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Field Theory Up: ENERGY-MOMENTUM TENSOR Previous: ENERGY-MOMENTUM TENSOR
Newton's second law of motion is
|
 |
(139) |
or in component form (for each
component Fi)
|
 |
(140) |
where
(with qi being the generalized
position coordinate) so that
. If
then
. For conservative forces
where V is the scalar potential. Rewriting
Newton's law we have
|
 |
(141) |
Let us define the Lagrangian
where T is the kinetic energy. In freshman physics
and V=V(qi) such
as the harmonic oscillator
. That is in freshman physics T is a function only
of velocity
and V is a function only of position
qi. Thus
. It follows that
and
. Thus Newton's law is
with the canonical momentum [1] defined as
|
 |
(142) |
The second equation of (4.4) is known as the
Euler-Lagrange equations of motion and serves as an alternative formulation of
mechanics [1].
It is usually written
|
 |
(143) |
or just
|
 |
(144) |
We have obtained the
Euler-Lagrange equations using simple arguments. A more rigorous derivation is
based on the calculus of variations [1] as discussed
in Section 7.3.
We now introduce the Hamiltonian H defined as a function of p
and q as
|
 |
(145) |
For the simple case
and
we have
so that
and
so that
which is the total energy. Hamilton's equations of motion
immediately follow from (4.8) as
|
 |
(146) |
because
and
so that from (4.4)
|
 |
(147) |
Next: Classical
Field Theory Up: ENERGY-MOMENTUM TENSOR Previous: ENERGY-MOMENTUM TENSOR
John Norbury
12/9/1997