Re: Newton Vs Lagrange Vs Hamilton
From: richard miller (richard_at_microscitech.freeserve.co.uk)
Date: 03/27/05
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Date: Sun, 27 Mar 2005 08:56:55 +0000 (UTC)
"Vonny N" <vonnyn@hotmail.com> wrote in message
news:df799de5.0503251424.2f339940@posting.google.com...
> What is the relationship between:
>
> 1. Newtonian Mechanics
> 2. Lagrangian Mechanics
> 3. Hamiltonian Mechanics
>
> Can they be declared to be *rigorously* mathematically equivalent?
>
> Does it suffice to say that they are reformulations of each other,
> each well-adapted to a particular class of problems?
>
> If so, can we describe reasonably clearly which type of problems are
> more easily represented and/or solved using each version of mechanics?
>
> Finally, is there a book on classical mechanics which treats all three
> of these approaches to mechanics, along with their
> inter-relationships, with mathematical rigour? (I'm afraid I don't
> much like Goldstein's widely cited book).
>
> Vonny N.
>
A very brief reply from me cos my girlfriend is due around in 5 mins, I
haven't cleaned the house like I promised, and I haven't cooked the dinner
like I promised!
Please forgive the rather lax presentation as well
If you take a Lagrangian function L = difference in Kinetic T and Potential
energy V
i.e. L = T - V. Apply a first order variation/Hamilton's primnciple and end
up with the standard Lagrange equations d/dt(dL/dv) - dL/dx = 0. (little v =
velocity dx/dt)
With the standard, one dimensional T = 1/2mv^2 (v=dx/dt), V = 0 (freely
moving particle no field) and so L = 1/2.m.v^2 you will get mdv/dt = 0, with
a little flippancy we have d(mv)/dt = 0, i.e. rate of change of linear
momemtum = 0 when no force acts.
In short, if the Lagrangian is the difference of kinetic and potential
energy, you should get back to Newtonian mechanics. The real beauty is that
the Lagrangian can be expanded to to other functionals L. Do this and you
can are in the realm of field theory.
Hamilton takes Lagranges equations, which are second order in time, and
converts them to two first order equations. The beauty in these latter
equations is that they are almost symmetric...
crikes my girlfriend has just turned up
I'm done for, back later
cheers
Richard Miller
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