2
0. Some Aspects of Theoretical Mechanics
I
or in the local coordinates as
t i—• qi(t) with qi(t0) = g°, i = 1,..., n.
Here physical principles must be found which allow one to give the curve as a
solution to a differential equation. The starting point for this determination
is the classical mechanical principle of least action. For this it is assumed
that the system has a Lagrange function L of the form
L = L(q,q,t),
which is gotten as the difference of the kinetic and the potential energies,
L = -Eton
_
^pot
which is also written as
L = T - V.
The principle of least action now says that the change in the system
proceeds so that the curve 7 minimizes the path integral
ti
Ldt.
to
The variational calculus now says (see
COURANT-HILBERT
[CH], p. 170)
that for the minimum curve 7 q(t) the system satisfies the Euler-Lagrange
equations
m d dL dL
dt dqi dqi
This can be seen as a system of ordinary differential equations in a
2n-dimensional space TQ with local coordinates
9i 9n? Qi, An
(which can be understood as the tangent bundle over the configuration space
Q (see Section A.3)). The desired curve 7 on Q is the projection of the
solution curve 7 of (1) onto TQ.
0.2. Hamilton's equations
Classical mechanics now takes the following formulation: for a given La-
grange function L the coordinates position and velocity, (g, g), are replaced
by the coordinates position and momentum (g, p) made possible by the
transformation
dL .
dqi
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