1
CHAPTER 13
LAGRANGIAN MECHANICS
13.1
Introduction
The usual way of using newtonian mechanics to solve a problem in dynamics is first of
all to draw a large
, clear
diagram of the system, using a ruler
and a compass
.
Then mark
in the forces on the various parts of the system with red arrows and the accelerations of
the various parts with green arrows.
Then apply the equation
F
=
ma
in two different
directions if it is a two-dimensional problem or in three directions if it is a three-
dimensional problem, or
θ
=
τ
&
&
I
if torques are involved.
More correctly, if a mass or a
moment of inertia is not constant, the equations are
p
F
&
=
and
.
L
&
=
τ
In any case, we
arrive at one or more
equations of motion
, which are differential equations which we
integrate with respect to space or time to find the desired solution.
Most of us will have
done many, many problems of that sort.
Sometimes it is not all that easy to find the equations of motion as described above.
There is an alternative approach known as lagrangian mechanics which enables us to find
the equations of motion when the newtonian method is proving difficult.
In lagrangian
mechanics we start, as usual, by drawing a large
, clear
diagram of the system, using a
ruler
and a compass
.
But, rather than drawing the forces and accelerations with red and
green arrows, we draw the
velocity
vectors (including angular velocities) with blue
arrows, and, from these we write down the
kinetic energy
of the system.
If the forces are
conservative
forces (gravity, springs and stretched strings), we write down also the
potential energy
.
That done, the next step is to write down the
lagrangian equations of
motion
for each coordinate.
These equations involve the kinetic and potential energies,
and are a little bit more involved than
F
=
ma
, though they do arrive at the same results.
I shall derive the lagrangian equations of motion, and while I am doing so, you will think
that the going is very heavy, and you will be discouraged.
At the end of the derivation
you will see that the lagrangian equations of motion are indeed rather more involved than
F
=
ma
, and you will begin to despair – but do not do so!
In a very short time after that
you will be able to solve difficult problems in mechanics that you would not be able to
start using the familiar newtonian methods, and the speed at which you do so will be
limited solely by the speed at which you can write.
Indeed, you scarcely have to stop and
think.
You know straight away what you have to do.
Draw the diagram.
Mark the
velocity vectors.
Write down expressions for the kinetic and potential energies, and
apply the lagrangian equations.
It is automatic, fast, and enjoyable.
Incidentally, when Lagrange first published his great work
La méchanique analytique
(the modern French spelling would be
mécanique
), he pointed out with some pride in his
introduction that there were no drawings or diagrams in the book – because all of
mechanics could be done
analytically
– i.e. with algebra and calculus.
Not all of us,
however, are as gifted as Lagrange, and we cannot omit the first and very important step