The chapter introduces the generalized Hamiltonian dynamics, a generalization of both Lagrangian and Hamiltonian formulations of dynamics. It discusses the advantages of each formulation and highlights the need for a more general theory that can accommodate cases where momenta are not independent functions of velocities. The chapter defines the Hamiltonian and explores the strong and weak equations, distinguishing between equations that remain valid to first order in variations and those that are violated by a small amount. It introduces the concept of Hamiltonian variables and the Hamiltonian equations of motion, which are derived from the Hamiltonian function and the Poisson bracket. The chapter also covers the case of homogeneous velocities, where the Lagrangian is homogeneous of the first degree in velocities, leading to simpler equations of motion. It discusses consistency conditions and supplementary conditions, and provides methods for transforming the Hamiltonian form of dynamics. Finally, it explores the application of this theory to relativistic dynamics and the quantization process, emphasizing the importance of first-class constraints in relativistic theories.The chapter introduces the generalized Hamiltonian dynamics, a generalization of both Lagrangian and Hamiltonian formulations of dynamics. It discusses the advantages of each formulation and highlights the need for a more general theory that can accommodate cases where momenta are not independent functions of velocities. The chapter defines the Hamiltonian and explores the strong and weak equations, distinguishing between equations that remain valid to first order in variations and those that are violated by a small amount. It introduces the concept of Hamiltonian variables and the Hamiltonian equations of motion, which are derived from the Hamiltonian function and the Poisson bracket. The chapter also covers the case of homogeneous velocities, where the Lagrangian is homogeneous of the first degree in velocities, leading to simpler equations of motion. It discusses consistency conditions and supplementary conditions, and provides methods for transforming the Hamiltonian form of dynamics. Finally, it explores the application of this theory to relativistic dynamics and the quantization process, emphasizing the importance of first-class constraints in relativistic theories.