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Since the equation of mechanics are of second order in time, we know that for $N$ degrees of freedom we have to specify $2N$ initial conditions. One of them is the initial time $t_0$ and the rest of them, $2N-1$ are initial positions and velocity. Any function of these initial condition is a constant of motion, by definition. Also, there should be exactly $2N-1$ algebraically independent constants of motion.

On the other hand, Noether's procedure gives us integrals of motion as a result of variational symmetries of the action. These integrals of motion are also conserved but they are not always $2N-1$ in number. In consequence, we classify the system by their integrability.

So, what is the difference between the constant of motion and integral of motion? Why do non-integrable systems have less integrals of motion when they should always have $2N-1$ constants of motion?

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If you like this question you may also enjoy reading this and this Phys.SE post. – Qmechanic Mar 4 '13 at 21:47
up vote 3 down vote accepted

1) A constant of motion $f(z,t)$ is a (globally defined, smooth) function $f:M\times [t_i,t_f] \to \mathbb{R}$ of the dynamical variables $z\in M$ and time $t\in[t_i,t_f]$, such that the map $$[t_i,t_f]~\ni ~t~~\mapsto~~f(\gamma(t),t)~\in~ \mathbb{R}$$ doesn't depend on time for every solution curve $z=\gamma(t)$ to the equations of motion of the system.

An integral of motion is a constant of motion $f(z)$ that doesn't depend explicitly on time.

2) In the following let us for simplicity restrict to the case where the system is a finite-dimensional autonomous$^1$ Hamiltonian system with Hamiltonian $H:M \to \mathbb{R}$ on a $2N$-dimensional symplectic manifold $(M,\omega)$.

Such system is called (Liouville/completely) integrable if there exist $N$ functionally independent, Poisson-commuting, globally defined functions $I_1, \ldots, I_N: M\to \mathbb{R}$, so that the Hamiltonian $H$ is a function of $I_1, \ldots, I_N$, only.

Such integrable system is called maximally superintegrable if there additionally exist $N-1$ globally defined integrals of motion $I_{N+1}, \ldots, I_{2N-1}: M\to \mathbb{R}$, so that the combined set $(I_{1}, \ldots, I_{2N-1})$ is functionally independent.

It follows from Caratheodory-Jacobi-Lie theorem that every finite-dimensional autonomous Hamiltonian system on a symplectic manifold $(M,\omega)$ is locally maximally superintegrable in sufficiently small local neighborhoods around any point of $M$ (apart from critical points of the Hamiltonian).

The main point is that (global) integrability is rare, while local integrability is generic.


$^1$ An autonomous Hamiltonian system means that neither the Hamiltonian $H$ nor the symplectic two-form $\omega$ depend explicitly on time $t$.

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Oh, I see, so the local integrability is a generic property of a system of differential equations, since we are always solving the equations localy. On the other hand, global integrability is connected to the symmetries of the system and Noether's theorem? – LBO Mar 15 '13 at 10:48

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