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?

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!