A One Hamiltonian formalism query - source is Goldstein's book

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In the Hamiltonian formulation of classical mechanics, the absence of constraint equations among coordinates is essential for the formulation's mathematical consistency and to ensure the independence of coordinates. This requirement contrasts with the Lagrangian formulation, where constraint equations are permissible, allowing for a broader range of systems to be analyzed. An example illustrating this difference can be seen in systems with holonomic constraints, which can be effectively handled in the Lagrangian framework but complicate the Hamiltonian approach. Although Legendre transformations can be applied to derive Hamiltonian equations from Lagrangian equations with constraints, the resulting Hamiltonian framework typically assumes independent coordinates for simplicity and clarity. Understanding these distinctions is crucial for correctly applying each formulation in classical mechanics.
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In 3rd edition of Goldstein's "Classical Mechanics" book, page 335, section 8.1, it is mentioned that :

In Hamiltonian formulation, there can be no constraint equations among the co-ordinates.

Why is this necessary ? Any simple example which will elaborate this fact ?

But in Lagrangian formulation, there can be constraint equations. Then why not in Hamiltonian formulation ?
 
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Actually you can make Legendre transformation for the Lagrange equations with constraints and obtain Hamiltonian-type equations with constraints. No problem
 
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