Non-inertial reference frame

Accelerated reference systems are all systems of reference that are not inertial.

Although in accelerated reference systems, the laws of physics in general look more complicated ( in mechanics must be taken into account eg in the preparation of the equations of motion inertia forces), these reference systems can simplify the solution of certain problems in some cases though.

This is usually the case when they are adapted to the movements, the movements in the accelerated reference system so simply:

• Rotating circular or spiral movements around a common center can be, for example, in about the center of uniformly rotating reference systems often describe good: The orbiting or spiral -border body then rests or moves along a straight line.

• The Foucault pendulum is usually calculated over a reference system that mitvollführt the Earth's rotation.

Kinematics

The position vector of a point P is equal to the inertial frame K. The origin of a moving reference frame K ' is K of view at. The position vector of the point P in K ' is.

In the following, the time derivatives, ie velocity and acceleration to be calculated. The position vector of P with respect to component-wise K. Write the components that are represented in K ' deleted.

Now, first, the time derivation of vectors that are shown in the moving frame of reference is calculated. Since K must rotate ', also the basis vectors change and thus must also be differentiated ( product rule ):

The length of the normalized basis vectors does not change its direction may change, however seen from K. This corresponds to a rotation around the. Specified by axis with angular velocity The relationship between Inertialbasis and base of the moving reference system is given by the linear ( length and orientation faithful ) image:

The components of the three-dimensional rotation matrix are (with the Kronecker delta and epsilon tensor )

The skew-symmetric matrix is the generator of the passive rotation about the axis.

As the basis vectors are rotated, there is a passive rotation; because of the negative sign in the matrix. The time derivative of the rotation matrix can be represented as a matrix multiplication:

This corresponds to the matrix representation of the cross product.

Thus, the time derivative of the coated base vectors in three dimensions can be represented as a cross product:

Does a derivation only on the painted components, so it write the Painted dissipation. This is the temporal change of the vector, so the speed is ' measured in a moving reference system C:

Together, we are therefore:

Thus, a time derivative corresponds to the K -painted time derivative in K ' plus the rotation of K':

Especially for the vector apply:

To calculate the second derivative used twice above relation:

The point P is finally obtained, the speed and acceleration:

Dynamics

The Newtonian equation of motion applies only to inertial frames and reads:

In all reference systems, an external force is equal, it must be expressed only with respect to the respective basis vectors. Used in the above equation for the acceleration and by changing provides:

Thus one obtains these four inertial forces as additional terms in the equation of motion with respect to K '. In the following, various special cases are discussed.

Accelerated translational motion

Here, too, so that the motion equation simplifies to:

This is, for example, the case of a connected with a straight line moving vehicle reference system. Be, so it should work no external force. Decelerates the vehicle, so the acceleration is negative (delay ) and thus. A to properties under vehicle body is therefore in the direction of travel speeds (eg driving a car: " sterno " the short, strong brakes).

Rotating frame of reference

It should apply, that is, the origin of K ' moves uniformly with respect to the origin of K:

Reference system at the surface

The angular velocity of the earth is constant, ie. Here rotates ( = vector from the earth center to the origin of K ' at the surface ) with the same angular velocity as K':

If, with respect to K ' is, we obtain the second time derivative ( with respect to K is ' constant):

This results in the equation of motion:

For movements that are in the vicinity of the surface, one can neglect the last term, then applies here.

Set as force, the gravitational force one:

Normally comprises the gravity acceleration ( acting in the radial direction ), and the centrifugal force ( acting perpendicular to the axis of the earth ) together in an effective gravitational acceleration ( for determining the direction of the vector sum of image ). Since the centrifugal force of the latitude dependent ( at the poles and zero at the equator maximum), the effective gravitational acceleration of the latitude dependent; the earth's surface is approximately an equipotential surface of the effective gravitational acceleration, same thing an ellipsoid that is flattened at the poles compared to the ball. determines the vertical from the surface, which deviates from the radial direction somewhat.

Consider a co-moving coordinate system K ' on the earth's surface that is oriented so that points towards the east, to the north and the zenith. The angular velocity of the earth is in K ', where the latitude is

Thus is the Coriolis acceleration

Example: Foucault pendulum

For further explanation, see Foucault 's Pendulum, here is the explicit calculation can be performed.

Consider a simple pendulum on the earth's surface. For small deflections ( deflection is much smaller than pendulum length ), the approximation is valid, respectively, and the pendulum mass swings in the - plane; Thus, one can consider the motion in two dimensions. It acts on the pendulum mass with the restoring force and the Coriolis force. The coercive force of the pendulum suspension means that from actually acting weight force results the restoring force and the component of the Coriolis force is compensated. The equation of motion of the pendulum mass is:

The two coupled ordinary differential equations of second order can be easily solved in the complex representation is being defined.

This differential equation has the general solution

The term in parentheses describe the "normal" oscillation of the pendulum with the light shifted frequency, in general, the trajectories of the normal two-dimensional ellipses vibration (depending on the initial condition, is also the movement along a straight line possible). This is overlaid by a further oscillation, namely the rotation of the plane of vibration. From the superposition of the two oscillations known rosette orbits come about. With 360 ° / day = 15 ° / hour, the plane of oscillation rotates at a latitude to

On the northern hemisphere ( and ) so clockwise ( in mathematics a positive angle in the counterclockwise direction is defined, ie negative angle means clockwise). For Germany with about 50 ° north latitude, the vibration level by about -11.5 ° per hour turns.

The constants are to be determined from the initial conditions: