may choose the to make the map linear. The second requirement for the G-action to be Hamiltonian is that the map be a Lie algebra homomorphism from to the algebra of smooth functions on M under the Poisson bracket. If the action of G on (M, ω) is Hamiltonian in this sense, then a momentum map is a map such that writing defines a Lie algebra homomorphism satisfying . Here is the vector field of the Hamiltonian , defined by Examples of momentum maps In the case of a Hamiltonian action of the circle , the Lie algebra dual is naturally identified with , and the momentum map is simply the Hamiltonian function that generates the circle action. Another classical case occurs when is the cotangent bundle of and is the Euclidean group generated by rotations and translations. That is, is a six-dimensional group, the semidirect product of and . The six components of the momentum map are then the three angular momenta and the three linear momenta. Let be a smooth manifold and let be its cotangent bundle, with projection map . Let denote the tautological 1-form on . Suppose acts on . The induced action of on the symplectic manifold , given by for is Hamiltonian with momentum map for all . Here denotes the contraction of the vector field , the infinitesimal action of , with the 1-form . The facts mentioned below may be used to generate more examples of momentum maps. Some facts about momentum maps Let be Lie groups with Lie algebras , respectively. Let be a coadjoint orbit. Then there exists a unique symplectic structure on such that inclusion map is a momentum map. Let act on a symplectic manifold with a momentum map for the action, and be a Lie group homomorphism, inducing an action of on . Then the action of on is also Hamiltonian, with momentum map given by , where is the dual map to ( denotes the identity element of ). A case of special interest is when is a Lie subgroup of and is the inclusion map. Let be a Hamiltonian -manifold and a Hamiltonian -manifold. Then the natural action of on is Hamiltonian, with momentum map the direct sum of the two momentum maps and . Here , where denotes the projection map. Let be a Hamiltonian -manifold, and a submanifold of invariant under such that