Why Lagrangian and Hamiltonian formalism

[Goldbeetle]

Dear all,
could please give my some links or references to material that justifies the mathematical and physical reasons for introducing these two formalisms in mechanics?

Thanks.

Goldbeetle

 

[Andy Resnick]

They make some problems easier to solve. Some people would add that they place the foundations of mechanics on more solid gound.

 

[charng]

The Lagrangian and Hamiltonian formalisms are two mathematical frameworks used to describe the dynamics of a mechanical system. They were developed to overcome the limitations of Newtonian mechanics and provide a more elegant and powerful approach to studying the behavior of physical systems.

The Lagrangian formalism, developed by Joseph-Louis Lagrange in the 18th century, is based on the principle of least action. This principle states that the true path of a system is the one that minimizes the action, which is a mathematical quantity that represents the total energy of the system. By using the Lagrangian formalism, we can describe the evolution of a system in terms of generalized coordinates, rather than the traditional Cartesian coordinates used in Newtonian mechanics. This allows for a more flexible and intuitive approach to solving complex problems.

The Hamiltonian formalism, developed by William Rowan Hamilton in the 19th century, is based on the concept of energy conservation. It introduces a new mathematical function called the Hamiltonian, which represents the total energy of a system. By using the Hamiltonian formalism, we can describe the dynamics of a system in terms of canonical coordinates and momenta, which simplifies the equations of motion and allows for more efficient calculations.

One of the main advantages of these formalisms is their ability to handle systems with constraints and degrees of freedom. In Newtonian mechanics, the equations of motion can become very complex and difficult to solve when dealing with systems with multiple constraints. However, the Lagrangian and Hamiltonian formalisms provide a more elegant and systematic way to handle these types of systems.

Moreover, these formalisms have been extended to other areas of physics, such as quantum mechanics and field theory, making them powerful and versatile tools in understanding the behavior of physical systems.

In conclusion, the Lagrangian and Hamiltonian formalisms were developed to provide a more elegant and powerful approach to studying the dynamics of physical systems. They offer a more intuitive and systematic way to handle complex systems and have been extended to other areas of physics, making them essential tools for scientists and researchers in various fields. I hope this helps and provides some insight into the importance of these formalisms in mechanics.