Lagrangian mechanics

[Sriman Dutta]

Hi everybody,

 

Can anyone tell me what is the actual purpose of the Lagrangian mechanics ? Lagrangian mechanics use a potential field to interpret Newtonian physics, but at the end of the day, it is just an approach to interpret Newton's laws in a different way. Why is this so ? Newtonian physics (as I think) is quite fine and can predict and define classical events. I understand how Lagrangian works but I don't know why it's needed ?

[studiot]

Hi everybody,

 

Can anyone tell me what is the actual purpose of the Lagrangian mechanics ? Lagrangian mechanics use a potential field to interpret Newtonian physics, but at the end of the day, it is just an approach to interpret Newton's laws in a different way. Why is this so ? Newtonian physics (as I think) is quite fine and can predict and define classical events. I understand how Lagrangian works but I don't know why it's needed ?

 

First why do you object to there being more than one way to do things?

 

What would it be like if there was only one road in and out of Calcutta?

Surely many roads are better?

 

Moving the discussion back to mechanics, in statics there is an equivalent distinction in methods of calculation of the mechanics of structures.

There are direct Newtonian Force-Displacement methods and Energy methods such as Castigliano's and Maxwell's theorems and the theorems of virtual work.

I can assure you that structural engineers are all too pleased to have alternative methods at their disposal for calculation, from which they can choose the easist one for the task in hand.

 

You say you understand Lagrangian mechanics, so what are the differences?

 

Lagrangian mechanics is an energy method.

As such it involves scalars, rather than the vectors in the Newtonian equations of motion.

Scalars are the same in all coordinate systems.

Newtonian methods are differential equations, which are local in nature and refer to points close by.

 

Lagrangian equations of motion are integral equations and global in nature.

They refer to the entire trajectory.

They are suitable to enter directly into Hamilton's Principle or the Principle of Least Action.

 

Further they feed directly into and connect more modern and advanced mechanics, that of symmetry (Noether's theorem), Relativity and Quantum Mechanics.

 

The derivation of the Schrodinger Equation is a particularly simple via this route.

[Sriman Dutta]

Oh I now see the its importance.