Hodograph

Hodograph (Gr. ὁδός, a way, and γράφειν, to write), a curve of which the radius vector is proportional to the velocity of a moving particle. It appears to have been used by James Bradley, but for its practical development we are mainly indebted to Sir William Rowan Hamilton, who published an account of it in the Proceedings of the Royal Irish Academy, 1846. If a point be in motion in any orbit and with any velocity, and if, at each instant, a line be drawn from a fixed point parallel and equal to the velocity of the moving point at that instant, the extremities of these lines will lie on a curve called the hodograph. Let PP1P2 be the path of the moving point, and let OT, OT1, OT2, be drawn from the fixed point O parallel and equal to the velocities at P, P1, P2 respectively, then the locus of T is the hodograph of the orbits described by P (see figure). From this definition we have the following important fundamental property which belongs to all hodographs, viz. that at any point the tangent to the hodograph is parallel to the direction, and the velocity in the hodograph equal to the magnitude of the resultant acceleration at the corresponding point of the orbit. This will be evident if we consider that, since radii vectores of the hodograph represent velocities in the orbit, the elementary arc between two consecutive radii vectores of the hodograph represents the velocity which must be compounded with the velocity of the moving point at the beginning of any short interval of time to get the velocity at the end of that interval, that is to say, represents the change of velocity for that interval. Hence the elementary arc divided by the element of time is the rate of change of velocity of the moving-point, or in other words, the velocity in the hodograph is the acceleration in the orbit.

Analytically thus (Thomson and Tait, Nat. Phil.):—Let x, y, z be the coordinates of P in the orbit, ξ, η, ζ those of the corresponding point T in the hodograph, then

ξ = | dx | ,  η = | dy | ,  ζ = | dz | ;  
---|---|---|---|---|---|---  
dt | dt | dt  
  
therefore

dξ | = | dη | = | dζ  
---|---|---|---|---  
d²x/dt² | d²y/dt² | d²z/dt²  
  
(1).

Also, if s be the arc of the hodograph,

ds | = v = √ [( | dξ | ) | ² | \+ ( | dη | ) | ² | \+ ( | dζ | ) | ² | ]  
---|---|---|---|---|---|---|---|---|---|---|---|---|---  
dt | dt |   | dt |   | dt |    
= √ [( | d²x | ) | ² | \+ ( | d²y | ) | ² | \+ ( | d²z | ) | ² | ]  
---|---|---|---|---|---|---|---|---|---|---|---|---  
dt² |   | dt² |   | dt² |    
  
(2).

Equation (1) shows that the tangent to the hodograph is parallel to the line of resultant acceleration, and (2) that the velocity in the hodograph is equal to the acceleration.

Every orbit must clearly have a hodograph, and, conversely, every hodograph a corresponding orbit; and, theoretically speaking, it is possible to deduce the one from the other, having given the other circumstances of the motion.

For applications of the hodograph to the solution of kinematical problems see Mechanics.

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