One Hamiltonian formalism query - source is Goldstein's book

In summary, the Hamiltonian formalism is a mathematical framework used in classical mechanics to describe the behavior of physical systems. It was developed by William Rowan Hamilton in the 19th century and is based on the concept of a Hamiltonian, which represents the total energy of the system. It differs from the Lagrangian formalism in the choice of variables used and is a fundamental quantity in classical mechanics. The Hamiltonian formalism has practical applications in various fields and provides a more intuitive approach to understanding and solving complex problems.
  • #1
swapnilp
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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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  • #2
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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1. What is the Hamiltonian formalism?

The Hamiltonian formalism is a mathematical framework used in classical mechanics to describe the behavior of a physical system. It is based on the concept of a "Hamiltonian", which is a function that represents the total energy of the system.

2. Who developed the Hamiltonian formalism?

The Hamiltonian formalism was developed by William Rowan Hamilton, an Irish mathematician and physicist, in the 19th century. He introduced the concept of the Hamiltonian in his famous paper "On a General Method in Dynamics" published in 1835.

3. How does the Hamiltonian formalism differ from the Lagrangian formalism?

The main difference between the Hamiltonian and Lagrangian formalisms is the choice of variables used to describe the system. The Lagrangian formalism uses generalized coordinates and velocities, while the Hamiltonian formalism uses generalized coordinates and momenta. The two formalisms are equivalent and can be used interchangeably to describe the same physical system.

4. What is the significance of the Hamiltonian in classical mechanics?

The Hamiltonian is a fundamental quantity in classical mechanics as it represents the total energy of a physical system. It is a conserved quantity, meaning that it remains constant throughout the motion of the system, as long as there are no external forces acting on it. The Hamiltonian also plays a crucial role in the Hamiltonian formalism, providing a set of equations (Hamilton's equations) that govern the motion of the system.

5. How is the Hamiltonian formalism used in practical applications?

The Hamiltonian formalism has many practical applications in various fields such as mechanics, electromagnetism, and quantum mechanics. It is used to study the behavior of physical systems, predict their future states, and solve complex problems that cannot be solved using other methods. The Hamiltonian formalism also provides a more intuitive and geometrical approach to classical mechanics, making it easier to understand and apply in real-world situations.

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