In physics, two objects are said to be coupled
when they are interacting with each other.
In classical mechanics, coupling is a connection
between two oscillating systems, such as pendulums
connected by a string. The connection affects
the oscillatory pattern of both objects. In
particle physics, two particles are coupled
if they are connected by one of the four fundamental
forces.
== Wave mechanics ==
=== Coupled harmonic oscillator ===
If two waves are able to transmit energy to
each other, then these waves are said to be
"coupled." This normally occurs when the waves
share a common component. An example of this
is two pendulums connected by a spring. If
the pendulums are identical, then their equations
of motion are given by
m
x
¨
=
−
m
g
x
l
1
−
k
(
x
−
y
)
{\displaystyle m{\ddot {x}}=-mg{\frac {x}{l_{1}}}-k(x-y)}
m
y
¨
=
−
m
g
y
l
2
−
k
(
x
−
y
)
{\displaystyle m{\ddot {y}}=-mg{\frac {y}{l_{2}}}-k(x-y)}
These equations represent the simple harmonic
motion of the pendulum with an added coupling
factor of the spring. This behavior is also
seen in certain molecules (such as CO2 and
H2O), wherein two of the atoms will vibrate
around a central one in a similar manner.
=== Coupled LC circuits ===
In LC circuits, charge oscillates between
the capacitor and the inductor and can therefore
be modeled as a simple harmonic oscillator.
When the magnetic flux from one inductor is
able to affect the inductance of an inductor
in an unconnected LC circuit, the circuits
are said to be coupled. The coefficient of
coupling k defines how closely the two circuits
are coupled and is given by the equation
M
L
p
L
s
=
k
{\displaystyle {\frac {M}{\sqrt {L_{p}L_{s}}}}=k}
Where M is the mutual inductance of the circuits
and Lp and Ls are the inductances of the primary
and secondary circuits, respectively. If the
flux lines of the primary inductor thread
every line of the secondary one, then the
coefficient of coupling is 1 and
M
=
L
p
L
s
{\displaystyle M={\sqrt {L_{p}L_{s}}}}
In practice, however, there is often leakage,
so most systems are not perfectly coupled.
== Chemistry ==
=== Spin-spin coupling ===
Spin-spin coupling occurs when the magnetic
field of one atom affects the magnetic field
of another nearby atom. This is very common
in NMR imaging. If the atoms are not coupled,
then there will be two individual peaks, known
as a doublet, representing the individual
atoms. If coupling is present, then there
will be a triplet, one larger peak with two
smaller ones to either side. This occurs due
to the spins of the individual atoms oscillating
in tandem.
== Astrophysics ==
Objects in space which are coupled to each
other are under the mutual influence of each
other's gravity. For instance, the Earth is
coupled to both the sun and the moon, as it
is under the gravitational influence of both.
Common in space are binary systems, two objects
gravitationally coupled to each other. Examples
of this are binary stars which circle each
other. Multiple objects may also be coupled
to each other simultaneously, such as with
globular clusters and galaxy groups. When
smaller particles, such as dust, which are
coupled together over time accumulate into
much larger objects, accretion is occurring.
This is the major process by which stars and
planets form.
== Plasma ==
The coupling constant of a plasma is given
by the ratio of its average Coulomb-interaction
energy to its average kinetic energy—or
how strongly the electric force of each atom
holds the plasma together . Plasmas can therefore
be categorized into weakly- and strongly-coupled
plasmas depending upon the value of this ratio.
Many of the typical classical plasmas, such
as the plasma in the solar corona, are weakly
coupled, while the plasma 
in a white dwarf star is an example of a strongly
coupled plasma.
== Quantum mechanics ==
Two coupled quantum systems can be modeled
by a Hamiltonian of the form
H
^
=
H
^
a
+
H
^
b
+
V
^
a
b
{\displaystyle {\hat {H}}={\hat {H}}_{a}+{\hat
{H}}_{b}+{\hat {V}}_{ab}}
which is the addition of the two Hamiltonians
in isolation with an added interaction factor.
In most simple systems,
H
^
a
{\displaystyle {\hat {H}}_{a}}
and
H
^
b
{\displaystyle {\hat {H}}_{b}}
can be solved exactly while
V
^
a
b
{\displaystyle {\hat {V}}_{ab}}
can be solved through perturbation theory.
If the two systems have similar total energy,
then the system may undergo Rabi oscillation.
=== Angular momentum coupling ===
When angular momenta from two separate sources
interact with each other, they are said to
be coupled. For example, two electrons orbiting
around the same nucleus may have coupled angular
momenta. Due to the conservation of angular
momentum and the nature of the angular momentum
operator, the total angular momentum is always
the sum of the individual angular momenta
of the electrons, or
J
=
J
1
+
J
2
{\displaystyle \mathbf {J} =\mathbf {J_{1}}
+\mathbf {J_{2}} }
Spin-Orbit interaction (also known as spin-orbit
coupling) is a special case of angular momentum
coupling. Specifically, it is the interaction
between the intrinsic spin of a particle,
S, and its orbital angular momentum, L. As
they are both forms of angular momentum, they
must be conserved. Even if energy is transferred
between the two, the total angular momentum,
J, of the system must be constant,
J
=
L
+
S
{\displaystyle \mathbf {J} =\mathbf {L} +\mathbf
{S} }
.
== Particle physics and quantum field theory
==
Particles 
which interact with each other are said to
be coupled. This interaction is caused by
one of the fundamental forces, whose strengths
are usually given by a dimensionless coupling
constant. In quantum electrodynamics, this
value is known as the fine-structure constant
α, approximately equal to 1/137. For quantum
chromodynamics, the constant changes with
respect to the distance between the particles.
This phenomenon is known as asymptotic freedom.
Forces which have a coupling constant greater
than 1 are said to be "strongly coupled" while
those with constants less than one are said
to be "weakly coupled
