Ahodograph is adiagram that gives avectorial visual representation of the movement of abody or afluid. It is thelocus of one end of a variable vector, with the other end fixed.^[1] The position of any plotted data on such a diagram is proportional to thevelocity of the moving particle.^[2] It is also called avelocity diagram. It appears to have been used byJames Bradley, but its practical development is mainly from SirWilliam Rowan Hamilton, who published an account of it in theProceedings of the Royal Irish Academy in 1846.^[2]

It is used inphysics,astronomy,solid andfluid mechanics to plot deformation of material, motion of planets or any other data that involves the velocities of different parts of a body.

Inmeteorology, hodographs are used to plotwinds from soundings of theEarth's atmosphere. It is a polar diagram where wind direction is indicated by the angle from the center axis and its strength by the distance from the center. In the figure to the right, at the bottom one finds values of wind at 4 heights above ground. They are plotted by thevectors${\displaystyle {\overrightarrow{V}}_0}$ to${\displaystyle {\overrightarrow{V}}_4}$. One has to notice that direction are plotted as mentioned in the upper right corner.

With the hodograph andthermodynamic diagrams like thetephigram, meteorologists can calculate:

• Wind shear: The lines uniting the extremities of successive vectors represent the variation in direction and value of the wind in a layer of the atmosphere.Wind shear is important information in the development ofthunderstorms and future evolution of wind at these levels.
• Turbulence: wind shear indicates possibleturbulence that would cause ahazard toaviation.
• Temperature advection: change oftemperature in a layer of air can be calculated by the direction of the wind at that level and the direction of the wind shear with the next level. In the northern hemisphere, warm air is to the right of the wind shear vector between levels in the atmosphere. The opposite is true in the southern hemisphere (seethermal wind). Using the example hodograph, the wind${\displaystyle {\overrightarrow{V}}_3}$ from the southwest meets the right side of the wind shear vector which means warmadvection is occurring and thus the air is warming at that level.

It is a method of presenting the velocity field of a point in planar motion. The velocity vector, drawn at scale, is shown perpendicular rather than tangent to the point path, usually oriented away from the center of curvature of the path.^[3]

Hodograph transformation is a technique used to transform nonlinearpartial differential equations into linear version. It consists of interchanging the dependent and independent variables in the equation to achieve linearity.^[4]

• Visual calculus, a related approach useful in solving a variety of integral calculus problems.
• Laplace–Runge–Lenz vector, for an example in solving the Kepler problem

• Hamilton, William Rowan. "The Hodograph, or a New Method of Expressing in Symbolic Language the Newtonian Law of Attraction",Proceedings of the Royal Irish Academy, Vol. 3 (1847), pp. 344–353. Edited by David R. Wilkins (2000).
• In his bookMatter and Motion, Maxwell writes:

  The study of the hodograph, as a method of investigating the motion of a body, was introduced by Sir W. R. Hamilton. The hodograph may be defined as the path traced out by the extremity of a vector which continually represents, in direction and magnitude, the velocity of a moving body. In applying the method of the hodograph to a planet, the orbit of which is in one plane, we shall find it convenient to suppose the hodograph turned round its origin through a right angle, so that the vector of the hodograph is perpendicular instead of parallel to the velocity it represents.

  and he applies these techniques to analyseKepler's first and second laws.
  Free "Matter and Motion" e-books are available on the Internet.
• Feynman's Lost Lecture: The Motion of Planets Around the Sun byDavid Goodstein &Judith R. Goodstein (ISBN 0-393-03918-8, W. W. Norton & Company: New York, 1996). In this book the hodograph is used to geometrically derive elliptical (Keplerian) orbits from Newton's laws of motion and gravitation.

• The Hodograph - Dr. James B. Calvert, University of Denver

• Classical mechanics
• Meteorological diagrams
• Wind

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