Newton Vs Lagrange Vs Hamilton
Vonny N
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.
ig...@bigbang.richmond.edu
"Vonny N" <von...@hotmail.com> wrote in message
news:df799de5.05032...@posting.google.com...
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
Neil Price
> What is the relationship between:
>
> 1. Newtonian Mechanics
> 2. Lagrangian Mechanics
> 3. Hamiltonian Mechanics
1. --> 2. In case of conservative forces, one can study a Lagrangian
of the form L = T - V with V the potential energy. This is equivalent
to the original Newtonian setup. There are, however, Lagrangians
which are not of the form L = T - V and therefore are not
representable in Newtonian form. (This should be taken with a grain
of salt).
2. --> 3. To construct a Hamiltonian from a given Lagrangian, one
studies a map called the Legendre transform (this expresses the p's as
derivatives of the Lagrangian). In order to do this construction, one
has to invert the Legendre transformation, so the transition to the
Hamiltonian formalism only works when this map is actually invertible.
If you know that p = dL/dv, it's easy to see that invertibility is
equivalent to the matrix d^2 L/dv^2 being invertible. Now, this is
always the case for a Lagrangian of mechanical type (one of the form L
= T - V, with T the kinetic energy).
The above should again be taken with a grain of salt. Even if the
Lagrangian is degenerate, there are more sophisticated ways of going
to the Hamiltonian formalism. In this case, some constraints will
arise and these will determine a subset of the phase space. Take the
example where the Lagrangian does not depend on a certain velocity
coordinate v_0. Then the associated momentum p_0 = dL/dv_0 will be
identically zero.
Furthermore, the above treatment only goes through in the case of
first-order theories, i.e. depending on x, v, but not the derivatives
of v. In that case, one can still make sense of a lot of these
things, but it is all a lot more involved. It certainly is no mere
extension of the first order case.
> 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?
The prevailing mathematical point of view is that Lagrangian mechanics
takes place on the tangent bundle of the configuration space, whereas
the Hamiltonian formalism is dealt with on the cotangent bundle.
There are various intrinsic objects defined on these bundles, and the
equations of motion can be expressed by use of them.
As to which formulation to use, I think it mostly depends (in the
regular case) on the formulation of the problem. Sometimes one is
preferred over the other: in the Lagrangian picture one has a
"variational principle"; i.e. the equations of motion are derived by
extremizing a certain functional. This can sometimes be exploited,
e.g. in the construction of numerical schemes which preserve (some of)
the geometrical content of the problem.
On the other hand, Hamiltonian mechanics is more directly amenable to
techniques from symplectic geometry, e.g. symplectic reduction,
definitions involving integrability, etc.
> 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).
>
See the book "foundations of mechanics" by Abraham and Marsden or
"introduction to mechanics and symmetry" by Marsden and Ratiu. These
make use of a lot of differential geometry so they might not be very
easy going on a first reading (especially FoM), but they tell you just
about everything you'll ever want to know on this topic. There's also
the book by V.I. Arnold "Mathematical Methods of Classical Mechanics",
which is very good.
N.
richard miller
A very brief reply from me cos my girlfriend is due around in 5 mins, I
Bob Hearn
> 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.
There are many excellent classical mechanics texts. But the only one
that I'm aware of that is completely clear on the exact meaning of the
equations, and avoids the formal ambiguity inherent in most
presentations of the Euler-Lagrange equations, is _Structure and
Interpretation of Classical Mechanics_, by Sussman and Wisdom. It is
both quite clear and quite rigorous. It uses a functional notation,
similar to Spivak (Calculus on Manifolds, Differential Geometry).
The book is most useful if you also install a freely available Scheme
variant, which has a symbolic classical mechanics library, so you can
do all the exercises. But it's still quite valuable without the
software. Disclaimer - Sussman is my advisor. But see the Amazon
reviews.
What did you not like about Goldstein?
whop...@csd.uwm.edu
> What is the relationship between:
> 1. Newtonian Mechanics
> 2. Lagrangian Mechanics
> 3. Hamiltonian Mechanics
Defining "Newtonian Mechanics" to mean a system with a 2nd order
equation of motion q^i''(t) = A^i(q(t),q'(t)) for i=1,...,N
(q=(q^1,...,q^N)), then the equivalence is:
Newtonian + Helmholz Conditions = Lagrangian
Lagrangian + Non-zero mass matrix -> Hamiltonian
Hamiltonian + Non-zero dispersion matrix -> Lagrangian
with the mass matrix m_{ij} and dispersion matrix W^{ij} defined by:
m_{ij} = d^2L/dv^idv^j; W^{ij} = d^2H/dp_idp_j.
(m and W are inverses when L and H are related by the Legendre
transform).
In Quantum Theory, if starting only with a 2nd order equation of motion
plus the equal time commutators
[q^i(t),q^j(t)] = 0 for all time t
then self-consistency will very nearly force the system to be
Lagrangian and Hamiltonian with the dispersion matrix being
W^ij = [q^i(t),q^j'(t)]/(i h-bar).
Strictly speaking, this is true in the classical limit as h-bar -> 0,
if the limiting value of W is non-singular. It is, however, probably
also true *before* going to the classical limit.
In the more general situation, what you have is roughly the following.
If W is singular, the singular modes can be factored out and yield
classical coordinates. The remaining modes gives you the quantum
coordinates, which must then comprise a quantum system that has a
Hamiltonian and non-singular Lagrangian as its classical limit.
If, further, the commutators [q,v] are c-numbers, then this forces the
Hamiltonian to be quadratic in the momenta. If, more generally, the
[q,v] only commute with the q's (so that [q,[q,v]]'s are 0), then the
same general conclusion may hold there, as well, but with the
corresponding mass matrix being non-constant functions of q. The
equations of motion will then take on the form of a combination of the
geodesic law (in configuration space) with the mass matrix proportional
to the metric and a Lorentz force law with respect to some Yang-Mills
potential (but in configuration space).
If further requiring that the system be a representation of the
appropriate space-time symmetry group (Galilei or Poincare') then it
may also be true that the configuration space factors into a number of
copies of single particle states, each satisfying a version of geodesic
+ Lorentz -- but this time with the laws holding in *ordinary*
spacetime. The "factorability conjecture" hasn't been proven, or even
properly formulated, as far as I know.
This is material, when fully developed, will form an essential core of
the treatise on Quantum Theory that I am currently working on. Stay
tuned for further developments ...
Jon Bell
Bob Hearn <bob....@gmail.com> wrote:
>
> [...] _Structure and
>Interpretation of Classical Mechanics_, by Sussman and Wisdom. It is
>both quite clear and quite rigorous. It uses a functional notation,
>similar to Spivak (Calculus on Manifolds, Differential Geometry).
>
>The book is most useful if you also install a freely available Scheme
I hadn't heard of that book before, but the title caught my eye because
I've read Abelson and Sussman's _Structure and Interpretation of Computer
Programs_. It's a true "classic" in computer science, taking a functional
approach using Scheme. Based on that experience, I'd expect SICM to be
very interesting. I'll have to try to carve out time to work through it
this summer.
--
Jon Bell <jtb...@presby.edu> Presbyterian College
Dept. of Physics and Computer Science Clinton, South Carolina USA
Arnold Neumaier
> von...@hotmail.com (Vonny N) wrote in message news:<df799de5.05032...@posting.google.com>...
>
>>What is the relationship between:
>>
>>1. Newtonian Mechanics
>>2. Lagrangian Mechanics
>>3. Hamiltonian Mechanics
1 is a special case of 3 which is a special case of 2.
>>Can they be declared to be *rigorously* mathematically equivalent?
No.
> On the other hand, Hamiltonian mechanics is more directly amenable to
> techniques from symplectic geometry, e.g. symplectic reduction,
> definitions involving integrability, etc.
There is also a symplectic view of Lagrangian mechanics, presented,
e.g., in the book on Mechanics and Symmetry by Marsden and Ratiu.
>>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?
Marsden and Ratiu give an almost rigorous approach.
> See the book "foundations of mechanics" by Abraham and Marsden or
> "introduction to mechanics and symmetry" by Marsden and Ratiu.
Marsden and Ratiu give an almost rigorous approach.
Arnold Neumaier
richard miller
news:d24a63$n3u$1...@news8.svr.pol.co.uk...
Three or so days later, I've looked at the other replies, don't have much
too add, except:
The Hamiltonian formulation gives you these gorgeous symmetric equations in
the generalised coordinates (q's), conjugate momenta (p's) i.e. your q's and
p's.
with H defined by p dq/dt - L, then for the p and q's you have
dH/dp = dq/dt, dH/dq = -dp/dt
Before you even mention the words 'quantum mechanics', you have conjugate
coordinates 'mometum and position'. Uncertainty anyone?
The beauty is that if you just looked at Hamilton's equations, the p and q
equations look rather symmetric. You might lose track of what is position
and what is momenta. Then there's energy and time too.
That aside, Hamilton's formulation leads to the concept of phase space. It
is quite neat in that usually we enquire afer the time evolution of a
system, but something like a velocity versus position (distance) plot can be
just as, if not more, interesting. For example, a pendulum swings (no big
deal); but the time evolution (angular position versus time) is of little
interest; it is the frequencies and amplitudes of the system that are of far
more interest. Via more 'Canonical transformations' one gets to action angle
variables, frequencies and even stability of the system. This is where it
all gets very interesting from the modern dynamics viewpoint. Stability of
the solar System, KAM Theorem (Kolmogrov-Arnold-Moser theorem, Mathworld)..
etc. This is Hamiltonian dynamics at its best - and lots of modern
'symplectic' stuff to go.
OK, I know you wanted a better answer than the above woffle. Here is a book
link (I hope it is not too elementary for you)
A very good book from a modern perspective is Neil S. Rasband, 'Dynamics',
ISBN 0471873985
Then simply Google 'kolmogrov-Arnold-Moser' etc. Stability of the Solar
System.
And finally, after much woffle,
Newtonian = Lagrangian (and Hamilton's Principle) with a special Lagrangian
T - V
Newtonian = Hamiltonian with Hamiltonian = T + V
Lagragian = Hamilton, but without the mathematical beauty or theoretical
advancement or laughs
I know this is simplistic, there are dissipative forces etc, but just for
simplicity...
Enjoy
Richard Miller
whop...@csd.uwm.edu
> The prevailing mathematical point of view is that Lagrangian
mechanics
> takes place on the tangent bundle of the configuration space, whereas
> the Hamiltonian formalism is dealt with on the cotangent bundle.
The velocity v = dq/dt resides on the tangent bundle TQ of the manifold
Q where q lives. But the state (q,v) lives on the first jet J^1(Q),
which is the manifold whose local coordinates are just (q,v)
themselves, with v in T_q(Q).
> As to which formulation to use, I think it mostly depends (in the
> regular case) on the formulation of the problem. Sometimes one is
> preferred over the other: in the Lagrangian picture one has a
> "variational principle"; i.e. the equations of motion are derived by
> extremizing a certain functional.
The Lagrangian is the one that's fundamental. An action is formulated
over a spacetime region, and (after applying the equations of motion)
the total variation will reduce to one on the spacetime boundary
comprising the integral (p.delta(q)) over the boundary (dR) of the
region R. For ordinary mechanics, with regions being intervals on the
time line, dR is the chain (T+) - (T-) comprising 2 points and the
total variation is just the value
p(T+).delta(q(T+)) - p(T-).delta(q(T-)).
In a Lorentzian space (as opposed to a Newtonian space), one can also
take the spacetime region R so that (a) it's compact and (b) its
boundary is EVERYWHERE spacelike, and of the form dR = (R+) - (R-),
with (R+) and (R-) connected by a homotopy, such that the common
boundary d(R+) = d(R-) (= A, the "anchor") remains fixed. The homotopy
parameter then plays the role of the time coordinate, the anchor A
playing the analogous role of asymptotic infinity. This way you can
avoid running all the integrals off to infinity and get consistent
theories, even in a quantum setting (noting that the essential block
provided by Haag's no-interaction theorem is directly connected with
the issue of using infinite volume integrals and global unitary
evolution operators that encompass all of space).
These issues, too, may find their way into the treatise mentioned
earlier that I'm working on.
For a Hamiltonian, you need an additional prerequisite: a time-like
vector flow, and the ability to do Lie derivatives on the fields in
question. I think Lecture Notes in Physics 107 is the edition that
covers the general issues surrounding all the variational pictures.
For field theories, Legendre transforms are not unique, since you can
choose any of the infinity of time-like vector fields to do the
variation on.
bjflanagan
Neil Price
> Neil Price wrote:
> > The prevailing mathematical point of view is that Lagrangian
> mechanics
> > takes place on the tangent bundle of the configuration space, whereas
> > the Hamiltonian formalism is dealt with on the cotangent bundle.
>
> The velocity v = dq/dt resides on the tangent bundle TQ of the manifold
> Q where q lives. But the state (q,v) lives on the first jet J^1(Q),
> which is the manifold whose local coordinates are just (q,v)
> themselves, with v in T_q(Q).
Some caution is in order here. The coordinates on the jet bundle are
(t, q, v); the first jet bundle is isomorphic to TM x R (but not by a
natural map). Furthermore, for autonomous problems the tangent
bundle will do just fine.
> > As to which formulation to use, I think it mostly depends (in the
> > regular case) on the formulation of the problem. Sometimes one is
> > preferred over the other: in the Lagrangian picture one has a
> > "variational principle"; i.e. the equations of motion are derived by
> > extremizing a certain functional.
>
> The Lagrangian is the one that's fundamental.
(...)
> For a Hamiltonian, you need an additional prerequisite: a time-like
> vector flow, and the ability to do Lie derivatives on the fields in
> question. I think Lecture Notes in Physics 107 is the edition that
> covers the general issues surrounding all the variational pictures.
> For field theories, Legendre transforms are not unique, since you can
> choose any of the infinity of time-like vector fields to do the
> variation on.
Could you give more information about this LNP 107?
There are ways of formulating what you could call a "covariant
Hamiltonian formalism". In the case of time-dependent mechanics for
example, the Hamiltonian becomes a map from J^1(E)^* to T*E. This
concept can be generalized to field theories as well. It becomes
equivalent to the normal Hamiltonian formalism when choosing a
timelike vectorfield and "integrating" this covariant Hamiltonian over
a space-like hypersurface. There also exists a unique Legendre
transform, which can be shown to yield the normal Legendre map after
breaking covariance.
There are two unpublished articles by Gotay, Marsden and Isenberg
floating around where they treat this formalism in greater detail.
Granted, even with this formalism the Hamiltonian side loses much of
its appeal, but the original poster talked about mechanics; in this
case there really is something to gain from the Hamiltonian formalism.
Think Hamilton-Jacobi, the Liouville-Arnold integrability theorem,
that sort of thing.
N.
von...@hotmail.com
wonderful Springer-Verlag Graduate Outline series) he mandates that
Newtonian Mechanics is properly contained in Lagrangian Mechanics which
in turn is properly contained in Hamiltonian Mechanics; the latter
therefore being the most general. Do you then disagree with this
structural assertion?
Vonny N.
Arnold Neumaier
> In V. Arnol'd 's highly respected Classical Mechanics text (in the
> wonderful Springer-Verlag Graduate Outline series) he mandates that
> Newtonian Mechanics is properly contained in Lagrangian Mechanics which
> in turn is properly contained in Hamiltonian Mechanics; the latter
> therefore being the most general.
Can you quote the context and outline his argument?
I can't believe that he wrote that.
> Do you then disagree with this
> structural assertion?
Yes. One can convert any Hamiltonian system into a Lagrangian system
in extended phase space. But one can convert a Lagrangian system into
a Hamiltonian one only if d^2L/dq^2 is nonsingular.
Arnold Neumaier
Frank Hellmann
> in extended phase space. But one can convert a Lagrangian system into
> a Hamiltonian one only if d^2L/dq^2 is nonsingular.
>
That is only correct if you ignore the possibility to work with
constraints. Without constraints you can't even handle simple
relativistic systems though, so this is usually assumed to be part of
the Hamiltonian method. See the appendix of gr-qc/0110034 or Diracs
Lecture on QM (Here you'll find the original ideas that allowed GR to
be cast into Hamiltonian form).
f
Arnold Neumaier
True. But one would usually refer to this as a 'constrained Hamiltonian
system' and not just as a 'Hamiltonian system'.
Constrained Hamiltonian systems and Lagrnaigian systems are almost
equivalent. One still needs a constant rank assumtpion to go from
a Lagraingian to a Hamiltonian.
Arnold Neumaier