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The Cartesian equations of motion of our system take
the form
 |
(7.9) |
for
, where
are each equal to the mass of the
first particle,
are each equal to the mass of the
second particle, and so forth. Furthermore, the kinetic energy of the
system can be written
 |
(7.10) |
Because
, we can write
 |
(7.11) |
for
.
Hence, it follows that
. According to the
preceding equation,
 |
(7.12) |
where we are treating the
and the
as independent
variables.
Multiplying Equation (7.12) by
, and then differentiating
with respect to time, we obtain
 |
(7.13) |
Now,
 |
(7.14) |
Furthermore,
 |
(7.15) |
and
where use has been made of Equation (7.14). Thus, it follows
from Equations (7.13), (7.15), and (7.16) that
 |
(7.17) |
Let us take Equation (7.17), multiply by
, and then sum over all
.
We obtain
 |
(7.18) |
where use has been made of Equations (7.9) and (7.10). Thus, it follows from Equation (7.6) that
 |
(7.19) |
Finally, making use of Equation (7.8), we get
 |
(7.20) |
It is helpful to introduce a function
, called the Lagrangian, which
is defined as the difference between the kinetic and potential energies of the dynamical system under investigation:
 |
(7.21) |
Because the potential energy
is clearly independent of the
, it follows from Equation (7.20) that
 |
(7.22) |
for
. This equation is known as Lagrange's equation.
According to the preceding analysis, if we can express the kinetic and
potential energies of our dynamical system solely in terms of our generalized
coordinates and their time derivatives then we can immediately write
down the equations of motion of the system, expressed in terms
of the generalized coordinates, using Lagrange's equation, Equation (7.22).
Unfortunately, this scheme only works for conservative systems.
As an example, consider a particle of mass
moving in two dimensions in the central potential
. This is clearly a two-degree-of-freedom dynamical system.
As described in Section 4.4, the particle's instantaneous position
is most conveniently specified in terms of the plane polar
coordinates
and
. These are our two generalized coordinates.
According to Equation (4.13), the square of the particle's velocity
can be written
 |
(7.23) |
Hence, the Lagrangian of the system takes the form
 |
(7.24) |
Note that
Now, Lagrange's equation, Equation (7.22), yields the equations of motion,
Hence, we obtain
or
where
, and
is a constant. We recognize Equations (7.31) and (7.32) as the equations
that we derived in Chapter 4 for motion in a central potential.
The advantage of the Lagrangian method of deriving these equations is
that we avoid having to express the acceleration in terms of the generalized
coordinates
and
.
Next: Generalized momenta
Up: Lagrangian mechanics
Previous: Generalized forces
Richard Fitzpatrick
2016-03-31