THE BRACHISTOCHRONE PROBLEM; CYCLOIDS
4. THE BRACHISTOCHRONE PROBLEM; CYCLOIDS
We have already mentioned this problem in Section 1. We are given the points (x 1 ,y 1 ) and (x 2 ,y 2 ); we choose axes through the point 1 with the y axis positive downward as shown in Figure
4.1. Our problem is to find the curve joining the two points, down which a bead will slide (from rest) in the least time; that is, we want to minimize
be our reference level for potential energy. Then at the point (x, y) we have
Figure 4.1
kinetic energy = mv 2 = 1 m
2 2 dt potential energy = −mgy.
The sum of the two energies is zero initially and therefore zero at any time since the total energy is constant when there is no friction. Hence we have
mv 2 − mgy = 0 or v =
Then the integral which we want to minimize is
1 x 2 1+y ′2 dt =
ds
ds
dx.
2gy
2g x 1 y
Section 4 The Brachistochrone Problem; Cycloids 483
This is the integral (3.5) in Example 3, Section 3. Then the first integral of the Euler equation is given by (3.6):
√ = c.
x ′2 +1
Solving for x ′ , we get (4.1)
1 − cy This simplifies if we let cy = sin 2θ 2 = 1 2 (1 − cos θ). We find (Problem 1)
The equations for x and y as functions of θ are parametric equations of the curve along which the particle slides in minimum time. Since we have chosen axes to make the curve pass through the origin, x = y = 0 must satisfy the equations of the curve, so c ′ = 0, and we have
We shall now show that these are the parametric equations of a cycloid. Imagine
a circle of radius a (say a wheel) in the (x, y) plane rolling along the x axis. Let it start tangent to the x axis at the origin O in Figure 4.2. Place a mark on the circle at O. As the circle rolls, the mark traces out a cycloid as shown in Figure 4.3. Let point P in Figure 4.2 be the position of the mark when the circle is tangent to the x axis at A; let (x, y) be the coordinates of P . Since the circle rolled, OA = P A = aθ with θ in radians. Then from Figure 4.2 we have
x = OA − P B = aθ − a sin θ = a(θ − sin θ), (4.4)
y = AB = AC − BC = a − a cos θ = a(1 − cos θ).
Figure 4.3
Figure 4.2
484 Calculus of Variations Chapter 9
Equations (4.4) are the parametric equations of a cycloid. Comparing (4.3), we see that the brachistochrone is a cycloid as we claimed. Note that, since we have taken the y axis positive down (Figures 4.1 and 4.4), the circle which generates the brachistochrone rolls along the under side of the x axis.
From either (4.3) or (4.4), we see that all cycloids are similar; that is they differ from each other only in size (determined by a or c) and not in shape. Figure 4.4 is
a sketch of a cycloid for arbitrary a. If the given endpoints for the wire along which
the bead slides are O and P 3 , we see that
the particle slides down to P 2 and back up
Figure 4.4
to P 3 in minimum time! At point P 2 the
circle has rolled halfway around so OA = 1 2 · 2πa = πa. For any point P 1 on arc OP 2 ,P 1 is below the line OP 2 , and the coordinates (x 1 ,y 1 ) of P 1 have
or x 1 /y 1 < π/2. For points like P 3 on P 2 3 /y B, x 3 > π/2, whereas at P 2 , we have x 2 /y 2 = π/2 (Problem 2). Then if the right-hand endpoint is (x, y) and the origin is the left-hand endpoint, we can say that the bead just slides down, or slides down and back up, depending on whether x/y is less than or greater than π/2 (Problem 2).
PROBLEMS, SECTION 4
1. Verify equations (4.2). 2. Show, in Figure 4.4, that for a point like P 3 ,x 3 /y 3 > π/2 and for P 2 ,x 2 /y 2 = π/2. 3. In the brachistochrone problem, show that if the particle is given an initial velocity
v 0 4. Consider a rapid transit system consisting of frictionless tun-
nels bored through the earth between points A and B on the earth’s surface (see figure). The unpowered passenger trains would move under gravity. Using polar coordinates, set up R dt to be minimized to find the path through the earth requiring the least time. See Chapter 6, Problem 8.21, for the potential inside the earth. Find a first integral of the Euler equation. Evaluate the constant
of integration using dr/dθ = 0 when r = r 0 (where r 0 is the deepest point of the tunnel—see figure). Now solve for θ ′ = dθ/dr as a function of r. Substitute this into the integral for t and evaluate the integral to show that the transit time is
s R 2 −r 0 2 Z T=π R . Hint: Find 2 dt.
gR
r=r 0
Evaluate T for r 0 = 0 (path through the center of the earth—see Chapter 8, Problem 5.35); for r 0 = 0.99R. [For more detail, see Am. J. Phys. 34 701–704 (1966).]
In Problems 5 to 7, use Fermat’s principle to find the path followed by a light ray if the index of refraction is proportional to the given function.
5. − x 1/2
6. − (y − 1) 1/2 7. (2x + 5) 1/2
Section 5 Several Dependent Variables; Lagrange’s Equations 485
Parts
» CONVERGENCE TESTS FOR SERIES OF POSITIVE TERMS; ABSOLUTE CONVERGENCE
» CONDITIONALLY CONVERGENT SERIES
» POWER SERIES; INTERVAL OF CONVERGENCE
» EXPANDING FUNCTIONS IN POWER SERIES
» TECHNIQUES FOR OBTAINING POWER SERIES EXPANSIONS
» ACCURACY OF SERIES APPROXIMATIONS
» COMPLEX POWER SERIES; DISK OF CONVERGENCE
» ELEMENTARY FUNCTIONS OF COMPLEX NUMBERS
» POWERS AND ROOTS OF COMPLEX NUMBERS
» THE EXPONENTIAL AND TRIGONOMETRIC FUNCTIONS
» LINEAR COMBINATIONS, LINEAR FUNCTIONS, LINEAR OPERATORS
» LINEAR DEPENDENCE AND INDEPENDENCE
» SPECIAL MATRICES AND FORMULAS
» EIGENVALUES AND EIGENVECTORS; DIAGONALIZING MATRICES
» APPLICATIONS OF DIAGONALIZATION
» A BRIEF INTRODUCTION TO GROUPS
» POWER SERIES IN TWO VARIABLES
» APPROXIMATIONS USING DIFFERENTIALS
» CHAIN RULE OR DIFFERENTIATING A FUNCTION OF A FUNCTION
» APPLICATION OF PARTIAL DIFFERENTIATION TO MAXIMUM AND MINIMUM PROBLEMS
» MAXIMUM AND MINIMUM PROBLEMS WITH CONSTRAINTS; LAGRANGE MULTIPLIERS
» ENDPOINT OR BOUNDARY POINT PROBLEMS
» DIFFERENTIATION OF INTEGRALS; LEIBNIZ’ RULE
» APPLICATIONS OF INTEGRATION; SINGLE AND MULTIPLE INTEGRALS
» CHANGE OF VARIABLES IN INTEGRALS; JACOBIANS
» APPLICATIONS OF VECTOR MULTIPLICATION
» DIRECTIONAL DERIVATIVE; GRADIENT
» SOME OTHER EXPRESSIONS INVOLVING ∇
» GREEN’S THEOREM IN THE PLANE
» THE DIVERGENCE AND THE DIVERGENCE THEOREM
» THE CURL AND STOKES’ THEOREM
» SIMPLE HARMONIC MOTION AND WAVE MOTION; PERIODIC FUNCTIONS
» APPLICATIONS OF FOURIER SERIES
» COMPLEX FORM OF FOURIER SERIES
» LINEAR FIRST-ORDER EQUATIONS
» OTHER METHODS FOR FIRST-ORDER EQUATIONS
» SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS AND ZERO RIGHT-HAND SIDE
» SECOND-ORDER LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS AND RIGHT-HAND SIDE NOT ZERO
» OTHER SECOND-ORDER EQUATIONS
» SOLUTION OF DIFFERENTIAL EQUATIONS BY LAPLACE TRANSFORMS
» A BRIEF INTRODUCTION TO GREEN FUNCTIONS
» THE BRACHISTOCHRONE PROBLEM; CYCLOIDS
» SEVERAL DEPENDENT VARIABLES; LAGRANGE’S EQUATIONS
» TENSOR NOTATION AND OPERATIONS
» KRONECKER DELTA AND LEVI-CIVITA SYMBOL
» PSEUDOVECTORS AND PSEUDOTENSORS
» VECTOR OPERATORS IN ORTHOGONAL CURVILINEAR COORDINATES
» ELLIPTIC INTEGRALS AND FUNCTIONS
» GENERATING FUNCTION FOR LEGENDRE POLYNOMIALS
» COMPLETE SETS OF ORTHOGONAL FUNCTIONS
» NORMALIZATION OF THE LEGENDRE POLYNOMIALS
» THE ASSOCIATED LEGENDRE FUNCTIONS
» GENERALIZED POWER SERIES OR THE METHOD OF FROBENIUS
» THE SECOND SOLUTION OF BESSEL’S EQUATION
» OTHER KINDS OF BESSEL FUNCTIONS
» ORTHOGONALITY OF BESSEL FUNCTIONS
» SERIES SOLUTIONS; FUCHS’S THEOREM
» HERMITE FUNCTIONS; LAGUERRE FUNCTIONS; LADDER OPERATORS
» LAPLACE’S EQUATION; STEADY-STATE TEMPERATURE IN A RECTANGULAR PLATE
» THE DIFFUSION OR HEAT FLOW EQUATION; THE SCHR ¨ODINGER EQUATION
» THE WAVE EQUATION; THE VIBRATING STRING
» STEADY-STATE TEMPERATURE IN A CYLINDER
» VIBRATION OF A CIRCULAR MEMBRANE
» STEADY-STATE TEMPERATURE IN A SPHERE
» INTEGRAL TRANSFORM SOLUTIONS OF PARTIAL DIFFERENTIAL EQUATIONS
» EVALUATION OF DEFINITE INTEGRALS BY USE OF THE RESIDUE THEOREM
» THE POINT AT INFINITY; RESIDUES AT INFINITY
» SOME APPLICATIONS OF CONFORMAL MAPPING
» THE NORMAL OR GAUSSIAN DISTRIBUTION
» STATISTICS AND EXPERIMENTAL MEASUREMENTS
Show more