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Subsections
In Lagrangian mechanics we start by writing down the Lagrangian of the system
where T is the kinetic energy and U is the potential energy. Both
are expressed in terms of coordinates (q,) where
q is the position vector and
is the
velocity vector.
An example is the physical pendulum (see Figure 1).
Figure 1:
The configuration space of the pendulum

The natural configuration space of the pendulum is the circle. The
natural coordinate on the configuration space is the angle
. If the mass of the ball is m and the length of the rod is
l then we have
T

=

m(l)^{2}
 (2) 
U

=

 mglcos
 (3) 
Thus, the Lagrangian in coordinates
(,) is

L = m(l)^{2} + mglcos .
 (4)

In Lagrangian mechanics the equations of motion are written in the
following universal form:
For example, for the pendulum we have:

=

ml^{ 2},
 (6) 

=

 mglsin .
 (7) 
Thus, the equations of motion are written as

(ml^{ 2}) =  mglsin .
 (8)

This equation can be written as second order equation

ml^{ 2} =  mglsin
 (9)

or in the traditional way
We should emphasize that has dual meaning. It is both a
coordinate and the derivative of the position. This traditional abuse
of notation should be resolved in favor of one of these
interpretations in every particular situation.
In Newtonian mechanics we represent the equations of motion in the
form of the second Newton's law:

m = f (q,t)
 (11)

where f (q,t) is the force applied to the particle.
This equation is identical to the equation obtained from Lagrangian
representation if f (q,t) is a conservative field, i.e. it has a
potential. A potential is a function U(q,t) such that

f (q,t) =  .
 (12)

Indeed, the Lagrangian can be written as

L = m()^{2}  U(q,t).
 (13)

According to 5 the equations
of motion reduce to 11.
Next: 3 The variational principle
Up: Lagrangian and Hamiltonian mechanics
Previous: 1 What does this
Marek Rychlik
9/2/1997