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Newton's second law of motion is
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(3.34) |
or in component form (for each component Fi)
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(3.35) |
where
(with
qi being the generalized position coordinate) so that
.
If 
then
.
For conservative forces
where
V is the potential energy. Rewriting Newton's law it become:
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(3.36) |
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
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with the canonical momentum defined as
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(3.37) |
The second equation of motion known as the Lagrange-Euler equations. It is usually written
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(3.38) |
or just
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(3.39) |
The Lagrange-Euler equations obtained using simple arguments. A more rigorous derivation is based on the calculus of variations.
