André Lichnerowicz discusses the deformation theory and quantization in the context of symplectic geometry and Poisson brackets, which are fundamental to classical mechanics. He collaborates with Bayen, Flato, Fronsdal, Sternheimer, and J. Vey to study deformations of the Poisson Lie algebra, which provide a new approach to quantum mechanics. The focus is on dynamical systems with a finite number of degrees of freedom, though the methods can be extended to physical fields. 1. **Lie Algebras Associated with a Symplectic Manifold:** - A smooth symplectic manifold \((W, F)\) of dimension \(2n\) is defined by a closed 2-form \(F\). The isomorphism \(\mu\) and the skew-symmetric 2-tensor \(\Delta\) are introduced. - Symplectic infinitesimal transformations (i.t.) are vector fields \(X\) such that \(L(X)F = 0\). The Lie algebra \(L\) of these transformations is defined, and its properties are discussed. - \(L^C\) is the Lie algebra of conformal symplectic i.t., where \(L\) and \(L^C\) are ideals. The structure of \(L^C\) depends on whether \(F\) is exact or not. - The space \(N\) of equivalence classes of functions modulo constants is equipped with a Poisson bracket, defining a Lie algebra structure. 2. **Classical Dynamics and Symplectic Manifolds:** - A dynamical system with \(n\) degrees of freedom has a configuration space \(M\) and a Hamiltonian \(H\) defined on the cotangent bundle \(T^*M\). - The phase space is a symplectic manifold \((W, F)\) of dimension \(2n\), where the Hamiltonian vector field \(X_H\) determines the dynamics. - The space \(N\) has two algebraic structures: an associative algebra structure by function multiplication and a Lie algebra structure by the Poisson bracket. - The evolution of a family of functions \(u_t\) satisfying \(\frac{d u_t}{d t} = \{H, u_t\}\) is related to the trajectories in the Hamiltonian formalism. This chapter provides a foundational understanding of the mathematical framework that connects classical mechanics to symplectic geometry and Poisson brackets, laying the groundwork for further developments in quantum mechanics.André Lichnerowicz discusses the deformation theory and quantization in the context of symplectic geometry and Poisson brackets, which are fundamental to classical mechanics. He collaborates with Bayen, Flato, Fronsdal, Sternheimer, and J. Vey to study deformations of the Poisson Lie algebra, which provide a new approach to quantum mechanics. The focus is on dynamical systems with a finite number of degrees of freedom, though the methods can be extended to physical fields. 1. **Lie Algebras Associated with a Symplectic Manifold:** - A smooth symplectic manifold \((W, F)\) of dimension \(2n\) is defined by a closed 2-form \(F\). The isomorphism \(\mu\) and the skew-symmetric 2-tensor \(\Delta\) are introduced. - Symplectic infinitesimal transformations (i.t.) are vector fields \(X\) such that \(L(X)F = 0\). The Lie algebra \(L\) of these transformations is defined, and its properties are discussed. - \(L^C\) is the Lie algebra of conformal symplectic i.t., where \(L\) and \(L^C\) are ideals. The structure of \(L^C\) depends on whether \(F\) is exact or not. - The space \(N\) of equivalence classes of functions modulo constants is equipped with a Poisson bracket, defining a Lie algebra structure. 2. **Classical Dynamics and Symplectic Manifolds:** - A dynamical system with \(n\) degrees of freedom has a configuration space \(M\) and a Hamiltonian \(H\) defined on the cotangent bundle \(T^*M\). - The phase space is a symplectic manifold \((W, F)\) of dimension \(2n\), where the Hamiltonian vector field \(X_H\) determines the dynamics. - The space \(N\) has two algebraic structures: an associative algebra structure by function multiplication and a Lie algebra structure by the Poisson bracket. - The evolution of a family of functions \(u_t\) satisfying \(\frac{d u_t}{d t} = \{H, u_t\}\) is related to the trajectories in the Hamiltonian formalism. This chapter provides a foundational understanding of the mathematical framework that connects classical mechanics to symplectic geometry and Poisson brackets, laying the groundwork for further developments in quantum mechanics.