generally, if two functions and have a zero Poisson bracket (cf. below), then is constant along the integral curves of , and similarly, is constant along the integral curves of . This fact is the abstract mathematical principle behind Noether's theorem. The symplectic form is preserved by the Hamiltonian flow. Equivalently, the Lie derivative Poisson bracket The notion of a Hamiltonian vector field leads to a skew-symmetric bilinear operation on the differentiable functions on a symplectic manifold M, the Poisson bracket, defined by the formula where denotes the Lie derivative along a vector field X. Moreover, one can check that the following identity holds: where the right hand side represents the Lie bracket of the Hamiltonian vector fields with Hamiltonians f and g. As a consequence (a proof at Poisson bracket), the Poisson bracket satisfies the Jacobi identity: which means that the vector space of differentiable functions on , endowed with the Poisson bracket, has the structure of a Lie algebra over , and the assignment is a Lie algebra homomorphism, whose kernel consists of the locally constant functions (constant functions if is connected). Remarks Notes Works cited See section 3.2. Hamiltonian mechanics Symplectic geometry William Rowan Hamilton