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Encyclopedia > Galilean transformation

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 point of view. The equations below, although apparently obvious, break down at speeds that approach the speed of light due to Einstein's theory of relativity. A frame of reference in physics is a set of axes which enable an observer to measure the aspect, position and motion of all points in a system relative to the reference frame. ... Classical mechanics is a model of the physics of forces acting upon bodies. ... 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. ... A line showing the speed of light on a scale model of Earth and the Moon The speed of light in a vacuum is an important physical constant denoted by the letter c for constant or the Latin word celeritas meaning swiftness. It is the speed of all electromagnetic radiation... Albert Einstein ( ) (March 14, 1879 – April 18, 1955) was a German-born theoretical physicist who is best known for his theory of relativity and specifically mass-energy equivalence, . He was awarded the 1921 Nobel Prize in Physics for his services to Theoretical Physics, and especially for his discovery of the... Two-dimensional analogy of space-time curvature described in General Relativity. ...



Galileo formulated these concepts in his description of uniform motion [1] The topic was motivated by Galileo's description of the motion of a ball rolling down a ramp, by which he measured the numerical value for the acceleration of gravity, at the surface of the Earth. The descriptions below are another mathematical notation for this concept. Galileo Galilei (15 February 1564 – 8 January 1642) was an Italian physicist, mathematician, astronomer, and philosopher who is closely associated with the scientific revolution. ... Galileo can refer to: Galileo Galilei, astronomer, philosopher, and physicist (1564 - 1642) the Galileo spacecraft, a NASA space probe that visited Jupiter and its moons the Galileo positioning system Life of Galileo, a play by Bertolt Brecht Galileo (1975) - screen adaptation of the play Life of Galileo by Bertolt Brecht... Balls are objects typically used in games. ... The word ramp can mean one of several things: Inclined plane A ramp is the area around an airport terminal where aircraft are loaded and unloaded. ... Acceleration is the time rate of change of velocity, and at any point on a velocity-time graph, it is given by the slope of the tangent to that point basicly. ... Gravity is a force of attraction that acts between bodies that have mass. ... Adjectives: Terrestrial, Terran, Telluric, Tellurian, Earthly Atmosphere Surface pressure: 101. ... Mathematical notation is used in mathematics, and throughout the physical sciences, engineering, and economics. ...

Contents

Translation (one dimension)

The Galilean transformation is nothing more than careful addition and subtraction of velocity vectors.


Unlike the Galilean transformation, the relativistic Lorentz transformation can be shown to apply at all velocities so far measured, and the Galilean transformation can be regarded as a low-velocity approximation to the Lorentz transformation. The special theory of relativity was proposed in 1905 by Albert Einstein in his article On the Electrodynamics of Moving Bodies. Some three centuries earlier, Galileos principle of relativity had stated that all uniform motion was relative, and that there was no absolute and well-defined state of rest... A Lorentz transformation (LT) is a linear transformation that preserves the spacetime interval between any two events in Minkowski space, while leaving the origin fixed (=rotation of Minkowski space). ...


The notation below describes the relationship of two coordinate systems (x′ and x) in constant relative motion (velocity u) in the x-direction. All other parameters (t, y, z) are unchanged in the transformation from x′ to x coordinates. The velocity of an object is its speed in a particular direction. ...

begin{align}t'&=t  x'&=x-ut  y'&=y  z'&=z end{align}
Diagram 1. Views of spacetime along the world line of a slowly accelerating observer.Vertical direction indicates time. Horizontal indicates distance, the dashed line is the spacetime trajectory ("world line") of the observer. The lower half of the diagram shows the events visible to the observer. Upper half shows those that will be able to see the observer. The small dots are arbitrary events in spacetime. The slope of the world line (deviation from being vertical) gives the relative velocity to the observer. Note how the view of spacetime changes when the observer accelerates. This caption probably should be rewritten, to be more relevant for the Galilean transform.
Diagram 1. Views of spacetime along the world line of a slowly accelerating observer.

Vertical direction indicates time. Horizontal indicates distance, the dashed line is the spacetime trajectory ("world line") of the observer. The lower half of the diagram shows the events visible to the observer. Upper half shows those that will be able to see the observer. The small dots are arbitrary events in spacetime.

The slope of the world line (deviation from being vertical) gives the relative velocity to the observer. Note how the view of spacetime changes when the observer accelerates. This caption probably should be rewritten, to be more relevant for the Galilean transform.

Image File history File links No higher resolution available. ... In physics, the world line of an object is the unique path of that object as it travels through 4-dimensional spacetime. ... In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. ... In physics, the world line of an object is the unique path of that object as it travels through 4-dimensional spacetime. ...

Galilean transformations

Under the Erlangen program, the space-time (no longer spacetime) of nonrelativistic physics is described by the symmetry group generated by Galilean transformations, spatial and time translations and rotations. An influential research programme and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen. ... In physics, spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. ... The symmetry group of an object (e. ...


The Galilean symmetries (interpreted as active transformations): 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. ...


Spatial translations:

trightarrow t ,!
vec{x}rightarrow vec{x}+vec{a} ,!

Time translations:

trightarrow t+tau ,!
vec{x}rightarrow vec{x} ,!

Boosts:

trightarrow t ,!
vec{x}rightarrow vec{x}+vec{v}t ,!

Rotations:

trightarrow t ,!
vec{x}rightarrow mathbf{R}vec{x} ,!

where R is an orthogonal matrix. In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // Overview An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ...


Central extension of the Galilean group

The Galilean group: Here, we will only look at its Lie algebra. It's easy to extend the results to the Lie group. The Lie algebra of L is spanned by E, Pi, Ci and Lij (antisymmetric tensor) subject to commutators (operators of the form [,]), where In nonrelativistic quantum mechanics, an account can be given of the existence of mass and spin as follows: The spacetime symmetry group of nonrelativistic mechanics is the Galilean group. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... In the mathematical subfield of linear algebra, the linear span, also called the linear hull, of a set of vectors in a vector space is the intersection of all subspaces containing that set. ... In mathematics and theoretical physics, an antisymmetric tensor is a tensor that flips the sign if two indices are interchanged: If the tensor changes the sign under the exchange of any pair of indices, then the tensor is completely antisymmetric and it is also referred to as a differential form. ... In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. ... In mathematics, an operator is a function that performs some sort of operation on a number, variable, or function. ...

[E,P_i]=0 ,!
[P_i,P_j]=0 ,!
[L_{ij},E]=0 ,!
[C_i,C_j]=0 ,!
[L_{ij},L_{kl}]=ihbar [delta_{ik}L_{jl}-delta_{il}L_{jk}-delta_{jk}L_{il}+delta_{jl}L_{ik}] ,!
[L_{ij},P_k]=ihbar[delta_{ik}P_j-delta_{jk}P_i] ,!
[L_{ij},C_k]=ihbar[delta_{ik}C_j-delta_{jk}C_i] ,!
[C_i,E]=ihbar P_i ,!
[C_i,P_j]=0 ,!

We can now give it a central extension into the Lie algebra spanned by E', P'i, C'i, L'ij (antisymmetric tensor), M such that M commutes with everything (i.e. lies in the center, that's why it's called a central extension) and In group theory, a central extension of a group G is an exact sequence of groups such that A is in Z(E), the center of the group E. Examples of central extensions can be constructed by taking any group G and any abelian group A, and setting E to... In mathematics, a tensor is (in an informal sense) a generalized linear quantity or geometrical entity that can be expressed as a multi-dimensional array relative to a choice of basis; however, as an object in and of itself, a tensor is independent of any chosen frame of reference. ... Mathematical meaning A map or binary operation is said to be commutative when, for any x in A and any y in B . ... The term center is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. ...

[E',P'_i]=0 ,!
[P'_i,P'_j]=0 ,!
[L'_{ij},E']=0 ,!
[C'_i,C'_j]=0 ,!
[L'_{ij},L'_{kl}]=ihbar [delta_{ik}L'_{jl}-delta_{il}L'_{jk}-delta_{jk}L'_{il}+delta_{jl}L'_{ik}] ,!
[L'_{ij},P'_k]=ihbar[delta_{ik}P'_j-delta_{jk}P'_i] ,!
[L'_{ij},C'_k]=ihbar[delta_{ik}C'_j-delta_{jk}C'_i] ,!
[C'_i,E']=ihbar P'_i ,!
[C'_i,P'_j]=ihbar Mdelta_{ij} ,!

Notes

  1. ^ Galileo 1638 Discorsi e Dimostrazioni Matematiche, intorno á due nuoue scienze 191 - 196, published by Lowys Elzevir (Louis Elsevier), Leiden, or Two New Sciences, English translation by Henry Crew and Alfonso de Salvio 1914, reprinted on pages 515-520 of On the Shoulders of Giants: The Great Works of Physics and Astronomy. Stephen Hawking, ed. 2002 ISBN 0-7624-1348-4

Events March 29 - Swedish colonists establish first settlement in Delaware, called New Sweden. ... Lodewijk Elzevir (c. ... Lodewijk Elzevir (c. ... The Discourses and Mathematical Demonstrations Relating to Two New Sciences (1638) was Galileos final book and a sort of scientific testament covering much of his work in physics over the preceding thirty years. ... 1914 (MCMXIV) was a common year starting on Thursday (see link for calendar). ... Stephen William Hawking, CH, CBE, FRS, FRSA, (born 8 January 1942) is a British theoretical physicist. ... For album titles with the same name, see 2002 (album). ...

See also


  Results from FactBites:
 
Galilean transformation - Wikipedia, the free encyclopedia (368 words)
The Galilean transformation is used to transform between the coordinates of two coordinate systems in a constant relative motion in Newtonian physics.
Unlike the Galilean transformation, the relativistic Lorentz transformation can be shown to apply at all velocities so far measured, and the Galilean transformation can be regarded as a low-velocity approximation to the Lorentz transformation.
Under the Erlangen program, the space-time (no longer spacetime) of nonrelativistic physics is described by the symmetry group generated by Galilean transformations, spatial and time translations and rotations.
  More results at FactBites »

 
 

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