A Poisson manifold is a type of geometric structure that combines aspects of both differential geometry and mathematical physics. It is defined as a smooth manifold equipped with a Poisson bracket, which is a bilinear operation that satisfies certain properties, allowing for the study of dynamical systems. This structure is particularly useful in the context of classical mechanics, where it helps describe the evolution of physical systems. In a Poisson manifold, the Poisson bracket provides a way to define the notion of observables and their relationships. This framework is essential for understanding the underlying symplectic geometry, which is closely related to Hamiltonian mechanics. Poisson manifolds also play a significant role in areas such as quantum mechanics and integrable systems, making them a vital concept in modern mathematical physics.