Galileo's transformations are in classical mechanics ( Newton's mechanics ) and non-relativistic quantum mechanics of transformation of coordinates and velocity during the transition from one inertial reference system (ISO) to another [1] . The term was proposed by Philip Frank in 1909 . [2] Galilean transformations are based on the Galilean principle of relativity , which implies that time is the same in all reference systems (“absolute time” [3] ).
Galilean transformations are a limiting (particular) case of Lorentz transformations for velocities that are small compared to the speed of light in a vacuum and in a limited volume of space. For speeds up to the order of the speeds of the planets in the Solar System (and even greater ones), the Galilean transformations are approximately correct with very high accuracy.
- The requirement (postulate) of the principle of relativity, together with Galileo's transformations, which seem to be quite intuitively obvious, can be considered in many respects defining the structure of Newtonian mechanics. Together with such additional ideas as the symmetry of space and the principle of superposition in one form or another (asserting the equivalence of the interaction of many bodies in a short period of time of the composition of imaginary sequential pairwise interactions of these bodies), Galileo's transformations can be practically sufficient grounds for the formulation of Newtonian mechanics ( its basic laws).
Content
View of transformations with collinear axes [4]
If the ISO S ' moves relative to the ISO S at a constant speed along the axis , and the origin of the coordinates coincide at the initial time in both systems, then the Galilean transformations have the form:
or using vector notation
(the last formula remains true for any direction of the axes of coordinates).
- As you can see, these are just formulas for shifting the origin of coordinates, linearly dependent on time (implied by the same for all reference systems).
From these transformations follow the relationship between the speeds of the point and its accelerations in both reference systems:
- {\ displaystyle {\ vec {a}} = {\ vec {a '}}}
- Galilean transformations are a limiting (particular) case of Lorentz transformations for low velocities. (much less than the speed of light).
Enough to differentiate in the Galilean transformation formula given above, and immediately get the speed conversion formula given in the same paragraph.
Let us give a more elementary, but also a more general conclusion - for the case of an arbitrary movement of the origin of one system relative to another (in the absence of rotation). For such a more general case, one can obtain a velocity conversion formula, for example, like this.
Consider the transformation of an arbitrary origin of origin by vector ,
where the radius vector of a bodyA in the reference frameK is denoted by , and in the frame of referenceK ' - for ,
implying, as always in classical mechanics, that time in both reference systems the same, and all radius vectors depend on this time: .
Then at any given time
and in particular, given
- ,
we have:
Where:
- - the average velocity of the bodyA relative to the systemK ;
- - the average velocity of the bodyA relative to the systemK ' ;
- - the average speed of the systemK ' relative to the systemK.
If a then the average speed coincides withinstant :
or shorter
- for both medium and instantaneous speeds (the formula of addition of speeds).
Thus , the velocity of the body relative to the fixed coordinate system is equal to the vector sum of the velocity of the body relative to the moving coordinate system and the speed of the reference frame relative to the fixed reference frame.
- (Similarly, you can get the formula for converting accelerations when moving from one coordinate system to another, true, provided that the motion of the systems relative to each other is equally accelerated and translational: ).
The Schrödinger equation in nonrelativisticquantum mechanics is invariant with respect to Galileo transformations. This fact implies a number of important consequences: the existence of a number of quantum mechanics operators associated with the Galilean transformations (Schrödinger group ), the impossibility of describing states with a mass spectrum or unstable elementary particles in nonrelativistic quantum mechanics (Bargmann's theorem ), the existence of quantum-mechanical invariants generated by the Galilean transformation.[five]
- ↑Being purely kinematic, Galileo transformations are also applicable to non-inertial reference systems - but only under the condition of their uniform rectilinear translational motion relative to each other - which limits their importance in such cases.Together with the privileged role of inertial reference systems, this fact leads to the fact that in the overwhelming number of cases Galileo’s transformations are spoken about in connection with the latter.
- ↑Frank P. / Sitz.Ber.Akad.WissWien. — 1909. — Ila, Bd 118. —S.373 (esp. P. 382).
- ВообщеGenerally speaking, physics had to give up absolute time at the beginning of the twentieth century — for the sake of preserving the principle of relativity in its strong formulation implying the requirement of uniformity of writing all the fundamental equations of physics in any (inertial; and later the principle of relativity was extended to non-inertial) reference system.
- ↑Of fundamental interest from the point of view of physics is only the case when the coordinate axes (if a coordinate representation is used at all; this question can be considered irrelevant to the symbolic vector record) between the inertial systems between which the transformation is performed, are directed in the same way.In principle, they can be directed in different ways, but transformations of this kind are from a physical point of view only of technical interest, since transformations with codirectional axes, considered in this article, and fixed (time-independent)rotation of the axes representing a purely geometric problem, and in addition, in principle, simple.The rotation of the axes, depending on time, would mean the rotation of the coordinate systems relative to each other, and at least one of them could not then be inertial.
- ↑Kempfer, 1967 , p.390.
- Kempfer F. The main provisions of quantum mechanics.-M .: Mir, 1967. - 391 p.
- Inertial frame of reference
- Einstein's principle of relativity
- Classical mechanics
- Lorentz transformations
- Velocity addition
- Difficult movement
- Physics in the notes - wiki-book
- Schrödinger Group
- Bargman's theorem