Take the 2-minute tour ×
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It's 100% free, no registration required.

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?

share|improve this question
    
If you like this question you may also enjoy reading this and this Phys.SE post. –  Qmechanic Mar 4 '13 at 21:47
add comment

1 Answer 1

up vote 2 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 $t$.

share|improve this answer
    
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? –  Little Brown One Mar 15 '13 at 10:48
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.