How Does Noether's Theorem Extend Beyond Conservation in Hamiltonian Systems?

[wrobel]

In Hamiltonian statement the Noether theorem is read as follows. Consider a system with the Hamiltonian function $$H=H(z),\quad z=(p,x),\quad p=(p_1,\ldots,p_m),\quad x=(x^1,\ldots,x^m)$$ and the phase space $M,\quad z\in M.$ Assume that this system has a one parametric group of symmetry $z\mapsto g^s(z)$. This group is generated by a Hamiltonian $F=F(z):$
$$\frac{d g^s(z)}{ds}=v_F( g^s(z)),\quad g^0(z)=z,$$
here $v_F$ stands for a Hamiltonian vector field that corresponds to the Hamiltonian function $F(z)$.

By definition, "group of symmetry" means $$H(g^s(z))=H(z),\quad \forall s.\qquad (1).$$
Then Noether says: $F$ is a first integral for the system with the Hamiltonian $H$. To prove this trivial observation one just should differentiate (1) in s and put s=0.

My question is little bit philosophical. The point is that the assertion of Noether's theorem is the least of what we can actually extract from the symmetry group given. Indeed,
assume that for some point $z_0$ one has $dF(z_0)\ne 0$ . Then in a neighborhood of this point there are local canonical coordinates $P,X$ such that in these new coordinates the function $F$ has the form $F=X^1.$
This fact is proved in [Olver, P. J. (1986), Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, 107, Springer] but there is a much simpler proof by means of the generating functions.
The coordinates $(P,X)$ can be constructed explicitly provided the group $g^s$ is given .Then by the Noether theorem in the coordinates $(P,X)$ we get
$$\{F,H\}=0=\frac{\partial H}{\partial P_1}.$$
So that the function $H$ does not depend on $P_1$ and the coordinate $X^1=const$ is a first integral. We obtain a Hamiltonian system with $m-1$ degrees of freedom $H=H(P_2,\ldots,P_m,const,X^2,\ldots,X^m)$. Observe that the first integral by itself does not give possibility to reduce the system explicitly to one with $m-1$ degrees of freedom.

So my question is why do physics textbooks never mention this consequence from the presence of symmetry group?

 

[sysprog]

So my question is why do physics textbooks never mention this consequence from the presence of symmetry group?

I like it; however, I'm a mere aficionado $-$ perhaps there is a tendency to compartmentalize at play $-$ while Noether's two symmetry theorems are important in Physics, extending from them further into Group Theory, as you seem to have done here, may by some be regarded as meandering afield into what belongs more in a textbook on Abstract Algebra, than in a Physics textbook.

 

[vanhees71]

But it's treated in most modern mechanics textbooks, because Noether's theorem is among the most important single achievements on the fundamentals of physics of the 20th century.

The idea to find enough "1st integrals" to make all variables cyclic is of course much older and leads to the Hamilton-Jacobi partial differential equation. The idea is to look for a generating function $F(q,P,t)$ (the version with "old generalized coordinates" and "new generalized momenta") which makes all $P$ conserved, i.e., all $Q$ cyclic. This leads to the demand
$$H'=H+\partial_t F \stackrel{!}{=}0,$$
i.e., to the partial differential equation
$$H(q,\partial_q F,t)+\partial_t F=0.$$
The $P=\text{const}$ occur as integration constants and then $Q=\partial_P F=\text{const}$ too.

 

[Dale]

So my question is why do physics textbooks never mention this consequence from the presence of symmetry group?

I don’t think it is true that it is never mentioned. However, it is certainly not used much. I think that is simply because such symmetries are hard to see unless they are specifically pointed out to you.

 

[wrobel]

But it's treated in most modern mechanics textbooks,

I have never seen a book that contains such a remark. Arnold writes few informal words just without any details about reduction of a Hamiltonian system with the help of a symmetry group.

The idea to find enough "1st integrals" to make all variables cyclic is of course much older and leads to the Hamilton-Jacobi partial differential equation.

it would be interesting to look at a historical treatment devoted to the origins of application of group theory to differential equations. I am an ignorant guy and know only two names: Sophus Lie and Emmy Noether.

 

[vanhees71]

I must admit that I'm also not very knowledgeable about the history of this most important development. I don't know, what you mean by that you've never seen the Noether theorem treated in physics textbooks. Most introductory modern textbook have this. There's some lack in treating the elegant method of the Hamilton-Jacoby partial differential equation in the more modern books, while it was very frequently found in older textbooks.

 

[wrobel]

by that you've never seen the Noether theorem treated in physics textbooks.

I did not say that. I have never seen this:

Indeed,
assume that for some point one has . Then in a neighborhood of this point there are local canonical coordinates such that in these new coordinates the function has the form
This fact is proved in [Olver, P. J. (1986), Applications of Lie Groups to Differential Equations, Graduate Texts in Mathematics, 107, Springer] but there is a much simpler proof by means of the generating functions.
The coordinates can be constructed explicitly provided the group is given .Then by the Noether theorem in the coordinates we get

So that the function does not depend on and the coordinate is a first integral. We obtain a Hamiltonian system with degrees of freedom . Observe that the first integral by itself does not give possibility to reduce the system explicitly to one with degrees of freedom.

 

[wrobel]

There's some lack in treating the elegant method of the Hamilton-Jacoby partial differential equation in th

we tell this in lectures it is a part of the classical mechanics course in our dept :)