Galilean transformation

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.

• Europa's Possible Ocean

  • Europa, discovered in 1610 by Galileo Galilei, is one of Jupiter's four moons (called the Galilean moons) .
  • In other words, in a similar fashion to the tides flowing in and out on Earth due to the moon's gravitational pull, the tides on Europa are affected due to its orbit around Jupiter and possibly also its orbital resonance with other Galilean moons.

• 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.

• Linear Transformations

  • Often it is necessary to transform data from one measurement scale to another.
  • To transform feet to inches, you simply multiply by 12.
  • Similarly, to transform inches to feet, you divide by 12.
  • The transformation consists of multiplying by a constant and then adding a second constant.
  • Such transformations are therefore called linear transformations.

• Transactional Versus Transformational Leaders

  • Transactional leaders are concerned about the status quo, while transformational leaders are more change-oriented.
  • Leadership can be described as transactional or transformational.
  • Transformational leaders work to enhance the motivation and engagement of followers by directing their behavior toward a shared vision.
  • Transactional leaders work within existing an organizational culture, while transformational leaders emphasize new ideas and thereby "transform" organizational culture.
  • Differentiate between transactional leaders and transformational leaders in a full-range approach, particularly from a behavioral perspective

• Exercises

  • If the arithmetic mean of log10 transformed data were 3, what would be the geometric mean?
  • Using Tukey's ladder of transformation, transform the following data using a λof 0.5: 9, 16, 25
  • In the ADHD case study, transform the data in the placebo condition (D0) with λ's of .5, 0, -.5, and -1.
  • Which transformation leads to the least skew?

• 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

• Log Transformations

  • State how a log transformation can help make a relationship clear
  • The log transformation can be used to make highly skewed distributions less skewed.
  • Figure 1 shows an example of how a log transformation can make patterns more visible.
  • The raw weights are shown in the upper panel; the log-transformed weights are plotted in the lower panel.
  • The comparison of the means of log-transformed data is actually a comparison of geometric means.