Angular momentum

The angular momentum outdated, even twist, momentum or angular momentum is a physical conserved quantity. It thus occupies a position similar to the energy or the pulse. At a rotating body to be proportional to the rotational frequency. To change the angular momentum of the body, a torque must act on the body. The unit used internationally for the angular momentum is Newton meter seconds or joules second.

The angular momentum is a pseudovector. Its specification always refers to a preselected point in space. For a freely rotating body, the focus is usually taken. With a body, which rotates about a predetermined axis, usually a point on the axis is selected, without this being explicitly mentioned.

In quantum mechanics, the angular momentum is a quantized variable which is described by the angular momentum operator. The angular momentum quantum number that indicates the amount of angular momentum can assume only integer or half-integer multiples of the reduced Planck's constant. The measurement of the orientation of the angular momentum along an axis subject to the direction of quantization. The angular momentum occurs in quantum mechanics in two forms: Derived from classical mechanics of rotation and circular motion angular momentum is called angular momentum. In addition, the spin angular momentum is not affiliated with rotational movements exist.

In astronomy, a distinction is a celestial body the orbital angular momentum due to movement of its center of gravity to a central star of its intrinsic angular momentum due to rotation about an axis running through its center of gravity axis.

Basics

In a circular motion, one can imagine the angular momentum with respect to the center of the circle as an arrow indicating the direction of the rotational axis of the movement. Its length indicates the momentum of rotation. The longer the arrow, the larger the angular momentum. The angular momentum increases with:

• Higher speed,
• Greater mass and
• Greater distance from the axis of rotation.

In a circular movement, the angular momentum is perpendicular to the plane in which the mass is moved, if there is also the reference point of the angular momentum in this plane. Its length is equal to the product of mass, radius, and velocity.

In mathematical description of the angular momentum of a particle is the cross product of its position vector with its pulse

, The angular momentum vector pointing in the direction which forms a so-called right-handed screw and with the location of the velocity. It is the right-hand rule: if the curved fingers of the right hand indicate the direction of rotation, the thumb is in the direction of the angular momentum. That the right and not the left hand has to be used for this control is related to the definition of the cross product.

Conservation of angular momentum

From the fact that the physical laws do not depend on the orientation in space, it follows that the angular momentum is a conserved quantity ( stating the Noether theorem). In other words, the angular momentum of an isolated physical system does not change, no matter which forces and interactions between the components of the system. This applies to the angular momentum with respect to arbitrary axes and is shortened designated by the term conservation of angular momentum. Almost perfectly isolated systems there are in the universe. The second law of Kepler describes the conservation of angular momentum geometrically.

The conservation of angular momentum leads to the pirouette effect known in figure skating and other sporting disciplines.

Displacement, rotation, mirroring

The magnitude and direction of the angular momentum depend on what point is chosen as reference point. On displacement of the reference point, the vector of each place and in the angular momentum changes in

Often is chosen as a reference point the center of gravity or the point of resting with the observed rotation, that is on the axis.

The cross product of two vectors is always perpendicular to the plane defined by them, a rotation of the system under consideration, but rotates both the radius vectors, as well as the web speeds by the same amount, whereby the angular momentum is also rotated in the same way.

At a point reflection at the reference point of the place goes to the opposite place. Even the sign of the velocity with respect to this point is reversed. In forming the cross product of these two sign changes compensate. Therefore, the angular momentum does not change under point reflection. He differs from the behavior of the speed or the position vector. It thus belongs to the class of pseudo- vectors.

Level path, face set

Retains the angular momentum of a particle (for example, the earth, which rotates the sun ) each time the initial value, the path of the particle and extending in a plane can be expressed in polar coordinates.

Because the cross product is perpendicular to its factors and applies at all times

Now, if the angular momentum is independent of time, then met each track point, the plane equation

Thus, it is a movement in the plane through the center of the system perpendicular to the angular momentum.

Furthermore, the second Kepler's law applies: the driving beam to the planet sweeps out equal areas in equal times.

Since in a short time, the driving beam change to and sweeps the surface of the triangle, with the two sides. The triangle is half the size of the plane defined by the two vectors parallelogram whose magnitude is given by the cross product. In the time of the scanning beam sweeps consequently the surface

When the angular momentum does not change with time, therefore, the surface speed is constant.

The area theorem also holds in relativistic physics, if, in addition, the energy is conserved. For, in relativistic physics

And

For flat tracks there is a connection between angular momentum and angular velocity, which is relevant to the Runge -Lenz vector:

For the proof one decomposes the velocity into a radial and an azimuthal component (see polar coordinates / velocity). In the cross product with the radial velocity falls away, and we obtain

Eulerian angular momentum

To change the pulse of a body, a force has to act more specifically, the temporal change of the impulse, the force

Quite analogously formulated in 1754 Leonhard Euler Euler's angular momentum, after the time change of the angular momentum is equal to the torque,

To change the angular momentum of the body, a torque must occur. The torque is the cross product of the position vector ( lever arm ) and strength:

The results from the angular momentum if one derives the angular momentum with respect to time, ie the time variation of the angular momentum examined:

Since the velocity and momentum are parallel, their cross product vanishes at.

If it is in the power of a central force, the angular momentum is obtained:

The angular momentum of a rigid body

When a rigid body (for example, toy top, car wheel, flywheel ) refers to the angular momentum mostly on the center of gravity of the body and then calls him also intrinsic angular momentum. It is determined by the rotational speed of the body, more precisely its angular velocity, and its inertia tensor:

Generally show and not in the same direction - a rotating body " wobbles " when he can move freely, or shows imbalance when the direction of the axis remains fixed. Only in rotation about one of the principal axes of inertia of the body and in parallel so that the conservation of angular momentum, also causes uniform direction of rotation axis, and therefore the constancy of the moment of inertia. From the matrix of inertia tensor may be the principal axes of inertia and on any axis of rotation Calculate the moment of inertia. The inertia tensor for the rotary movement has similar meaning as the material for the translational movement (see rotation (Physics) # compared with the translational movement).

Derivation

The angular momentum of a rigid body is the sum of the angular momenta of the mass points, of which it consists. We denote the individual mass points with the places where they are located, and their speeds. The total angular momentum is

It has to sum ​​over all mass points that make up the body. His focus was chosen as the coordinate origin.

When the center of gravity rests, the movement of the rigid body is a rotation about the centroid. The speed of the individual mass points is the cross product of angular velocity and position vector,

Used, we obtain

The double cross product are modeled using the BAC -CAB- formula,

So the angular momentum is linear in the angular velocity and can therefore be as inertia tensor times in the form

. Write Here, the inertia tensor is the matrix

In a continuous instead of the distribution of mass is the sum of the mass points, a volume integral of the mass density of which is weighted according to the matrix element with different products of the coordinates,