First integral

2020 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

[Ar1]	V.I. Arnold, "Ordinary Differential Equations", Springer Verlag (1992)
[Ar2]	V.I. Arnold, "Mathematical methods of classical mechanics", Springer Verlag (1989)
[Po]	L.S. Pontryagin, "Ordinary differential equations" , Addison-Wesley (1962)