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The trivial case of the angular momentum of a body in an orbit is given by = where is the mass of the orbiting object, is the orbit's frequency and is the orbit's radius.. The angular momentum of a uniform rigid sphere rotating around its axis, instead, is given by = where is the sphere's mass, is the frequency of rotation and is the sphere's radius.
The law of conservation of angular momentum states that in the absence of applied torques, the angular momentum vector is conserved in an inertial reference frame, so =. The angular momentum vector L {\displaystyle \mathbf {L} } can be expressed in terms of the moment of inertia tensor I {\displaystyle \mathbf {I} } and the angular velocity ...
In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of mass-energy, conservation of linear momentum, conservation of angular momentum, and conservation of electric charge.
As another example, if a physical process exhibits the same outcomes regardless of place or time, then its Lagrangian is symmetric under continuous translations in space and time respectively: by Noether's theorem, these symmetries account for the conservation laws of linear momentum and energy within this system, respectively.
A form of Newton's second law, that force is the rate of change of momentum, also holds, as does the conservation of momentum. However, the definition of momentum is modified. Among the consequences of this is the fact that the more quickly a body moves, the harder it is to accelerate, and so, no matter how much force is applied, a body cannot ...
Euler's second law states that the rate of change of angular momentum L about a point that is fixed in an inertial reference frame (often the center of mass of the body), is equal to the sum of the external moments of force acting on that body M about that point: [1] [4] [5]
For reference and background, two closely related forms of angular momentum are given. In classical mechanics, the orbital angular momentum of a particle with instantaneous three-dimensional position vector x = (x, y, z) and momentum vector p = (p x, p y, p z), is defined as the axial vector = which has three components, that are systematically given by cyclic permutations of Cartesian ...
For example, every central force is symmetric under the rotation group SO(3), leading to the conservation of the angular momentum L. Classically, an overall rotation of the system does not affect the energy of an orbit; quantum mechanically, rotations mix the spherical harmonics of the same quantum number ℓ without changing the energy.