}{\partial x} f = 0\, . \end{equation} A solution of \eqref{e:first_integral} always exists in a neighborhood of a point where $f$ is Lipschitz. In fact, if $f$ is Lipschitz on $\mathbb R \times \mathbb R$, then there is a unique global solution of \eqref{e:first_integral} subject to any initial condition of the Cauchy type $f (0, x) = f_0 (x)$ $\forall x$, see Transport equation.

The knowledge of a first integral "reduces" the number of unknowns by $1$. A particularly interesting case is when the system is completely integrable, i.e. when there are $n$ functionally independent first integrals $\Psi_1, \ldots , \Psi_n$. This condition is equivalent to the existence of a general formula for solutions of \eqref{e:ODE} in implicit form. The knowledge of $n$ functionally independent first integrals guarantees also that any other first integral $\Lambda$ must take the form \[ \Lambda (t, x) = F ( \Psi_1 (t,x), \ldots , \Psi_n (t,x))\, . \] First integrals of motions are particularly studied in the theory of Hamiltonian systems. For phsically relevant cases the first integrals are also called "constants of motions" and some of them correspond to conservation laws for physically relevant quantities. The primary example is the system of equations governing the motion of a particle in a potential field: \begin{equation}\label{e:potential_field} \ddot{x} (t) = - \nabla U (x(t))\, . \end{equation} Introducing the new variables $v(t) := \dot{x} (t)$ we can turn \eqref{e:potential_field} into a first order system \begin{equation}\label{e:Hamilton} \left\{ \begin{array}{ll} \dot{x} (t) = v (t)\\ \dot{v} (t) = - \nabla U (x(t))\, . \end{array}\right. \end{equation} Well-known first integrals of motion are then the total energy $\Psi (x, v) = \frac{|v|^2}{2} + U (x)$ and the components of the angular momentum $\Psi_{ij} (x,v) = x_j v_i -x_i v_j$.

References

[Ar] V.I. Arnold, "Mathematical methods of classical mechanics", Springer Verlag (1989)
[Fi] L.S. Pontryagin, "Ordinary differential equations" , Addison-Wesley (1962)
How to Cite This Entry:
First integral. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=First_integral&oldid=12974
This article was adapted from an original article by N.N. Ladis (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article