When the complexity of the system gets increased, it becomes very hard to analyze the problem using vectors, hence we need "a more sustainable technique" to deal with the problems using scalers.
Newtonian mechanics is essentially vectorial in nature (deals with forces and constraints). The Lagrangian and Hamiltonian formulations, on the other hand, are cast in terms of kinetic and potential energies, which are scalar functions. The equations of motion are thus derived from the scalar variables, which reduces the mathematical complexity. The Hamiltonian method is further advantageous as it involves only first-order derivatives instead of second-order derivatives involved in the Lagrangian approach.
The Lagrangian and Hamiltonian methods are conceptually more simpler and efficient when the mechanical system is more complicated
It is actually necessary to study Newtonian mechanics to truly understand the Lagrangian and Hamiltonian mechanics since their underlying foundation is Newtonian mechanics. These are essentially different formulations of the same mechanics.
Some of the advantages of Lagrangian formalism over Newtonian mechanics are:
1. As a problem solving tool, it takes constraint forces into account i.e, eliminating redundant coordinates takes care of constraints automatically.
2. It is possible to generalize Lagrangian formalism to relativistic mechanics whereas Newtonian mechanics is not applicable in that regime.
3. Lagrangian formalism can be extended to case of continuous degree of freedom, i.e. fields.
(Source: Prof. V. Balakrishnan Video lectures)
When the complexity of the system gets increased, it becomes very hard to analyze the problem using vectors, hence we need "a more sustainable technique" to deal with the problems using scalers.
Newtonian mechanics is essentially vectorial in nature (deals with forces and constraints). The Lagrangian and Hamiltonian formulations, on the other hand, are cast in terms of kinetic and potential energies, which are scalar functions. The equations of motion are thus derived from the scalar variables, which reduces the mathematical complexity. The Hamiltonian method is further advantageous as it involves only first-order derivatives instead of second-order derivatives involved in the Lagrangian approach.
The Lagrangian and Hamiltonian methods are conceptually more simpler and efficient when the mechanical system is more complicated
I think question should be
Why we need Newtonian if we have Hamiltonian and Lagrangian mechanics 🤔
It is actually necessary to study Newtonian mechanics to truly understand the Lagrangian and Hamiltonian mechanics since their underlying foundation is Newtonian mechanics. These are essentially different formulations of the same mechanics.
Are you trying to say one can't understand the mechanics of system using Lagrangian or Hamiltonian mechanics,only.
It is necessary to understand the Newtonian??
I don't think so
Because Newtonian deals with force system
And Lagrangian and Hamiltonian use energy systems
So always energy systems is more general as compared to force system
So we can understand better using energy systems compare to force system
Think about this 😌
i agree with Reet
We don't "need" them, but they are convenient in different problems. They do not introduce any new physics.