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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Balance of angular momentum | 2/2 | https://en.wikipedia.org/wiki/Balance_of_angular_momentum | reference | science, encyclopedia | 2026-05-05T10:01:14.393868+00:00 | kb-cron |
must hold. This is actually the statement of the equality of corresponding shear stresses in the xy plane.
== Cosserat Continuum == In addition to the torque-free classical continuum with a symmetric stress tensor, cosserat continua (polar continua) that are not torque-free have also been defined. One application of such a continuum is the theory of shells. Cosserat continua are not only capable to transport a momentum flux but also an angular momentum flux. Therefore, there also may be sources of momentum and angular momentum inside the body. Here the Boltzmann Axiom does not apply and the stress tensor may be skew-symmetric. If these fluxes are treated as usual in continuum mechanics, field equations arise in which the skew-symmetric part of the stress tensor has no energetic significance. The balance of angular momentum becomes independent of the balance of energy and is used to determine the skew-symmetric part of the stress tensor. American mathematician Clifford Truesdell saw in this the "true basic sense of Euler's second law".
== Area rule ==
The area rule is a corollary of the angular momentum law in the form: The resulting moment is equal to the product of twice the mass and the time derivative of the areal velocity. It refers to the ray
r
→
{\displaystyle {\vec {r}}}
to a point mass with mass m. This has the angular momentum with the velocity
r
→
˙
{\displaystyle {\dot {\vec {r}}}}
and the momentum
p
→
=
m
r
→
˙
{\displaystyle {\vec {p}}=m{\dot {\vec {r}}}}
L
→
=
r
→
×
p
→
=
m
r
→
×
r
→
˙
=
m
r
→
×
d
r
→
/
d
t
{\displaystyle {\vec {L}}={\vec {r}}\times {\vec {p}}=m{\vec {r}}\times {\dot {\vec {r}}}=m{\vec {r}}\times \mathrm {d} {\vec {r}}/\mathrm {d} t}
. In the infinitesimal time dt the trajectory sweeps over a triangle whose content is
d
A
→
=
1
2
r
→
×
d
r
→
{\displaystyle \mathrm {d} {\vec {A}}={\tfrac {1}{2}}{\vec {r}}\times \mathrm {d} {\vec {r}}}
, see image, areal velocity and cross product "×". This is how it turns out:
L
→
=
m
r
→
×
d
r
→
/
d
t
=
2
m
d
A
→
/
d
t
=
2
m
A
→
˙
{\displaystyle {\vec {L}}=m{\vec {r}}\times \mathrm {d} {\vec {r}}/\mathrm {d} t=2m\,\mathrm {d} {\vec {A}}/\mathrm {d} t=2m{\dot {\vec {A}}}}
. With Euler's second law this becomes:
M
→
=
L
→
˙
=
2
m
A
→
¨
{\displaystyle {\vec {M}}={\dot {\vec {L}}}=2m{\ddot {\vec {A}}}}
. The special case of plane, moment-free central force motion is treated by Kepler's second law, also known as the area rule.
== See also == Conservation of angular momentum
== References ==