This sums it up:
The
equations of motion are obtained by means of an
action principle, written as:
where the
action,
, is a
functional of the dependent variables
with their derivatives and
s itself
and where
denotes the
set of
n independent variables of the system, indexed by
The equations of motion obtained from this
functional derivative are the
Euler–Lagrange equations of this action. For example, in the
classical mechanics of particles, the only independent variable is time,
t. So the Euler-Lagrange equations are
Dynamical systems whose equations of motion are obtainable by means of an action principle on a suitably chosen Lagrangian are known as
Lagrangian dynamical systems. Examples of Lagrangian dynamical systems range from the classical version of the
Standard Model, to
Newton's equations, to purely mathematical problems such as
geodesic equations and
Plateau's problem.
[edit] An example from classical mechanics[edit] In the rectangular coordinate systemSuppose we have a
three-dimensional space and the Lagrangian
.
Then, the Euler–Lagrange equation is:
where
i = 1,2,3.
The derivation yields:
The Euler–Lagrange equations can therefore be written as:
where the time derivative is written conventionally as a dot above the quantity being differentiated, and
is the
del operator.
Using this result, it can easily be shown that the Lagrangian approach is equivalent to the Newtonian one.
If the force is written in terms of the potential
; the resulting equation is
, which is exactly the same equation as in a Newtonian approach for a constant mass object.
A very similar deduction gives us the expression
, which is Newton's Second Law in its general form.
[edit] In the spherical coordinate systemSuppose we have a three-dimensional space using
spherical coordinates r,θ,φ with the Lagrangian
Then the Euler–Lagrange equations are:
Here the set of parameters
si is just the time
t, and the dynamical variables φ
i(
s) are the trajectories
of the particle.
Despite the use of standard variables such as
x, the Lagrangian allows the use of any coordinates, which do not need to be orthogonal. These are "
generalized coordinates".
[edit] Lagrangian of a test particleA test particle is a particle whose mass and charge are assumed to be so small that its effect on external system is insignificant. It is often a hypothetical simplified point particle with no properties other than mass and charge. Real particles like electrons and up-quarks are more complex and have additional terms in their Lagrangians.
[edit] Classical test particle with Newtonian gravitySuppose we are given a particle with mass
kilograms, and position
meters in a Newtonian gravitation field with potential
joules per kilogram. The particle's world line is parameterized by time
seconds. The particle's kinetic energy is:
and the particle's gravitational potential energy is:
Then its Lagrangian is
joules where
Varying
in the integral (equivalent to the Euler–Lagrange differential equation), we get
Integrate the first term by parts and discard the total integral. Then divide out the variation to get
and thus
is the equation of motion — two different expressions for the force.
[edit] Special relativistic test particle with electromagnetismIn special relativity, the form of the term which gives rise to the derivative of the momentum must be changed; it is no longer the kinetic energy. It becomes:
(In special relativity, the energy of a free test particle is
)
where
meters per second is the
speed of light in vacuum,
seconds is the proper time (i.e. time measured by a clock moving with the particle) and
The second term in the series is just the classical kinetic energy. Suppose the particle has electrical charge
coulombs and is in an electromagnetic field with
scalar potential 
volts (a volt is a joule per coulomb) and
vector potential 
volt seconds per meter. The Lagrangian of a special relativistic test particle in an electromagnetic field is:
Varying this with respect to
, we get
which is
which is the equation for the
Lorentz force where
[edit] General relativistic test particleIn
general relativity, the first term generalizes (includes) both the classical kinetic energy and interaction with the Newtonian gravitational potential. It becomes:
The Lagrangian of a general relativistic test particle in an electromagnetic field is:
If the four space-time coordinates
are given in arbitrary units (i.e. unit-less), then
meters squared is the rank 2 symmetric
metric tensor which is also the gravitational potential. Also,
volt seconds is the electromagnetic 4-vector potential. Notice that a factor of
c has been absorbed into the square root because it is the equivalent of
Note that this notion has been directly generalized from special relativity
[edit] Lagrangians and Lagrangian densities in field theoryThe time integral of the Lagrangian is called the action denoted by
S.
In
field theory, a distinction is occasionally made between the Lagrangian
L, of which the action is the time integral:
and the
Lagrangian density 
, which one integrates over all
space-time to get the action:
The Lagrangian is then the spatial integral of the Lagrangian density. However,
is also frequently simply called the Lagrangian, especially in modern use; it is far more useful in
relativistic theories since it is a
locally defined,
Lorentz scalar field. Both definitions of the Lagrangian can be seen as special cases of the general form, depending on whether the spatial variable
is incorporated into the index
i or the parameters
s in
.
Quantum field theories in
particle physics, such as
quantum electrodynamics, are usually described in terms of
, and the terms in this form of the Lagrangian translate quickly to the rules used in evaluating
Feynman diagrams.
