In classical mechanics, the Hamiltonian and Lagrangian formulations are two equivalent ways to describe the dynamics of a physical system. The relation between the Hamiltonian and Lagrangian becomes more involved when dealing with systems that have constraints.
In the Lagrangian formulation, the dynamics of a system with constraints are described using generalized coordinates, and the Lagrangian function, denoted by L(q, q̇, t), where q represents the generalized coordinates, q̇ represents their corresponding velocities, and t denotes time. The Lagrangian captures the kinetic and potential energy of the system and any other forces acting on it. The equations of motion are derived by applying the principle of least action, known as Hamilton's principle or the principle of stationary action, to the Lagrangian.
When constraints are present in the system, they can be classified into two types: holonomic and non-holonomic constraints. Holonomic constraints are equations that restrict the possible positions of the system but not its velocities. Non-holonomic constraints, on the other hand, impose restrictions on both the positions and velocities.
To incorporate constraints into the Lagrangian formulation, one can use methods such as the method of Lagrange multipliers or generalized coordinates. The Lagrange multipliers introduce additional terms into the Lagrangian that enforce the constraints. By solving the resulting Euler-Lagrange equations, which include the Lagrange multipliers, one can obtain the equations of motion for the system.
The Hamiltonian formulation, on the other hand, describes the system's dynamics in terms of generalized coordinates and their conjugate momenta. The Hamiltonian function, denoted by H(q, p, t), is defined as the Legendre transform of the Lagrangian, where p represents the generalized momenta. For a system without constraints, the Hamiltonian is given by H(q, p, t) = Σpᵢq̇ᵢ - L(q, q̇, t), where the q̇ᵢ are the time derivatives of the generalized coordinates.
When constraints are present, the Hamiltonian formulation is modified to include the constraints. The constraints are incorporated into the Hamiltonian using the method of Lagrange multipliers or generalized coordinates, similar to the Lagrangian formulation. The resulting Hamiltonian equations of motion include the Lagrange multipliers or additional terms that account for the constraints.
In summary, the relation between the Hamiltonian and Lagrangian for a system with constraints involves incorporating the constraints into the respective formulations using methods like Lagrange multipliers or generalized coordinates. The resulting equations of motion in both formulations will account for the constraints and describe the dynamics of the system.