This page contains an extremely simple but (hopefully!) informative
introduction to Lagrangian
mechanics.
"Lagrangian mechanics" is, fundamentally, just
another way of looking at Newtonian mechanics. Newtonian
mechanics, in a nutshell, says:
(1a)
I've
labeled them with their common names: the second and third laws.
The "first law", which I didn't show, can be derived from the
other two laws, if we assume all forces arise from interactions between
objects. The "Second law" as shown here assumes
the mass of a body is constant (unless it ejects a second body or
merges with a second body); that's true for Newtonian mechanics but not
in relativity theory. From these two (or three) laws one can
derive conservation of energy, momentum, and angular momentum.
The fundamental forces in the universe are all
conservative,
and many forces we deal with in everyday life are conservative as well
(friction being one obvious exception). A conservative force can
be represented as the gradient of a
potential; when an object
is being affected only by conservative forces, we can rewrite the
second law as:
(
1b)
or, in vector form, using
r as the object's position vector,
(
1c)
where φ is the potential function.
The "Lagrangian formulation" of Newtonian mechanics is based on
equation (
1c), which, again, is just an alternate
form of Newton's laws which is applicable in cases where the forces are
conservative. Lagrangian mechanics adds no new "semantics" 
it's just a
mathematical change, not a change in the physics.
So why use it? Because...
... Newtonian
mechanics has a problem: It works very nicely in Cartesian
coordinates, but it's difficult to switch to a different coordinate
system. Something as simple as changing to polar coordinates is
cumbersome; finding the equations of motion of a particle acting under
a "central force" in polar coordinates is tedious. The Lagrangian
formulation, in contrast, is
independent of the coordinates,
and the equations of motion for a nonCartesian coordinate system can
typically be found immediately using it. That's (most of) the
point in "Lagrangian mechanics".
Before we go on I should hasten
to add that the Lagrangian formulation also generalizes very nicely to
handle situations which are outside the realm of basic Newtonian
mechanics, including electromagnetism and relativity. But at the
moment we're primarily concerned with the coordinateindependent
feature.
The Lagrangian and the Principle of Least Action
Newton's
laws are relationships among vectors, which is why they get so messy
when we change coordinate systems. The Lagrangian formulation, on
the other hand, just uses scalars, and so coordinate transformations
tend to be much easier (which, as I said, is pretty much the whole
point). Given a
Lagrangian,
,
which is a function of the location in space and the velocity, we
define the
action:
(2)
Given particular starting and ending positions, the system follows a
path between the start and end points which
minimizes the
"action". Let's call the coordinates in space "
q_{1}"
through "
q_{n}". On the
shortest path page, we
showed visually that, on the path of least action, for each coordinate,
we must have:
(3) 
The Lagrangian,
, is
chosen
so that the the path of least action will be the path along which
Newton's laws are obeyed. That's it; fundamentally, it's all
there is to it. For Newtonian mechanics, the Lagrangian is chosen
to be:
(4) 
where
T is kinetic energy, (1/2)mv
^{2}, and
V
is potential energy, which we wrote as φ in equations (
1b)
and (
1c). For example, for gravity
considered in a
small region, we might use
V=mgh.
For gravity considered over a larger volume, we might use
V=
Gm
_{1}m
_{2}/
r.
We'll stick with the convention of using "
q_{i}" for the
spatial coordinates on the rest of this page. With that
convention,
and with the assumption that we're working in an
inertial frame with Cartesian coordinates, let's write out the kinetic
energy and some of its derivatives (watch out for those "flyspec" dots
over the q's  they almost shrank out of existence when LaTex typeset
the equations!):
(5)
For an ordinary potential function, which doesn't depend on velocity,
we can also write:
(6)
(Note that we wrote "F
_{i}" for the i
^{th} component of
the force vector there, which is a little different from what we were
using subscripts for back in (1a).)
Now, plugging the derivatives in (5) and (6) into equation
(4) for the Lagrangian, we see that:
(7)
And so, for Cartesian coordinates, using equation (
4)
for our Lagrangian, equation (
3)
is equivalent to Newton's second law. To obey the second law, a
particle must travel a path which minimizes the "action"; if we can
find that path, then we'll know what the particle is going to do.
The
nice thing is that the path of least action is the same, no matter what
coordinates you use. It's just the integral over time of a scalar
 that won't change, no matter how you choose to measure distances in
space. So, to find the equations of motion in an arbitrary
coordinate system
K, we just need to figure out what the
kinetic and potential energy must be expressed in the
K
coordinates. Then we write equation (
4),
take the derivatives used in equation (
3)  still
in
K coordinates  and we'll obtain the equations of
motion.
Friction and other NonConservative Forces
Friction
is not conservative, and so it doesn't fit neatly into the scheme we've
outlined so far. Certain nonconservative forces, including some cases of
"constant friction" (which doesn't depend on velocity), can be
integrated to obtain a "psuedopotential" function, often called a "
generalized potential", which can then be added to
V
in the Lagrangian formulation already given. But it's sometimes difficult
to do that. In particular, velocitydependent
friction is difficult to model that way.
For cases where we
can't find a generalized potential function, if we can determine a
function for the force in the coordinate system in which we're working,
we can simply add the force to equation (
3), which then becomes:
(
8)
where
Q_{i} is the nonconservative force (which is most often friction).
As we just mentioned, friction often has a
constant
component. That's what causes your car to "jerk" back slightly
when it finally comes to a stop if you don't lift your foot
up from the brake pedal as it finally stops. It's also why
the braking force
feels roughly the same no matter what speed
the car is traveling at when you press on the brake pedal. But
friction frequently also has a velocity dependent component; in viscous
drag that's the only component. Both components typically depend
linearly on the force between the surfaces, which often depends on the
mass of a sliding object, and both depend on the direction of the
object's motion. Overall, then, in many cases we can write the
frictional force as
(
9)
Keep in mind that the frictional force, as I just wrote it, is
coordinate dependent. If you change coordinates, you need to recompute it; there's nothing "automatic" about it.
A Simple Example: Gravity in 2 dimensions
We'll use
r and
θ
for the coordinates. To find the Lagrangian we need the kinetic
and potential energies. The straightline velocity of a particle in
polar coordinates is d
r/dt in the radial direction,
and
r(d
θ/dt) in the tangential
direction. The (Newtonian) gravitational potential is m
K/
r,
where
K=G
M (which I take to be positive), and
M
is the mass of the
gravitating body (e.g., the Earth or the Sun). So, after a small
amount of effort figuring out the kinetic and potential energy in polar
coordinates, we have:
(10)
From here on, it's purely mechanical. Taking the derivatives one
by one, we obtain:
(
11a)
(
11b)
(
11c)
(
11d)
(11e)
(
11f)
Note equation (
11b). That's the angular
momentum, and if we combine (
11a) and (
3),
we see that its time derivative is zero  so, it must be conserved.
Conservation of angular momentum (for this case!) just fell out
of the analysis "for free".
Plugging the derivatives from (
11a), (
11c), (
11d), and (
11f) into equation (
3), we find
the equations of motion:
(
12a)
(
12b)
The first term on the right side of (
12b) is
what's commonly called the "centrifugal force". We can see from
equation (
12b)
that we can have a circular orbit if the "centrifugal force" balances
the gravitational attraction; in that case there won't be any radial
acceleration. So, if we replace
r(d
θ/dt)
with
v in equation (
12b), we can
solve immediately for the velocity needed for a circular orbit at any
particular altitude:
(13)
Some Additional Examples
Here are a few additional simple examples which may help in seeing how
one can actually use the Lagrangian in a mechanics problem:
Rotating Polar Coordinates  A particle's motion in a rotating frame
Rotating Rectangular Coordinates  A rotating frame again, this time in rectangular coordinates
Two Masses, a Ramp, and a String  A simple example problem, done with and without friction
Other Lagrangians
It's
common to define the Lagrangian for a particular situation as being the
function whose path integral must be minimized to get the right
equations of motion. This is useful, again, because it makes it
so easy to switch to an arbitrary coordinate system, where the problem
may be simpler the but the equations of motion would otherwise be
difficult to find.
In electromagnetism, if we set
c=1, then the Lagrangian of a
particle is given as:
(14)
where
A is the vector magnetic potential;
.
Minimizing the action then produces the right motion according to the
Lorentz force law, f = q(E+vxB). Note that in this case the
"potential" actually depends on the velocity as well as the position,
so it's not really a Newtonian "potential" function at all! But
that doesn't matter; what matters is that, in Cartesian coordinates,
the least action principle works with this Lagrangian function.
Since it's a scalar, once we know that, we also know that we can
use
the same approach to attack the problem using any coordinate system.
In relativity, if we set
c=1, we can define the Lagrangian of a
free particle to be:
(15)
The path of least action then becomes a worldline which follows a
geodesic.
A few words about Hamiltonian mechanics
Equation
(
3) is a second order differential equation.
The Hamiltonian
formulation, which is a simple transform of the Lagrangian formulation,
reduces it to a system of first order equations, which can be easier to
solve. It's heavily used in quantum mechanics.
The Hamiltonian is the "Legendre transform" of the Lagrangian,
but we could just as well say the Lagrangian is (part of) what we get
when we integrate the Hamiltonian by parts  or we could say we just
use the product rule (the "Leibnitz rule") to transform between them.
Suppose we have a function
u(v), where
u
is monotonic increasing. Then it's invertible; in that case we
can talk sensibly about the function
v(u). See
figure 1. We've shown a curve representing
either
u(v) or
v(u). The blue area
"under" the curve is the integral of
u as a function of
v,
which we've called
f. The pink area to the left of
the curve, which we've labeled
g, is the integral of
v
as a function of
u.
figure 1  The
transformation between f and g
In
figure 1, consider the function
uv.
Its derivative, written in differential form, is:
(h.1)
Integrating both sides:
(h.2)
This is obvious from
figure 1. Equally
obvious from figure 1 is:
(
h.3)
That's integration by parts, of course, which is just a different way
of
looking at the product rule. From the definitions of
f,
g,
v, and
u as illustrated
in figure 1, we also have:
(h.4)
Now, let's replace
f with
, and replace
g
with
. We'll
define
such that it's a
function of the
q_{i} (the position) and the
p_{i}
(the momenta). In contrast,
is a function of
position and
velocity. To do this, in the above
discussion we let
Figure 2
 The transformation between and :

(h.5)
As we see in
figure 2, the transformation in
equation (
h.3) then becomes:
(
h.6)
Just as with
u and
v, the roles of d
q_{i}/dt
and
p_{i} are swapped vis a vis
and
. But the
dependency of
on position is
the same that of
(save that it is
negated): we didn't transform the position variables (but did flip the
sign). So we have:
(h.7) 
Those
are the Hamiltonian equations of motion. Instead of a single
secondorder equation for each coordinate, we have two firstorder
equations, which may be easier to solve.
The derivation I gave
above was hardly airtight. However, it's easy to verify
that in Newtonian mechanics using Cartesian coordinates, the
Hamiltonian we obtain from equation (
h.6) reduces
to:
(h.8) 
which is just the total energy. In that case it's also easy to
verify equations (
h.7)
directly.
As with the Lagrangian formulation, however, much of the value of
the Hamiltonian formulation lies in the fact that equations (h.7) are
true regardless of the coordinates we're using. Also keep in mind
that equation (h.8) is only necessarily true when the Lagrangian chosen
is the "pure Newtonian"
TV. For
other Lagrangians, the Hamiltonian won't
necessarily be the total energy of the system.
Page created on 11/13/06; last updated, with additions
and different equation numbers, on 12/1/06. Corrected equation
numbers, 10/4/07.