thomas malloy wrote:
Jones Beene wrote:
Attn: Perpmos and other eccentric swingers.
SAM’s simplicity of construction and operation is
evident in the Hamiltonian, and it can be modeled
easily
Ah, Jones, you've raised a matter that I've been wondering about. What
are Hamiltonians?
In this context, we're talking about mechanics, and in the context of
mechanics the concept of Hamiltonians is simpler than it seems. Like
the Lagrangian formulation, the Hamiltonian formulation of mechanics is
just another way to package up Newton's equations. Also like Lagrangian
mechanics, Hamiltonian mechanics is normally only used in cases where
the system is conservative: if the system isn't fully described by its
current kinetic energy, along with forces which can be described as the
derivative of a potential, then you can't easily describe the system's
behavior with a Lagrangian or a Hamiltonian. A realistic model of
friction, in particular, is a major pain in the neck to incorporate into
the Lagrangian.
In general, the Lagrangian is the kinetic energy minus the potential
energy: L = T - V. That is not the way it is *defined*, but it's what
it usually works out to be equal to. (BTW "T" typically means kinetic
energy, and "V" typically is used to mean potential energy, and don't
ask me how the letters T and V got in here.)
The Hamiltonian is generally the total energy of the system:
H = T + V
As with the Lagrangian, the Hamiltonian is not *defined* that way, but
it typically works out that way.
The actual *definition* of the Lagrangian more obscure. Define the
'action' as the line integral of the Lagrangian along the system's path
through phase space. Then the conditions on the chosen path that
guarantee that the 'action' is *minimized* also assure that Newton's
equations will be obeyed along the path. The Lagrangian is chosen to
make this last statement true. In other words, we pick the Lagrangian
such that choosing a path along which its integral is minimized will
produce the "correct" path, which is the one the system will actually
follow.
The *definition* of the Hamiltonian is, likewise, more obscure than just
saying it's the total energy. The Hamiltonian is typically *defined* as
the Legendre transform of the Lagrangian, if the system has a Lagrangian.
One major reason for using both the Lagrangian and Hamiltonian
formulations of mechanics is that they are both independent of the
coordinate system chosen. Consequently, restating a problem in
different coordinates is typically very easy using Lagrangian mechanics,
while it is often very hard using straight-up Newtonian mechanics.
Since problems are often far easier to solve in one coordinate system
than another, this is a significant advantage.
Here's a little more on the subject, which might (or might not) help:
http://physicsinsights.org/lagrange_1.html
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