3.6 Transformation of Galilean coordinates and the mechanical principle of relativity

Lecture



  3.6 Transformation of Galilean coordinates and the mechanical principle of relativity
Consider two reference systems: fixed (K) and moving relative to the first along the X axis with a constant X with a constant speed   3.6 Transformation of Galilean coordinates and the mechanical principle of relativity (K '). The coordinates of the body M in the system K x: y: z, and in the system K '- x': y ': z'. These coordinates are related to each other by the relations that are called the Galilean transformation

  3.6 Transformation of Galilean coordinates and the mechanical principle of relativity

Differentiating these equations over time and considering that   3.6 Transformation of Galilean coordinates and the mechanical principle of relativity , we find the relationship between speeds and accelerations:

  3.6 Transformation of Galilean coordinates and the mechanical principle of relativity   3.6 Transformation of Galilean coordinates and the mechanical principle of relativity
  3.6 Transformation of Galilean coordinates and the mechanical principle of relativity   3.6 Transformation of Galilean coordinates and the mechanical principle of relativity
  3.6 Transformation of Galilean coordinates and the mechanical principle of relativity   3.6 Transformation of Galilean coordinates and the mechanical principle of relativity

Thus, if in system K a body has an acceleration a, then it has the same acceleration in system K '.

According to Newton's second law:

  3.6 Transformation of Galilean coordinates and the mechanical principle of relativity

those. Newton's second law is the same in both cases.

With   3.6 Transformation of Galilean coordinates and the mechanical principle of relativity motion by inertia, thus, the Newton's first law is also valid, i.e. the mobile system under consideration is inertial. Consequently, Newton's equations for the material point, as well as for an arbitrary system of material points are the same in all inertial reference systems - invariant with respect to the Galilean transformations. This result is called the mechanical principle of relativity (Galilean principle of relativity), and is formulated as follows: uniform and straight-line movement (relative to any inertial reference system) of a closed system does not affect the patterns of mechanical processes in it. Consequently, in mechanics all inertial reference systems are completely equal. Therefore, no mechanical experiments inside the system can detect whether the system moves evenly and rectilinearly or is at rest.


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Physical foundations of mechanics

Terms: Physical foundations of mechanics