Lagrangian Mechanics: Purpose, Advantages, Hamiltonian Reformulation

[yango_17]

What exactly was the purpose for the development of Lagrangian mechanics? Does it describe physical systems and situations that Newtonian mechanics cannot? I would also like to know why the Hamiltonian reformulation of mechanics occurred after the development of Lagrangian mechanics.

 

[Dale]

What exactly was the purpose for the development of Lagrangian mechanics?

It was primarily designed to allow physics to be expressed in terms of generalized coordinates.

 

[andrewkirk]

I can give a couple of reasons. No doubt others can add more.

Firstly, the Newtonian equations of motion are predicated on a Cartesian coordinate system. They do not hold if a non-Cartesian system is used, without a change of form that in many cases makes them unrecognisable. In Lagrangian mechanics the coordinates can be anything at all, eg polar, spherical or a very situation-dependent set of coordinates that reflects system constraints, yet the equations always have the same form.

Secondly, problems are often much easier to solve in Lagrangian form rather than Newtonian. I think the reason is something to do with the fact that Lagrangian is more focused on energy whereas Newtonian is focused on forces and acceleration. For many problems, an energy-based approach is quicker and easier.

Thirdly, they allow uses in electromagnetics where the Lagrangian is not simply an energy-based function and hence, I suspect, not a direct derivation from the Newtonian approach.

I haven't used Hamiltonian approaches much but the impression I get is that one advantage is that they allow an even greater choice of coordinate systems to use. One can choose $p$ ('momentum') coordinates that are not defined as simply $m\dot{q}$, provided the canonical equations are satisfied.

In my texts Hamiltonian mechanics was introduced principally in order to lay the ground for the standard quantum mechanical formalism, which is built squarely around the Hamiltonian. (although does the Feynman path integral formulation use the Lagrangian? I can't remember).
In a sense the Hamiltonian formulation is more intuitive because the Hamiltonian

 

[OldEngr63]

The energy based approach used in Lagrangian mechanics is often much, much easier to apply correctly than the vector based approach of Newtonian mechanics. It is usually pretty easy to write the energy expressions; no derivatives higher than the first are required. From that point on, it is rather "mechanical" in the "turn the crank" sense. The process is the same every time.

There is a bit of ambiguity associated with the name Hamilton. There is Hamilton's Principle and there is the Hamiltonian formulation of the equations of motion; they are not the same thing.

Hamilton's Principle is a minimum principle, and often used as a starting point to derive the Lagrange equations. But it is particularly useful in dealing with mixed systems, such as electro-mechanical systems. The advantage to using Hamilton's Principle in such a case is that the coupling terms all fall out automatically, particularly the boundary conditions on the interface between the different phenomena. See Dynamics of Mechanical and Electromechanical Systems by Crandall, Karnopp, Kurtz and Pridmore-Brown for more on this.

The Hamiltonian formulation provides a system of 2n first order differential equations for a system with n-degrees of freedom, whereas the Lagrange formulation provides a system of n second order differential equations of motion for the same system. For some purposes, the system of 2n first order equations is preferred, but the fact that momenta are taken as coordinates in the Hamiltonian formulation limits its utility in many classical mechanics applications.

I personally use the Lagrange equations often and Hamilton's Principle as well; only very, very rarely do I use Hamilton's equations of motion.

 

[lightarrow]

What exactly was the purpose for the development of Lagrangian mechanics? Does it describe physical systems and situations that Newtonian mechanics cannot?

Lagrangian mechanics is a powerful tool in physics essentially for three reasons: the first is that in problems of mechanics you can operate on active forces only and avoid having to consider constraints (in the most of cases); the second is the fact it comes from a principle, that is "Hamilton's principle" which allows to use energies (kinetic and potential) which are easier to find in most of cases and so it's easier to find the lagrangian function and then the differential equations to solve (Lagrange equations); the third is that Hamilton's principle is very useful in theoretical physics because it's a "guiding line" to find models and theories describing phenomena: when you have found a lagrangian of the system you are at half way.

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