3 arxiv tag
2 tried to clean up; edited title

Poisson brackets play the very important roles in Symplectic geometry and Dynamical system. I'm interested in some conserved quantities of Poisson brackets.

Let's say we are working on T^n x R^n (T^n is the torus in R^n, T^n = R^n/Z^n). Assume that I have the Hamiltonian H: T^n x R^n \mapsto R, where H=H(x,p) and H is smooth for simplicity.

Every function here is T^n-periodic in x. For K=K(x,p) then we define the Poisson bracket {H,K} = D_p H.D_x K - D_x H.D_p K.

The question is, in general setting or in some particular case, can you find some conserved quantities S such that {H,S}=0?

One easy and important example is for S= f(H) for f is smooth then we have {H,f(H)}=0.

But can you have some other conserved quantities S?

Thanks a lot

Edit: here's one specific example: If H(x,p)=H(p)+V(x_1+x_2+...+x_n) for V: R \mapsto R is T-periodic then this kind of Hamiltonian has some other conserved quantities S=p_i - p_j for any i \ne j.Bear!

Any deeper intuitions or physical examples?

1

Poisson brackets play the very important roles in Symplectic geometry and Dynamical system. I'm interested in some conserved quantities of Poisson brackets.

Let's say we are working on T^n x R^n (T^n is the torus in R^n, T^n = R^n/Z^n). Assume that I have the Hamiltonian H: T^n x R^n \mapsto R, where H=H(x,p) and H is smooth for simplicity.

Every function here is T^n-periodic in x. For K=K(x,p) then we define the Poisson bracket {H,K} = D_p H.D_x K - D_x H.D_p K.

The question is, in general setting or in some particular case, can you find some conserved quantities S such that {H,S}=0?

One easy and important example is for S= f(H) for f is smooth then we have {H,f(H)}=0.

But can you have some other conserved quantities S?

Thanks a lot. Bear!