Poisson Bracket

Poisson Bracket of Vector Fields

Poisson Bracket of Vector Fields

The Poisson bracket of two vector fields and is defined as minus the Lie bracket of the two vector fields: .

Poisson Bracket of Functions

Poisson Bracket of Functions

Suppose is a symplectic manifold. The Poisson bracket of two functions is defined as a function in :

Note that the above definition is equivalent to say that , because

Proposition

Suppose is a symplectic manifold. For any , there holds

Proof By definition of a Hamiltonian vector field, it suffices to show that . We start from the RHS. Recall the proposition, we can write Note that a Hamiltonian vector field preserves the symplectic form, so . Moreover, by definition. Hence, The LHS can be computed as follows: it implies

Proposition

The Poisson bracket gives functions on a symplectic manifold the structure of a Lie algebra.

Proof Bilinearity comes from the linearity of . Indeed, satisfies so is linear. Anti-symmetry inherits from the anti-symmetry of the symplectic form; It also satisfies the Jacobi identity: $${{F,G},H}=\L_{X_{H}}{F,G}={\L_{X_{H}}F,G}+{F,\L_{X_{H}}G}={{F,H},G}+{F,{G,H}}.$$$\square$

Noether’s Theorem

Noether's Theorem

A function is -symmetric iff is preserved under . In fact,

Proof It is easy to see that $$\L_{X_{G}}H={H,G}=-{G,H}=-\L_{X_{H}}G.$$$\square$

Remark

Noether’s theorem states that every differentiable symmetry of the action of a physical system has a corresponding conservation law. In the context of Hamiltonian mechanics, this means that if the Hamiltonian is invariant under a continuous symmetry transformation, then there exists a conserved quantity associated with that symmetry. (See Noether’s Theorem.)

Cotangent Lift Flow

Any global flow on a smooth manifold induces a cotangent lift flow on the cotangent bundle . For any (where and ), this flow is defined by where the covector is determined by the condition

Proof We shall verify that is a valid global flow on . Clearly, is a diffeomorphism as is, and So is the identity map. Moreover, to show the group property , consider , we have Thus, Therefore, is a valid global flow on .

Coordinate Form of Cotangent Lift Flow

Let be local coordinates on in coordinate form, and be the corresponding canonical coordinates on . Then the above lifted flow has the following coordinate form: where represents the components of the Jacobian matrix of the inverse transformation evaluated at .

e.g. Rotation around the axis in determines a Hamiltonian flow on . What is the corresponding Hamiltonian vector field on , and what is its Hamiltonian function ? What are and , the Hamiltonians generating rotations around the and axis and what is ? For a rotationally symmetric Hamiltonian , why is preserved by the flow of ? We use the standard coordinates for , and for the cotangent bundle . A rotation around the -axis can be expressed as a flow (of unit speed): We can lift this flow to the cotangent bundle according to the lemma. Note that the lifted covector transform via the inverse transpose of the Jacobian matrix of . In this case, it transforms exactly like the coordinates:Hence, the Hamiltonian vector field on can be derived by differentiating with respect to at :Similarly, , , , soWe can therefore find the associated Hamiltonian function by solving the Hamilton’s equations:Integrating these equations, we obtain (up to some constant)which is recognizable as the -component of the angular momentum. By symmetry, the Hamiltonians generating rotations around the and axes are Furthermore, we can derive the Poisson bracket of and : A Hamiltonian is rotationally symmetric (specifically, around the -axis) if it remains unchanged under rotations around the -axis. This invariance implies that the Lie derivative of along the Hamiltonian vector field is zero: . Observe that which implies that , i.e., is preserved by the flow of .