# Difference between Newtonian, Langrange, and Hamiltonian Mechanics

• Feldoh
In summary, the conversation discusses the differences between the three forms of mechanics: Newtonian, Lagrangian, and Hamiltonian. While they are all related and build upon each other, they approach problem-solving in different ways. Lagrangian mechanics uses energies and Hamiltonian mechanics simplifies equations by making them one-dimensional differential equations. While these concepts can be interesting and helpful, they are not necessary to fully understand mechanics.
Feldoh
Hey I was just wondering what the differences between the three forms of mechanics were. I've only studied basic Newtonian mechanics so I'm not really sure about the other two. Could anyone elaborate?

They are not different in the sense you imply. Lagrangian mechanics is a further development of Newtonian dynamics, and Hamilton's work is based on Lagrange principles. So they form a progression.

It is a very large, important and fascinating field of study so it wouldn't be practical to teach it here. I recommend any decent book on classical mechanics, which will take you through the progression.

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Mechanics is mechanics. Lagrange and Hamilton mechanics are both known as analytical mechanics. The treatment is just more conceptual and more mathematical. It depends if you like formalism. Anyway, give it a try: sometimes it is easier (especially for systems with several degrees of freedom) to solve a problem with these tools rather than with a (in fact: several coupled) Newton equation, which quickly becomes cumbersome. Have a look to Landau or Goldstein books. To be honest, I should however say that analytical mechanics is nice, it gives a lot of insights and concepts, but it is not so useful in the end. Don't get too fascinated by this kind of stuff, and don't loose too much time (physics is wide !).

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diff. between these mechanics

see Newtonion mechanics deals with vectors like mom. and acc. but then came lagrangian mech. which deals in terms of energies which are scalars and hence equations are easy to solve. but these eqns. are second order differential eqns. .to improve on that and mke eqn. of motion a one dimensional diff. eqn. hamiltonian mech. is introduced.

Thanks, everyone, especially nandan!

## What is the main difference between Newtonian, Langrange, and Hamiltonian Mechanics?

The main difference between these three mechanics is the approach they use to describe the motion of a system. Newtonian mechanics is based on the laws of motion and forces, while Lagrange mechanics uses the concept of energy and generalized coordinates, and Hamiltonian mechanics uses the concept of energy and momentum.

## Which mechanics is considered the most comprehensive?

Hamiltonian mechanics is considered the most comprehensive as it can be applied to both classical and quantum mechanical systems, and it takes into account the concept of phase space and the symmetries of a system.

## What is the significance of generalized coordinates in Lagrange mechanics?

Generalized coordinates allow for a more efficient and concise description of the motion of a system, as they are independent of the specific physical coordinates and can be chosen to simplify the equations of motion.

## What is the advantage of using Hamiltonian mechanics over Newtonian mechanics?

One advantage of using Hamiltonian mechanics is that it provides a more elegant and unified description of a system's motion, as it combines the concepts of energy and momentum. It also allows for the use of symmetries to simplify the equations of motion.

## Can these mechanics be applied to all types of systems?

Yes, all three mechanics can be applied to a wide range of systems, from simple particles to complex systems such as the motion of planets in a solar system. However, the most appropriate mechanics to use may depend on the specific characteristics of the system being studied.

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