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The Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. This is the passive transformation
Active and passive transformation

In the physics, an active transformation is one which actually changes the physical state of a system, and makes sense even in the absence of a coordinate system whereas a passive transformation is merely a change in the coordinate system of no physical significance....
 point of view. The equations below, although apparently obvious, break down at speeds that approach the speed of light
Speed of light

The speed of light in an free space is an important physical constant usually written as c, with a value of 299,792,458 metres per second....
 due to physics described by Einstein's
Albert Einstein

Albert Einstein was a Germany-born theoretical physics. He is best known for his theory of relativity and specifically mass?energy equivalence, expressed by the equation E = mc2....
 theory of relativity
Special relativity

Special relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "Annus Mirabilis Papers#Special relativity"....
.

Galileo
Galileo Galilei

Galileo Galilei was a Grand Duchy of Tuscany physicist, mathematician, astronomer, and philosopher who played a major role in the Scientific Revolution....
 formulated these concepts in his description of uniform motion The topic was motivated by Galileo's description of the motion of a ball
Ball

A ball is a round object with various uses. It is usually sphere but can be ovoid. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players....
 rolling down a ramp
Ramp

Ramp may refer to:Gravitational:* Inclined plane, a physical structure that is a simple machine* Airport ramp, the area around an airport terminal where aircraft are loaded and unloaded...
, by which he measured the numerical value for the acceleration
Acceleration

File:Acceleration.JPGFile:Acceleration components.JPGIn physics, and more specifically kinematics, acceleration is the change in velocity over time....
 of gravity, at the surface of the Earth
Earth

Earth is the third planet from the Sun. Earth is the largest of the terrestrial planets in the Solar System in diameter, mass and density. It is also referred to as the World and Wiktionary:Terra.Note that by International Astronomical Union convention, the term "Terra" is used for naming extensive land masses, rather...
.






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The Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. This is the passive transformation
Active and passive transformation

In the physics, an active transformation is one which actually changes the physical state of a system, and makes sense even in the absence of a coordinate system whereas a passive transformation is merely a change in the coordinate system of no physical significance....
 point of view. The equations below, although apparently obvious, break down at speeds that approach the speed of light
Speed of light

The speed of light in an free space is an important physical constant usually written as c, with a value of 299,792,458 metres per second....
 due to physics described by Einstein's
Albert Einstein

Albert Einstein was a Germany-born theoretical physics. He is best known for his theory of relativity and specifically mass?energy equivalence, expressed by the equation E = mc2....
 theory of relativity
Special relativity

Special relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "Annus Mirabilis Papers#Special relativity"....
.

Galileo
Galileo Galilei

Galileo Galilei was a Grand Duchy of Tuscany physicist, mathematician, astronomer, and philosopher who played a major role in the Scientific Revolution....
 formulated these concepts in his description of uniform motion The topic was motivated by Galileo's description of the motion of a ball
Ball

A ball is a round object with various uses. It is usually sphere but can be ovoid. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players....
 rolling down a ramp
Ramp

Ramp may refer to:Gravitational:* Inclined plane, a physical structure that is a simple machine* Airport ramp, the area around an airport terminal where aircraft are loaded and unloaded...
, by which he measured the numerical value for the acceleration
Acceleration

File:Acceleration.JPGFile:Acceleration components.JPGIn physics, and more specifically kinematics, acceleration is the change in velocity over time....
 of gravity, at the surface of the Earth
Earth

Earth is the third planet from the Sun. Earth is the largest of the terrestrial planets in the Solar System in diameter, mass and density. It is also referred to as the World and Wiktionary:Terra.Note that by International Astronomical Union convention, the term "Terra" is used for naming extensive land masses, rather...
. The descriptions below are another mathematical notation
Mathematical notation

A mathematical notation is a system of symbolic representations of mathematical objects and ideas. Mathematical notations are used in mathematics and the physical sciences, engineering and economics....
 for this concept.

Translation (one dimension)

In essence, the Galilean transformations embody the intuitive notion of addition and subtraction of velocities. The assumption that time can be treated as absolute is at the heart of the Galilean transformations.

This assumption is abandoned in the Lorentz transformation
Lorentz transformation

In physics, the Lorentz transformation converts between two different observers' measurements of space and time, where one observer is in constant motion with respect to the other....
s. These relativistic
Special relativity

Special relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "Annus Mirabilis Papers#Special relativity"....
 transformations are deemed applicable to all velocities, whilst the Galilean transformation can be regarded as a low-velocity approximation to the Lorentz transformation.

The notation below describes the relationship of two coordinate systems (x' and x) in constant relative motion (velocity
Velocity

In physics, velocity is defined as the Derivative of Position vector. It is a vector physical quantity; both speed and direction are required to define it....
 v) in the x-direction according to the Galilean transformation:

Note that the last equation expresses the assumption of a universal time independent of the relative motion of different observers.

Galilean transformations

Under the Erlangen program
Erlangen program

An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen ?ber neuere geometrische Forschungen....
, the space-time (no longer spacetime
Spacetime

In physics, spacetime is any mathematical model that combines space and Time in physics into a single continuum . Spacetime is usually interpreted with space being Three-dimensional space and time playing the role of a fourth dimension that is of a different sort than the spatial dimensions....
) of nonrelativistic physics is described by the symmetry group
Symmetry group

The symmetry group of an object is the group of all isometries under which it is invariant with Function composition as the operation. It is a subgroup of the isometry group of the space concerned....
 generated by Galilean transformations, spatial and time translations and rotations.

The Galilean symmetries (interpreted as active transformations):

Spatial translations:

Time translations:

Shear mappings:

Rotations and Reflections:

where R is an orthogonal matrix
Orthogonal matrix

In matrix theory, a real number orthogonal matrix is a Matrix #Square matrices Q whose transpose is its inverse matrix:A special orthogonal matrix is an orthogonal matrix with determinant +1:...
.

Central extension of the Galilean group

The Galilean group
Representation theory of the Galilean group

In nonrelativistic quantum mechanics, an account can be given of the existence of mass and spin as follows:The spacetime symmetry group of nonrelativistic quantum mechanics is the Galilean group....
: Here, we will only look at its Lie algebra
Lie algebra

In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds....
. It's easy to extend the results to the Lie group
Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the Differential structure....
. The Lie algebra of L is spanned
Linear span

In the mathematics subfield of linear algebra, the linear span, also called the linear hull, of a Set of vector space in a vector space is the intersection of all Linear subspace containing that set....
 by E, Pi, Ci and Lij (antisymmetric tensor
Antisymmetric tensor

In mathematics and theoretical physics, a tensor is antisymmetric on two indices i and j if it flips sign when the two indices are interchanged:...
) subject to commutator
Commutator

In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory....
s (operator
Operator

In mathematics, an operator is a function which operates on another function. Often, an "operator" is a function which acts on functions to produce other functions ; or it may be a generalization of such a function, as in linear algebra, where some of the terminology reflects the origin of the subject in operations on the functions which ar...
s of the form [,]), where

We can now give it a central extension into the Lie algebra spanned by E', P'i, C'i, L'ij (antisymmetric tensor
Tensor

A tensor is an object which extends the notion of Scalar , Vector , and Matrix . The term has slightly different meanings in mathematics and physics....
), M such that M commutes with everything (i.e. lies in the center
Center (algebra)

The term center or centre is used in various contexts in abstract algebra to denote the set of all those elements that commutative operation with all other elements....
, that's why it's called a central extension) and

See also


  • Representation theory of the Galilean group
    Representation theory of the Galilean group

    In nonrelativistic quantum mechanics, an account can be given of the existence of mass and spin as follows:The spacetime symmetry group of nonrelativistic quantum mechanics is the Galilean group....
  • Lorentz group
    Lorentz group

    In physics , the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical field theory setting for all physics....
  • Poincaré group
    Poincaré group

    In physics and mathematics, the Poincar? group, named after Henri Poincar?, is the group of isometry of Minkowski spacetime. It is a 10-dimensional compact space Lie group....
  • Lagrangian and Eulerian coordinates
    Lagrangian and Eulerian coordinates

    In fluid dynamics and finite-deformation plasticity the Lagrangian reference frame is a way of looking at fluid motion where the observer follows individual fluid particles as they move through space and time....