Conservation laws, Noether's theorem and initial conditions

[Rocky Raccoon]

Hello, everybody!

During the whole of my undergraduate study of physics, this one thing always bothered me. It concerns the interplay of conserved quantities, symmetries, Noether's theorem and initial conditions.

For a system of N degrees of freedom, governed by the usual Newton's laws, one has to give 2N initial conditions (e.g. position and momenta) to arrive at the complete solution of the problem. Given that one of the initial condition refers to the initial time, we are left with 2N-1 constant describing the trajectory of the system. We can now form 2N-1 independent functions of these 2N-1 constant which will be, by construction, also constant in time. Smartly chosen, these functions can represent energy, momentum... of the system. This shows that there are, in general, (at least!) 2N-1 constants of motion for a system od N degrees of freedom. The only thing that was used in this derivation was the assumption of Newtons law that the accelaration is uniquely given by some function of position and velocity. Note that the 2N-1 conserved quantities should(?) corespond to the space-time properties of the system.

On the other hand, Noether's theorem clearly states the connection between the conserved quantities and the symmetries of the physical system. Since we usually deal with Galilei invariance (in classical mechanics), it is usualy stated that any closed system has 10 conserved quantities - enegry (1), momentum (3), angular momentum (3) and vector of center of mass (3). Since clearly 2N-1 is not equal to 10, it is obvious that in some cases (N < 5), the before mentioned 10 Galileian invariants are not independent, and in other cases (N > 5), there are further space-time symmetries besides the Galileian.

Is this correct? If so, how do other symmetries (e.g. gauge invariance) fit into the "initial conditions" picture?

 

[lalbatros]

Hello RR,

I think that the case N<5 is not interresting to discuss. When N<5, you cannot take for granted the existence of 3 linear momentum and 3 angular momentum conserved quantities.

The case N>>5 is very interresting and has been discussed intensively by many people.
It is at the roots of statistical mechanics.

Maybe you could be interrested by a paper about integrable system like this one:
http://www.maths.ed.ac.uk/~s0567465/talks/PGtalk-int.pdf .

Have also a look at this book: http://books.google.com/books?id=WhJc9kt-_ZUC&printsec=frontcover&dq=Introduction+to+Classical+Integrable+Systems&hl=en&src=bmrr&ei=C74oTcubCYiIhQepnYSOAg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CCcQ6AEwAA#v=onepage&q&f=false .

Michel

 

[Rocky Raccoon]

Hello lalbatros,

You definitely have a point about the case N<5. About the paper - it confuses me why isn't every system completely integrable since, as we have shown, it always has 2N-1 constants of motion.

 

[lalbatros]

RR,

After all, I would re-phrase your question in this way:

What is the difference between

an initial condition and
a constant of the motion ?​

(is there any, what are the assumed definitions?)​

In addition I would also like to understand
why the number of "constants of the motion" could be less that the number of initial conditions?

I was never able to have simple and clear explanation of these things (before the web and still now).
Nevertheless, the "other constant of the motion" seem to play little if no role at all in statistical mechanics or thermodynamics, as far as I know. This must indicate that either they don't exist or they are at least very different than the set comprising energy, linear and angular momentum.

Thanks,

Michel


PS:
===
Arnold has written one of the best books on classical mechanics.
On page 291 you may find some hint to your question:
http://books.google.com/books?id=Pd...=onepage&q=arnold classical mechanics&f=false

I guess that when trajectories can fill in volumes in the phase space, then constant of the motion are lost. If this was true, then the question could be translated again to: why could trajectories fill volumes in the phase space?

 

[Rocky Raccoon]

As far as I know, the "other constants of motion" are not additive (e.g. Laplace - Runge vector). Therefore, they are no good in thermodynamics. But I must say, I've never seen the proof of the statement above. It was mentioned in Landau's Statistical physics as an explanation for considering only E, P and L.

 

[lalbatros]

As far as I know, the "other constants of motion" are not additive (e.g. Laplace - Runge vector). Therefore, they are no good in thermodynamics. But I must say, I've never seen the proof of the statement above. It was mentioned in Landau's Statistical physics as an explanation for considering only E, P and L.

You are right, and your reference is one of my preffered books!

 

[Rocky Raccoon]

So, for now we can agree that initial conditions imply 2N-1 constant of motion of which only 10 are additive (yet, we still need a proof of this).

We can also agree that the "other constants" correspond to some complicated (and more or less irrelevant) symmetries of the Lagrangian.

It is still not clear (to me, at least) the whole deal with integrability and completely integrable systems and their relation to initial conditions.

Also, I would like to know could non space-time symmetries be interpreted in terms of the "initial conditions picture".

 

[Rocky Raccoon]

I guess that when trajectories can fill in volumes in the phase space, then constant of the motion are lost. If this was true, then the question could be translated again to: why could trajectories fill volumes in the phase space?

I don't see how could constant of motion be lost. Perhaps only the additive ones. I'm pretty sure that there are constants of motion determined by initial conditions even when the Lagrangian totally lacks Galilean symmetry.

 

[lalbatros]

When trajectories tend to fill a volume,
then different initial conditions can lead to arbitrary close final conditions,
or conversly abitrarilly close initial conditions could also lead to totally different outcomes.

Intuitively speaking, this makes the initial conditions less relevant.
How could we check -practically- that these are constants of motion?

 

[Rocky Raccoon]

Oh, I see - technically, everything I said still holds, but it's more convenient to work in the "chaos framework". If we put an uncertainty in the initial conditions by hand as an assumptions, we lose the "other constants of motion".

Thank you for clarification on the subject.

 

[lalbatros]

RR,

I don't think I really clarified this topic!
I would really like to find a simple paper explaining all these things.
If I find one, I will post here.

Michel

 

[pavel1]

Hi guys !

I am sure that these statements hold:

1) Darboux theorem - locally (if $\mathrm{d}H\neq 0$), it is always possible to find $2N$ independent functions $H_2, H_3, \ldots, H_N$, ($H_1 = H$) and $G_1, G_2, \ldots, G_N$ such that $(H_1,\ldots,H_N,G_1,\ldots,G_N)$ constitute the canonical coordinates. Notice that the $N$ independent functions $H_2,\ldots, H_N$ and $H$ are in involution and thus provide a complete set of commuting integrals. By the way this implies that Hamilton's equations are trivial, but again, only locally.

2) There are always $2N$ independent constants of motion, however some of them are time dependent. I mean if I start with $2N$ initial conditions (a point in a phase space) at $t=0$ then I can remember all initial conditions at any time $t$. (an arbitrary function $F$ on the phase space composed with the reversed flow $\varphi_{-t}$ of the Hamiltonian vector field $X_H$, i.e. we consider the function $F_t = F\circ \varphi_{-t}$, is a time dependent constant of motion).

Explanation:

A) Integrability is implied by the existence of $N$ independent first integrals, i.e. globally defined functions on the phase space commuting with Hamiltonian.
The most important consequence is that we can locally define the so called action-angle variables $(I_1,\ldots,I_N,\alpha_1,\ldots,\alpha_n)$, such that $I_i$ are first integrals, $H = H(I_1,\ldots,I_N)$ and the bounded trajectories are periodic (span the Liouville tori). It is quite different from (1).

B) Let's take the 1D harmonic oscillator. The phase-space curves are circles. However, if you imagine time as the axis perpendicular to the phase space plane, the trajectories become helixes. At any time you can go along the helix to the phase space plane at $t=0$ and recover the value of any function $F$, as mentioned in (2). This is how time dependent constants of motion arise. Nevertheless, you cannot "project" all the $F_t$ to the phase-space plane, otherwise you get some multivalued functions. Time dependent constants of motion are well defined only on the phase space extended by time. The only projectable time dependent constant of motion is the distance to the time axis. In fact it is time independent and proportional to the conserved energy.

C) There are maximally $2N-1$ first integrals (including Hamiltonian) because of the non degeneracy of canonical Poisson brackets. Such systems are called superintegrable (Kepler problem, spherical symmetric harmonic oscillator). For example spherical symmetric systems in 3D are always integrable (there is a globally defined action of the rotation group which gives rise to the two well known commuting integrals - the squared angular momentum and one of its components), but only Kepler problem and isotropic oscillator are superintegrable. There is an interesting connection between the time dependent constants of motion, first integrals, closed trajectories and the transformations bringing one trajectory to the another while not preserving the energy... ( just like a motivation for further studying ;-) )

I hope it will help and please correct me if I was wrong (math and physics, not my terrible English :-)). I apologize for the symplectic formalism as well, but I consider it the best to describe classical mechanics...

 

[Rocky Raccoon]

So, in layman's terms - what's the difference between constants of motion and first integrals?

 

[pavel1]

From my point of view, "first integral" or "integral of motion" is more specific
than "constant of motion". You can find the mathematical definiton of "first integral" in every textbook on classical Hamiltonian mechanics:

"Any function depending only on dynamical states of the system which is constant along every trajectory."

I would use the term "constant of motion" when I talk about some general conserved quantity, possibly depending on time. When you observe the motion of a free particle and measure its velocity, you find that the velocity is the same at all times. What else can you observe in the framework of Newtonian mechanics (equivalent to Hamiltonian mechanics) ? Only dynamical states - positions and velocities - and any arbitrary function of them. Of course, in the context of other theories there can be several conserved quantities like charge, etc... But even if you stay in Newtonian mechanics, you can also form another conserved quantities, constants of motion. For example F_t = x_0 + x - v.t , where x_0 is the initial position, v velocity, t time and x position. In general, you can set F_t(x,v) = F(x,v,t) = F_0(x-vt,v) for every observable F_0(x,v). Then F is conserved, because d/dt F = 0 along every trajectory. However, it is not measurable directly - as an independent observer you have to count time and track the particle back to its initial state x_0 to get the value of F_0(x_0,v)=F(x,v,t). Moreover, there can be other conserved quantities which depend on time explicitely.

First integral is a mathematically precise term which came to physics from the theory of differential equations and their symmetries thanks to Nother's theorem.