Galilean transformation

The Galilean transformation, named after Galileo Galilei, the simplest coordinate transformation, can be converted with the physical observations from one reference system into another, which differs by a straight - uniform motion and / or a shift in the time from the first. All observations of distances, angles and time differences are consistent in both reference systems. All observed velocities differ by a constant relative velocity of the two reference systems.

The Galilean transformation is fundamental to classical mechanics, because there it describes the transformation between two inertial frames. With respect to the sequential execution form the Galilean transformations a group. According to the relativity principle of classical mechanics, the laws of nature with respect to this group must be covariant.

The covariance with respect to the Galilean transformation is not satisfied with the Maxwell's equations of electromagnetism. This example say one in all inertial equal the speed of light before, which is in agreement with the observations. Therefore, the Galileo transformation, must be replaced by the Lorentz transformation. This is the starting point for the theory of special relativity. The Lorentz transformation approaches in the case of low velocities of the Galilean transformation.

• 4.1 displacement along an axis
• 4.2 More general forms
• 4.3 Vectorial notation
• 4.4 Uniform speeds in any direction
• 4.6 Differences in the time
• 4.7 Rotated reference systems

Limit of validity of the Galilean transformation

Classical Mechanics

Classical mechanics begins before Newton by the ideas of René Descartes. In contrast to the scholasticism of the Middle Ages describes space and time coordinate systems relative to an arbitrary origin.

The independence of the laws of mechanics the state of motion with uniform movement was first recognized by Galileo Galilei and Isaac Newton formulated by in his Principia. Forces are at Newton only depends on the accelerations which do not change under Galilean transformations. In classical mechanics, the principle retains full force and it was thought for a long time given a priori and unassailable.

Lorentz transformation

The electrodynamics to the end of the 19th century went from an ether as a carrier of electromagnetic waves, including light out. Maxwell's equations and the resulting speed of light c as the velocity of propagation of electromagnetic waves, however, were not compatible with the Galileo transformation. Hendrik Antoon Lorentz, Joseph Larmor, and Henri Poincaré realized that they could solve this problem by the Galilean transformation is replaced by the Lorentz transformation. This eventually led to the special theory of relativity by Albert Einstein, but which requires a modification of the notions of time and space.

For speeds that are much smaller than the light speed of about 300,000 km / s, the Galileo transformation is a good approximation of the Lorentz transformation. The Galilean transformation is the limiting case of the Lorentz transformation for small velocities.

Practical Application

In everyday life, the Galilean transformation can be applied to mechanical problems almost always because the correction in the Lorentz transformation in earthly speeds is very small. The correction factor is often Immeasurably. Even in the celestial mechanics of our solar system, this factor is, for example, under 10-8 for the quite large rotational speed of the earth around the sun ( about 30 km / s).

The scope, however, completely elude electrodynamic phenomena of everyday life. A simple example of a charged body, which flies by a current-carrying conductor, shows the defect at:

• A charge q is flying with the initial velocity v at a straight current-carrying, but charge neutral conductor over ( see picture). The current in the conductor generates a magnetic field which deflects the moving charge q by the Lorentz force of their rectilinear motion. If we introduce now a Galilean transformation by which changes in the inertial frame of the charge at the initial time, acts in this system no Lorentz force, because the charge is at rest. The charge should therefore remain at their starting point. The reverse transformation leads to a rectilinear movement in opposition to the above statement, according to which the charge is accelerated by the Lorentz force. Can be explained this apparent paradox only with the Lorentz transformation in which contracts the length of the conductor in the inertial frame of the charge and the head thus obtains a relative electric charge. The resulting electric field replaces the original magnetic. Here, the relationship between the two fields is a little clearer.

Galilei transformation and conservation laws

The laws of nature do not change under Galilean transformation. The outcome of the experiment remains the same when one pulls his place a Galilei transformation. A shift of the place or the time, or even the orientation change anything. Such invariance is also called symmetry. According to the Noether theorem, every such symmetry is linked to a conservation law. From the invariance of physical laws under Galilean transformation so follow the conservation laws of classical mechanics. In particular:

• From the invariance under displacement of the local conservation of momentum follows.
• From the invariance under time shift in the energy conservation follows.
• From the invariance under rotation, the angular momentum conservation follows.

Mathematical Description

Displacement along an axis

A particularly simple Galileo transformation, the translation of the origin along the x- axis of a Cartesian coordinate system. It shows how it is necessary to change the coordinates, to describe a physical system of a shifted by a distance in the coordinate origin. The coordinates on the axis, the axis and in the time remain the same. To coordinate the value is added. In formulas, ie:

More general forms

The Galilean transformation is also

• For uniform speeds in any direction,
• If the zero points of reference systems coincide at any time,
• For different time points,
• And for each twisted ( non-rotating ) reference systems.

Vector notation

The three formulas for the transformation of the local coordinates can be summarized to an equation for a position vector. This is at first only an abbreviation, but turns out to be more complicated transformations in handy.

Substituting

Then we can write the four equations for the uniform shift of the coordinate origin along the axis also like:

The first equation in this case contains the three equations for the spatial coordinates in vector notation.

Uniform speeds in any direction

The vectorial form applies immediately for the generalization to speeds that do not occur in parallel to the x axis, if one sets:

The velocity and its components must be constant over time.

The constant vector

Is introduced. At the time this is the displacement vector of the zero points of the reference systems. The Galilean transformation is then

This clearly means that I can watch the game of table tennis in the train even from a distance.

Differences in the time

Time need not be the same but can be different in order. Clearly, the two systems clocks, which are set according to different time zones.

The laws of mechanics do not change with time.

Turned reference systems

The coordinate axes of the reference systems do not point in the same direction. Mathematically, the coordinates must be converted, which results without vector notation to rather long formulas.

Vectorially you can simply use a rotation matrix with nine figures. Since the lengths are not changed, certain conditions must be imposed on this matrix, so that only three parameters (for example, angle) are independent. The spelling with nine figures, however, is still the simplest:

A valid rotation matrix is, for example (45 ° angle about the z-axis)

The Galileo transformation, in its most general form is in rotation according to the translation then

Or in the case of a rotation before the translation to

The general form has 10 parameters ( three rotation angles, three distance coordinates, three velocity components and the time shift).

This clearly means that I take up the game of table tennis in the train through a window of a plane flying at an angle on film and I can look at, for example, a week later. The laws of physics have not changed in that time.

• Classical Mechanics
• Symmetry (Physics)
• Transformation