Using FullSimplify to simplify an expression with assumptions

Question

I am working with Poisson brackets of Angular momentum, and I want Mathematica to be able to return the angular momentum function if the Poisson bracket yields that result. I.e. provided

PoissonBracket[a_ , b_] := 
  Module[{position, momentum}, 
    position = {x, y, z};
    momentum = {px, py, pz};
    result = 
      D[a, {position}].D[b, {momentum}] - D[a, {momentum}].D[b, {position}]]

FLx[x_, y_, z_, px_, py_, pz_] := y*pz - z*py;
Lx = FLx[x, y, z, px, py, pz];
FLy[x_, y_, z_, px_, py_, pz_] := z*px - x*pz;
Ly = FLy[x, y, z, px, py, pz];
FLz[x_, y_, z_, px_, py_, pz_] := x*py - y*px;
Lz = FLz[x, y, z, px, py, pz];

(I did this to be able to compute actual values later on and to take Jacobians in an easier way)

If I compute

PoissonBracket[Lx, Ly] 

I get pz y - py z, but I want to get Lz straight away instead. I have tried using simplify with assumptions in various ways, but

FullSimplify[PoissonBracket[Ly, Lz], Lx]

returns True

 FullSimplify[PoissonBracket[Ly, Lz], Lx == pz y - py z]

returns pz y - py z, so it does nothing.

What am I doing wrong?

Answer 1

Clear["Global`*"]

PoissonBracket[a_, b_] := Module[{position, momentum}, position = {x, y, z};
  momentum = {px, py, pz};
  result = 
   D[a, {position}].D[b, {momentum}] - D[a, {momentum}].D[b, {position}]]

FLx[x_, y_, z_, px_, py_, pz_] := y*pz - z*py;
FLy[x_, y_, z_, px_, py_, pz_] := z*px - x*pz;
FLz[x_, y_, z_, px_, py_, pz_] := x*py - y*px;

If you Set (=) the values of Lx, Ly, and Lz then these can never appear in a result since they will always be evaluated to their Set values. Perhaps you want replacement rules.

rules = {FLx[x, y, z, px, py, pz] -> Lx,
  FLy[x, y, z, px, py, pz] -> Ly,
  FLz[x, y, z, px, py, pz] -> Lz}

(* {pz y - py z -> Lx, -pz x + px z -> -pz x + px z, py x - px y -> Lz} *)

PoissonBracket[FLx[x, y, z, px, py, pz], FLy[x, y, z, px, py, pz]] /. rules

{* Lz *)