How can the Poisson bracket {H, A} be computed directly without components.
H is the Hamiltonian for the inverse square force, H=p^2/m- k/|r| , and A is the integration constant called the Laplace vactor; A = pxrxp/m-mkr/|r|
At first glance the issue is that traditionally the brackets are defined for scalar valued functions of scalar values , but I don't see why the can't be interpreted as being defined on functions of vector variables and then using vector calculus and vector gradients instead of partial derivatives . I'm also fine with translating this to differential forms or using geometric algebra, or both .
The point is of course to show that the Poisson bracket {A,H} is zero since A is a constant of motion, but the usual way of using the levi Civita tensors is quite tedious , and I think it could be shortcuted vastly if computed directly , I just can't get it quite right