Natural Science Forum / Physics / Research / March 2008

What about the Hamiltonian of a charged particle in an EM field. In
this case the Hamiltonian is not p^2/2m + q\phi but rather the
momentum enters via minimal coupling: (p - qA/c)^2/2m + q\phi. (Note:
A & \phi are in general functions of space and time). Is this new
Hamiltonian the energy of the system or is it just an object that
correctly reproduces Newton's laws ?

-Mehul

Pmb schrieb:

In mechanics, time is a point in a 1-dimensional manifold,
and diffeomorphisms are just smooth reparameterizations of the time.

For any Lagrangian of the form
    L(q,qdot,t) := U(q(t)) qdot(t),
where q is an n-dimensional column vector and U an n-dimensionaler
row vector, the action
    S = integral L(q,qdot,t) dt
is diffeomorphism invariant. As a consequence, the Noether energy
(the formal Hamiltonian constructed in the transition from a Lagrangian
to a Hamiltonian formulation) vanishes identically and has no physical
content. For one can bring an arbitrary Hamiltonian system
    xdot=H_p(p,x) , pdot=-H_x(p,x),
where H is the physically relevant energy, into the above form by
putting
     q^T = (x^T,p^T,s),
    U(q) = (p^T,0^T,-H(p,x)).

For a careful discussion see Section 4.3 of
    PJ Olver,
    Applications of Lie groups to differential equations,
    Springer, New York 1993.

It shows that every conservative theory can be rewritten in a form where
the Hamiltonian is not equal to the total energy.

Arnold Neumaier

The action S = integral_M L transforms under a diffeomorphism f: M ->
M to integral_M f^*(L), where f^* is the pull back on the Lagrangian 4-
form. Generally this means that f^*(L) = L, though one may allow
f^*(L) = L + d(phi) since d(phi) will integrate to 0 if M is compact
with no boundary. Hedges have to be made if M is not compact or has a
boundary (e.g. when considering a diffeomorphism invariant theory of a
submanifold M of space-time).

The Hamiltonian for such a theory will always be homogeneous to the
first degreei in the conjugate momenta. As a result, when the
equations of motion are substituted into the Hamiltonian it is 0. That
excludes it being the total energy (unless the total energy is 0 too).

Martin Ouwehand schrieb:

Yes, but only with a physically meaningless Hamiltonian.

Indeed, _any_ system of ordinary differential equations can be brought
into an artificial Lagrangian form, by first rewriting it in first
order form
      F(q,qdot)=0
doubling the degrees of freedom by introducing conjugate variables p,
and then considering the Lagrangian
      L(p,q)= p^T F(q,qdot).

Unfortunately, the Hamiltonian in such a formulation has
nothing to do with the physical energy
     E = (m q'^2 + k q^2)/2

The same holds for various other representations for the damped
harmonic oscillator found in the literature.

Lagrangians for the damped harmonic oscillator go back to
H. Bateman, Phys. Rev. 38, 815-819 (1931); the treatise
   P.M. Morse and H. Feshbach,
   Methods of Theoretical Physics
   MacGraw-Hill, Boston 1953
discusses the procedure in Chapter 3 in terms of 'mirror images'
= additional dynamical variables needed to absorb the missing energy,
and remarks on p 313:
   ''The introduction of the mirror image ... is probably too artificial
     a prcedure to expect to obtain much of physical significance from
     it.''
And indeed, the book doesn't make use of it anywhere.

Having a formal Lagrangian or Hamiltonian is no virtue in itself.
In particular, for a _quantum_ system, the Hamiltonian _must_ be the
energy. Playing around with alternative Lagrangians and Hamiltonians
may be amusing, but does not produce relevant physics.

Since dissipative equations (like the diffusion equation or the damped
harmonic oscillator) describe open systems (where energy is lost to an
unspecified environment), they cannot be described by a Schroedinger
equation.

Classically, dissipative systems are described by stochastic
differential equations (and their equivalent deterministic
Fokker-Planck equations) or master equations;
the diffusion equation is the particular case of a Fokker-Planck
equation for Brownian motion.

Quantum mechanically, dissipative systems are described by stochastic
Schroedinger equations or, corresponding to the Fokker-Planck level,
by quantum Liouville equations with Lindblad terms. This gives correct
physics in a dissipative environment.

Chris H. Fleming schrieb:

If you model the dissipaticve environment explicitly, you have a
bigger conservative system, not a dissipative system. Of course, the
conservative system is Hamiltonian, but it does not describe the
dissipative system alone. Already a simple damped harmonic oscillator
becomes a huge and unwieldy dynamical system which is no longer
equivalent to the damped harmonic oscillator, but includes unwanted
memory terms.

Wen you contract it to the degrees of freedoms of the original system,
you get an integro-differential equation with memory, which is no longer
described by a Hamiltonian or Lagrangian framework.

If you then remove the memory by employing the Markov approximation
you get again a differential equation, which defines the Lindblad (or,
classicallally, the Focker-Planck) dynamics. Aghain, this is no longer
described by a Hamiltonian or Lagrangian framework.

Arnold Neumaier