Rotational symmetry Rotational symmetry In physics the angular momentum of an object with respect to a reference point is a measure for the extent to which, and the direction in which, the object rotates about the reference point. ...more on Wikipedia about "Angular momentum"
In quantum mechanics, the orbital and spin angular momentum of bodies can interact in angular momentum coupling. These interactions in atoms are used in spectroscopy. ...more on Wikipedia about "Angular momentum coupling"
In atomic physics, an angular quantum momentum number is any of the quantum numbers that quantize an angular momentum. They express an angular momentum as an integer multiple of (the reduced Planck's constant divided by two). ...more on Wikipedia about "Angular momentum quantum number"
The angular velocity of a point particle or rigid body describes the rate at which its orientation changes. It is analogous to translational velocity, and is defined in terms of the derivative of orientation with respect to time, just as translational velocity is the derivative of displacement with respect to time. It is customary to introduce the concept of velocity by first defining average velocity as displacement divided by time. There the analogy with angular velocity is less useful: for example, if a body is rotating at a constant angular velocity of one revolution per minute, then over a one-minute period the 'average angular velocity' of the body is zero, because the orientation is exactly the same at the beginning of the time period as it is at the end. ...more on Wikipedia about "Angular velocity"
The Azimuthal quantum number (or orbital angular momentum quantum number) symbolized as l is a quantum number for an atomic orbital which determines its orbital angular momentum. The azimuthal quantum number is the second of a set of quantum numbers which describe the unique quantum state of an electron and is designated by the letter l. ...more on Wikipedia about "Azimuthal quantum number"
> In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a 2-level quantum mechanical system. Alternatively, it is the pure state space of a 1 qubit quantum register. The Bloch sphere is actually geometrically a sphere and the correspondence between elements of the Bloch sphere and pure states can be explicitly given. ...more on Wikipedia about "Bloch sphere"
In mathematics, the special orthogonal group in three dimensions, otherwise known as the rotation group SO(3), is a naturally occurring example of a manifold. The various charts on SO(3) set up rival coordinate systems: in this case there cannot be said to be a preferred set of parameters describing a rotation. There are three degrees of freedom, so that the dimension of SO(3) is three. In numerous applications one or other coordinate system is used, and the question arises how to convert from a given system to another. ...more on Wikipedia about "Charts on SO(3)" I wish I had a www.shortopedia.com.
In physics, the Clebsch-Gordan coefficients are sets of numbers that arise in calculations involving addition of angular momentum under the laws of quantum mechanics. ...more on Wikipedia about "Clebsch-Gordan coefficients"
(Conversion between quaternions and Euler angles) The orthogonal matrix corresponding to a rotation by the unit quaternion q is given by ...more on Wikipedia about "Conversion between quaternions and Euler angles"
In linear algebra and geometry, a coordinate rotation is a type of transformation from one system of coordinates to another system of coordinates such that distance between any two points remains invariant under the transformation. In other words, a rotation is a type of isometry – note however that there are isometries other than rotations, such as translations, reflections, and glide reflections. ...more on Wikipedia about "Coordinate rotation"
Euler angles are the classical way of representing rotations in 3-dimensional Euclidean space, named after Leonhard Euler. ...more on Wikipedia about "Euler angles"
In physics, Euler's equations govern the rotation of a rigid body. We choose the body fixed axes to be principal axes of inertia. This will make the calculations easier, since we can now split the change in angular momentum into a component that describes the change of the size of and another component that compensates for the change in direction of . ...more on Wikipedia about "Euler's equations"
In mathematics, Euler-Rodrigues parameters, also called just Euler parameters, are ...more on Wikipedia about "Euler-Rodrigues parameters"
Flight dynamics is the study of orientation of air and space vehicles and how to control the critical flight parameters, typically named pitch, roll and yaw. ...more on Wikipedia about "Flight dynamics"
In gyroscopic devices controlled by Euler mechanics or Euler angles, gimbal lock is caused by the alignment of two of the three gimbals together so that one of the rotation references ( pitch/yaw/roll, often yaw) is cancelled. This would require a reset of the gimbals using an outside reference. It may also be described as the situation when all three gyros hit the limits of their ability to move within the sensing mechanism - they hit hard stops and stop moving around. ...more on Wikipedia about "Gimbal lock"
In atomic physics, the magnetic quantum number is the third of a set of quantum numbers which describe the unique quantum state of an electron and is designated by the letter m. The magnetic quantum number denotes the energy levels available within a subshell. ...more on Wikipedia about "Magnetic quantum number"
The Pauli matrices are a set of 2 × 2 complex Hermitian matrices developed by Wolfgang Pauli. They are: ...more on Wikipedia about "Pauli matrices"
This result can easily be proven by splitting the vectors into components and applying the product rule. Now using the definitions of velocity v = dr/dt, acceleration a = dv/dt and linear momentum p = mv, we can see that: ...more on Wikipedia about "Proof of angular momentum"
In mathematics, the quaternions are a non-commutative extension of the complex numbers. They were first described by the Irish mathematician Sir William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. At first, the quaternions were regarded as pathological, because they disobeyed the commutative law ab = ba. Although they have been superseded in most applications by vectors, they still find uses in both theoretical and applied mathematics, in particular for calculations involving three-dimensional rotations. ...more on Wikipedia about "Quaternion"
5. Math. The quotient of two vectors ... Such is the view of the inventor, Sir Wm. Rowan Hamilton, and his disciple, Prof. P. G. Tait; but authorities are not yet quite agreed as to what a quaternion is or ought to be....more on Wikipedia about "Quaternions and spatial rotation"
Then pass to the complex Lie algebra (i.e. complexify the Lie algebra). This doesn't affect the representation theory. The Lie algebra is spanned by three elements e, f and h with the Lie brackets ...more on Wikipedia about "Representation theory of SU(2)"
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The spacetime symmetry group of nonrelativistic mechanics is the Galilean group. Since we are interested in projective representations of this group, which is equivalent to unitary representations of the nontrivial central extension of the universal covering group of the Galilean group by the one dimensional Lie group R, refer to the article Galilean group for the central extension of its Lie algebra. We will focus upon the Lie algebra here because it is simpler to analyze and we can always extend the results to the full Lie group thanks to the Frobenius theorem. ...more on Wikipedia about "Representation theory of the Galilean group"
In physics, a rigid body is an idealisation of a solid body of finite size in which deformation is neglected. In other words, the distance between any two given points of a rigid body remains constant regardless of external forces exerted on it. ...more on Wikipedia about "Rigid body"
In physics, Rigid body dynamics differs from particle dynamics in that the body takes up space and can rotate, which introduces other considerations. Equations from particle dynamics can be generalized to rigid body dynamics as follows: ...more on Wikipedia about "Rigid body dynamics"
In mechanics and geometry, the rotation group is the set of all rotations about the origin of 3-dimensional Euclidean space, R3. By definition, a rotation about the origin is a linear transformation that preserves the length of vectors, and also preserves the orientation, or handedness, of space. A transformation that preserves length but reverses orientation is sometimes called an improper rotation. ...more on Wikipedia about "Rotation group"
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