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CHAPTER 14
HAMILTONIAN MECHANICS
14.1
Introduction
The hamiltonian equations of motion are of deep theoretical interest.
Having established
that, I am bound to say that I have not been able to think of a problem in classical
mechanics that I can solve more easily by hamiltonian methods than by newtonian or
lagrangian methods.
That is not to say that real problems cannot be solved by
hamiltonian methods.
What I have been looking for is a problem which I can solve easily
by hamiltonian methods but which is more difficult to solve by other methods.
So far, I
have not found one.
Having said that, doubt not that hamiltonian mechanics is of deep
theoretical significance.
Having expressed that mild degree of cynicism, let it be admitted that Hamilton theory –
or more particularly its extension the HamiltonJacobi equations
−
does have applications
is celestial mechanics, and of course hamiltonian operators play a major part in quantum
mechanics, although it is doubtful whether Sir William would have recognized his
authorship in that connection.
14.2
A Thermodynamics Analogy
Readers may have noticed from time to time – particularly in Chapter 9
−
that I have
perceived some connection between parts of classical mechanics and thermodynamics.
I
perceive such an analogy in developing hamiltonian dynamics.
Those who are familiar
with thermodynamics may also recognize the analogy.
Those who are not can skip this
section without seriously prejudicing their understanding of subsequent sections.
Please do not misunderstand:
The hamiltonian in mechanics is not at all the same thing
as enthalpy in thermodynamics, even though we use the same symbol,
H
.
Yet there are
similarities in the way we can introduce these concepts.
In thermodynamics we can describe the state of the system by its internal energy, defined
in such a way that when heat is supplied
to
a system and the system does external work,
the
increase
in internal energy of the system is equal to the heat supplied
to
the system
minus the work done
by
the system:
.
dV
P
dS
T
dU
−
=
14.2.1
From this point of view we are saying that the internal energy is a function of the entropy
and the volume:
)
,
(
V
S
U
U
=
14.2.2
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This note was uploaded on 10/25/2010 for the course MECHANIC Mechanic taught by Professor Monfered during the Fall '10 term at MIT.
 Fall '10
 monfered

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