Galilean transformation

(noun)

a transformation used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics.

Related Terms

Examples of Galilean transformation in the following topics:

  • Gallilean-Newtonian Relativity

    • Galilean invariance or Galilean relativity states that the laws of motion are the same in all inertial (or non-accelerating) frames.
    • This transformation of variables between two inertial frames is called Galilean transformation .
    • That is, unlike Newtonian mechanics, Maxwell's equations are not invariant under a Galilean transformation.
    • Newtonian mechanics is invariant under a Galilean transformation between observation frames (shown).
    • This is called Galilean invariance.

  • Relativistic Momentum

    • Relativistic momentum is given as $\gamma m_{0}v$ where $m_{0}$ is the object's invariant mass and $\gamma$ is Lorentz transformation.
    • This gives rise to Galilean relativity, which states that the laws of motion are the same in all inertial frames.
    • As a result, position and time in two reference frames are related by the Lorentz transformation instead of the Galilean transformation.
    • The Galilean transformation gives the coordinates of the moving frame as
    • Newton's second law [with mass fixed in the expression for momentum (p=m*v)], is not invariant under a Lorentz transformation.

  • Einstein's Postulates

    • (See our previous lesson on "Galilean-Newtonian Relativity. ") One issue, however, was that another well-established theory, the laws of electricity and magnetism represented by Maxwell's equations, was not "invariant" under Galilean transformation—meaning that Maxwell's equations don't maintain the same forms for different inertial frames.

  • Satellites

    • The large gas giants have extensive systems of natural satellites, including half a dozen comparable in size to Earth's Moon: the four Galilean moons, Saturn's Titan, and Neptune's Triton.
    • The seven largest natural satellites in the Solar System (those bigger than 2,500 km across) are Jupiter's Galilean moons (Ganymede, Callisto, Io, and Europa), Saturn's moon Titan, Earth's moon, and Neptune's captured natural satellite Triton.

  • Transformers

    • Transformers transform voltages from one value to another; its function is governed by the transformer equation.
    • Transformers change voltages from one value to another.
    • A step-up transformer is one that increases voltage, whereas a step-down transformer decreases voltage.
    • A symbol of the transformer is also shown below the diagram.
    • Apply the transformer equation to compare the secondary and primary voltages

  • Energy Transformations

    • Energy transformation occurs when energy is changed from one form to another.
    • Often it appears that energy has been lost from a system when it simply has been transformed.
    • When analyzing energy transformations, it is important to consider the efficiency of conversion.
    • Some energy transformations can occur with an efficiency of essentially 100%.
    • Other energy transformations occur with a much lower efficiency of conversion.

  • Lorentz Transformations

  • Some Basic Theorems for the Fourier Transform

    • It is very useful to be able think of the Fourier transform as an operator acting on functions.
    • Convolutions are done often and by going to the frequency domain we can take advantage of the algorithmic improvements of the fast Fourier transform algorithm (FFT).
    • Start by multiplying the two Fourier transforms.
    • If we put the symmetric $1/\sqrt{2 \pi}$ normalization in front of both transforms, we end up with a left-over factor of $1/\sqrt{2 \pi}$ because we started out with two Fourier transforms and we ended up with only one and a convolution.
    • On the other hand, if we had used an asymmetric normalization, then the result would be different depending on whether we put the $1/(2 \pi)$ on the forward or inverse transform.

  • The Discrete Fourier Transform

    • This is the discrete version of the Fourier transform (DFT).
    • In the handout you will see some Mathematica code for computing and displaying discrete Fourier transforms.
    • The reason is that Mathematica uses a special algorithm called the FFT (Fast Fourier Transform).

  • Introduction to The DFT from the Fourier Integral

    • Now that we are considering discrete transforms and real data, we need to make this distinction since we will generally have both the sampled data and its transform stored in arrays on the computer.
    • So for this section we will follow the convention that if $h=h(t)$ then $H=H(f)$ is its Fourier transform.
    • A sensible numerical approximation for the Fourier transform integral is thus:
    • The inverse transform is:
    • So in Mathematica, the forward and inverse transforms are, respectively: