The Hodograph

Contents

Introduction

The hodograph is a plot of the velocity of a particle as a function of time. The velocity is represented by the vector from the origin to a point on the hodograph. If the velocity is known as a function of the time, then it is easy to pass from the hodograph to the trajectory by a simple integration, since dx/dt = v x (t) and dy/dt = v y (t). The hodograph is the solution of the first-order equation dv/dt = f, which is Newton's Law. If f, the ratio of the force to the mass of the particle, is given as a function of t, then v is easily found. In most cases, f is a function of the position of the particle, which makes the problem more difficult. If the trajectory is known, then it is easy to find the hodograph by differentiation with respect to time. In most Physics courses, particle mechanics is based on the trajectory, while the hodograph is ignored, which is a shame. In this article, I shall present as many uses of the hodograph as I can find, beginning with the simplest problems and adding others later as I find them.

Hodograph is from Greek "hodos" and "grapho," meaning path-writing. Hodos is the general word for road or path, beginning with an "o" with rough breathing (if you want to look it up in a lexicon). In form, it looks masculine, but is of feminine gender. It is not a particularly well-chosen term ("tachygraph" might have been better) but it is the traditional one. The hodograph was invented and named by William Rowan Hamilton, the brilliant Irish mathematical physicist, inventor of quaternions and Hamiltonian mechanics, in 1847, and was used by James Clerk-Maxwell to analyze orbital motion and to show in an easy way that Kepler's Laws imply an inverse-square law of force.

The hodograph is not mentioned in introductory Physics texts (for example, Halliday and Resnick or Sears, Zemansky and Young), nor in advanced classical mechanics texts (Goldstein, Corben and Stehle). It is defined, but not further used, in A. P. French and in the engineering dynamics text of Beer and Johnston (see References). It does not appear in the McGraw-Hill Encyclopedia of Physics, 2nd ed., or even in compendious British sixth-form General Physics texts (Brown). It is not defined in standard dictionaries (Concise Oxford, 4th ed.; Webster's Seventh New Collegiate Dictionary). Webster's does not even bother with "hodometer." The Cambridge Dictionary of Science and Technology has a short definition on p. 431, which can be quoted in its entirety: "A curve used to determine the acceleration of a particle moving with known velocity on a curved path. The hodograph is drawn through the ends of vectors drawn from a point to represent the velocity of the particle at successive instants." There is not much danger of learning anything from this definition. Therefore, this article may well supply a need.

Motion with Constant Acceleration

The usual case of motion with constant acceleration is motion under the influence of gravity, an elementary problem of great importance that is treated in every introductory Physics course. The equation of the hodograph is then dv/dt = g, which integrates to the straight line gt, whose length is proportional to the time of the motion. If the initial velocity v o is drawn from the origin to a point P, then the hodograph is a vertical line lengthening downwards. The velocity at any time is easily read off, and if the motion ends at a point Q, the corresponding velocity is v f = v o + gt, making a triangle OPQ with side PQ vertical. The trajectory, of course, is a parabolic arc that is more difficult to work with than the simple triangle. These things are illustrated in the diagram at the right.

The area of the triangle is half the base, gt, times the altitude v o cos θ, where θ is the angle between the initial velocity and the horizontal, the angle of projection. Also, gt/2 is v o sin θ. Eliminating v o , we find the altitude to be gt/(2 tan θ), and its area (gt/2)2/tan θ. This area is proportional to the horizontal distance covered by the trajectory, since the altitude is the (constant) horizontal component of the initial velocity, and the base is proportional to the time. The proportionality constant of the area of the triangle to the horizontal distance covered is g/2, of course.

To find the condition for the maximum range as a function of the angle of projection, differentiate the area with respect to θ and set the derivative equal to zero. This gives cos 2θ = 0, or 2θ = 90°, or θ = 45°, the well-known result. Therefore, the initial and final velocities are perpendicular, so that the hodograph is an isosceles triangle.

For a particle projected directly upwards, the hodograph will be a vertical line of length 2v with its centre at the origin. The time of flight t is given by 2v = gt, or t = 2v/g. At time t/2, we are at the origin, and the velocity is zero. The area of the triangle is zero, meaning that the particle goes nowhere horizontally, as we know. The hodograph of a particle moving with constant velocity is a point, and the corresponding trajectory is a straight line. These simple relations are easy to visualize, and help to illustrate the meaning of the hodograph.

The hodograph of a uniform circular motion with angular velocity ω and radius r is a circle of radius v = rω, the peripheral velocity, with its centre at the origin. The hodograph is a quarter-revolution ahead of the trajectory, which is also a circle, of radius r. The circular hodograph implies steady rotation. These relations are shown in the figure at the right. The constant magnitude of the acceleration is v2/r = ω2r, directed toward the centre of the trajectory circle and perpendicular to v. Note that this centripetal acceleration is in the same direction in the hodograph and the trajectory. The radii of the circles differ by a factor of ω.

The hodograph of a simple harmonic motion is shown at the left, with the time dependence displayed. The amplitude of the motion is a and its angular frequency is ω = 2π/T = 2πf, where T is the period and f is the frequency. The velocity also varies sinusoidally, with an amplitude ωa. Points P and Q correspond to the passage of the particle through y = 0, and the centre of PQ to the extreme positions.

The Brachistochrone

This is a famous problem of constrained motion under gravity. The problem was published by Johann Bernoulli (1667-1748) in 1696, and was solved by him, Isaac Newton (1642-1727) and G. W. von Leibniz (1646-1716) shortly afterwards. It was the stimulus for the development of the calculus of variations by Euler and Lagrange, in which an unknown function is determined that minimizes a definite integral. The name is well-chosen from Greek, from "brachistos," shortest, and "chronos," time.

The particle is imagined to move from point A to a second point B at the same level, but some distance away, sliding on a certain curve without friction. It is clear that the particle will move away from A if the path slants downward in the direction of B, speeding up until it reaches the lowest point, then with decreasing velocity until it just reaches B. If the path is the straight horizontal line AB, the particle will never start moving. The particle could be dropped from A, and when it attained a certain velocity by dropping a distance h, be deflected horizontally in the direction of B. When it was directly under B, it would again be deflected upwards, finally stopping when it just reaches B. In this case, the minimum time required is T = 2√(2s/g), if s is the horizontal distance from A to B. T is infinite for h = 0 or for h = ∞, so there must be a minimum in between. The value h = s/4 gives the minimum T, which can be found by minimizing the expression for T with respect to h.

A more practical path would be a straight inclined ramp for half the distance on each end. The effective acceleration on a ramp making an angle θ with the horizontal is g sin θ, so it is easy to calculate the total time in this case as well. The answer is the same, 2√(2s/g), for a depth at the centre of h = s/2 (a 45° ramp). Suppose we were using this for a rapid transit system, and two stations were 1000 m apart. In this case, √(2s/g) = 14.3 s, so the total time would be 28.6 s, and the average speed 35 m/s or 142 km/h or 89 mph. This would be rapid transit indeed, with the maximum speed no less than 223 mph (the speed attained by falling 500 m). Friction, however, stands in the way of this energy-efficient transport system. Proposals have been made to use gravity for this purpose, but no bold ones have been realized.

If the path is given by the function y = y(x), then the speed at any point x is v = √(2gy), and the path length ds = √(1 + y'2) dx, where y' = dy/dx, so the time required to go from 0 to x can be expressed as the definite integral T = ∫(0,x) [√(1 + y'2)/√(2gy) dx. We search for the path y(x) that gives a minimum value for the integral.

What sort of path would give us the absolute minimum time? We can anticipate that it would be a smooth path, replacing the linear ramps and abrupt falls and rises, and that the minimum depth would be between s/4 and s/2. Bernoulli, Newton and Leibniz found the surprising result that the path is a cycloid, and the maximum depth is s/π. The time required is √π√(2s/g), so √π = 1.7725 replaces 2. The time for our rapid transit is then 25.3 s instead of 28.6 s. This path is the brachistochrone, which gives a shorter time than any neighboring path. It is clear that since the time is a minimum for the brachistochrone, the time will not be greatly different for any approximate path. Even the ramps give only a 13% difference.

A parametric equation for the cycloidal path is x = (s/2π)(θ - sin θ), y = -(s/2π)(1 - cos θ). The angle θ is the angle of rotation of a circle of radius s/2π rolling on the underside of the x-axis, and goes from 0 to 2π. This curve is shown at the right, with the generating circle indicated. It is also the path of an electron starting from rest at A in an electric field in the direction of -x, and a magnetic field out of the plane, in the z-direction. It is not easy to find this result with the usual methods of the calculus of variations because of algebraic complexity. However, the hodograph offers an easier solution.

The hodograph will begin at the origin, sloping downward as the particle picks up speed. Note that the acceleration is parallel to the path, not directly downward as in unconstrained motion. The hodograph will then loop around, reaching the horizontal axis at the minimum of the curve, where the velocity will be greatest, then looping around above as velocity decreases, and finally approaching the origin with a downward slope. Integrals can be written for the area of the hodograph (time of the motion), as well as for the horizontal distance covered, in terms of a hodograph u(θ). [This is not the parameter θ of the preceding paragraph, but the inclination of the velocity vector.] The area of the hodograph will be proportional to the time of the motion, so we look for a curve that will minimize the area inside the hodograph, while covering the necessary horizontal distance s. The first integral is to be minimized, with the second as a condition, for which the method of the Lagrange multiplier is convenient. Apostulatos (see References) shows that the hodograph will be a circle tangent to the vertical axis at the origin. It is, in fact, v(θ) = √(2gs/π) cos θ. The coefficient of the cosine is the diameter of the circular hodograph.

If we add a constant velocity to the left so that the hodograph becomes a circle centred at the origin, that is, a velocity √(gs/2π), we get the hodograph of a uniform circular motion. Therefore, the actual motion is a circular motion translating to the right at a speed equal to the peripheral speed of the circular motion. This is, of course, a cycloid. The total time is then the distance s covered by the rolling circle divided by its peripheral velocity, √(gs/2π), which gives the expression quoted above.

Planetary Orbits

The first two of Kepler's Laws of Planetary Motion are: (1) the planet moves in an ellipse with the sun at one focus; and (2) the area swept out by the radius vector is the same in equal time intervals. An elliptical orbit is shown at the right, in which the planet is located by its radius vector r and the true anomaly θ. The velocity is greatest at perihelion and least at aphelion, and in both cases is perpendicular to the radius, which attains a minimum and maximum value, respectively. In between, there is a component of the velocity dr/dt parallel to the radius. The quantity p is the semi-latus-rectum, and the mean distance, or semimajor axis of the ellipse, is a, given by p = a (1 - e2), where e is the eccentricity, and 0 ≤ e < 1. The focus is displaced by a distance c = ae from the centre of the ellipse.

The polar equation of the orbit is shown at the left, together with the mathematical expression of the Law of Areas. The Law of Areas is equivalent to the conservation of angular momentum, which is proportional to the constant h. These two expressions are derived directly from Kepler's Laws. If we differentiate r with respect to time, we can find the radial and transverse components of the velocity. The parameter θ is the true anomaly, and the radial and transverse components are relative to this direction. Plotting these components will give us the hodograph of the motion.

The hodograph is shown at the right. It is, remarkably, a circle with centre at O, which does not coincide with the origin unless e = 0 and the orbit is a circle. To draw the hodograph, all that is necessary are the velocities at perihelion and aphelion. These are laid off above and below the velocity origin, and form the diameter of the hodograph. The velocities marked 90° and 270° are the velocities at the ends of the latus rectum. Note that they fall at the intersections of a horizontal line through O with the hodograph. If we add a downward velocity of he/p, the centre O is brought to the origin, and the hodograph of this motion is a uniform circular motion. This fact can be used to solve the problem of the position in orbit as a function of time (Kepler's Problem), and the angle of rotation is the mean anomaly, which increases uniformly with time. The details will not be mentioned here, but the relation of the method to the hodograph is clear from what has been said.

Bringing in Kepler's Third Law, that the cubes of the mean distances are proportional to the squares of the orbital periods, allows the determination of the law of force, which turns out to be the familar inverse square. To obtain this result, note that the acceleration in the circular motion is the total acceleration, since the velocity he/p is a constant, and it is easy to find the force for a uniform circular motion. The reader is encouraged to solve the problem of a circular orbit under an inverse-square force, and from it to show that Kepler's Third Law results. Then, it is not difficult to see how to prove the inverse, which is the desired conclusion.

References

T. A. Apostulatos, Hodograph: A useful geometrical tool for solving some difficult problems in dynamics, American Journal of Physics, 71(3) 261-266 (2003).

R. Courant, Differential and Integral Calculus (London: Blackie & Son, 1936). Vol. II, pp. 491-521. Calculus of variations and the brachistochrone problem.

C. Lanczos, The Variational Principles of Mechanics (Toronto: University of Toronto Press, 1949). Still the best explanation of finding extrema and the calculus of variations. The brachistochrone is only mentioned, not analyzed.

F. P. Beer and E. R. Johnston, Jr., 4th ed., Vector Mechanics for Engineers (New York: McGraw-Hill, 1984), p. 447.

A. P. French, Newtonian Mechanics (New York: W. W. Norton, 1971), p. 103.

Composed by J. B. Calvert

Created 8 March 2003

Last revised 10 March 2003