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We can specify the instantaneous configuration of a conservative dynamical system
with
degrees of freedom in terms of
independent generalized coordinates,
, for
. Let
and
represent the kinetic and potential energies of the
system, respectively, expressed in terms of these generalized coordinates.
The Lagrangian of the system is defined

(B.65) 
Finally, the
Lagrangian equations of motion of the system take the form

(B.66) 
for
.
Note that the previous equations of motion have exactly the same mathematical form as the EulerLagrange equations, (B.62). Indeed, it is clear
that the
Lagrangian equations of motion, (B.66),
can all be derived from a single equation: namely,

(B.67) 
In other words, the motion of the system in a given time interval is such as to maximize or
minimize the time integral of the Lagrangian, which is known as the
action integral. Thus, the laws of Newtonian dynamics can be summarized in a
single statement:
The motion of a dynamical
system in a given time interval is such as to maximize or minimize the action integral.
(In practice, the action integral is almost always minimized.) This statement is known as Hamilton's principle, and was first formulated in 1834 by the Irish
mathematician William Hamilton.
Next: Hamilton's Equations
Up: Classical Mechanics
Previous: MultiFunction Variation
Richard Fitzpatrick
20160125