Week 14 page 1 of 5
CT14-1. Which of these problems could be solved using the calculus of
variations?
1. Find the period of small oscillations for a particle sliding (without
friction) on the inside of a sphere.
2. Find the surface with fixed area that encloses the maximum volume.
3. Find the path between two points that minimizes the time for a
particle to slide (without friction) between the points.
4. Find the path of a projectile (with no air resistance) that leads to the
maximum range.
A) None
D) Exactly three
B) Only one
E) All four
C) Exactly two
D. 1 is not a calculus of variations problem, because the answer is not a
path – 1 can be solved easily using regular mechanics. 2 and 3 are classic
CoV problems, where the answer is a path. 4 can be solved either using
regular calculus (after calculating the range) or using CoV.
CT14-2. Which of these paths could be a solution to the brachistochrone
problem?
y
A
B
C
x2,y2
x
D
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Week 14 page 2 of 5
C. The initial steep drop of the path accelerates the particle quickly, but
the path isn’t too long overall. The brachistochrone (the fastest path)
optimizes the balance between a longer path, which allows a steeper initial
drop, and a shorter path, which makes the overall transit time not too long.
CT14-3. If the particle starts
from rest at x=y=0 and slides
under the gravitational force
(without friction), what is the
relationship between x, y, and
the speed of the particle?
1 2
mv  mgy  const.
2
1 2
B) mv  mgy  const.
2
1 2
C) mv  mgx  const.
2
y
x2,y2
A)
x
D)
1 2
mv  mgx  const.
2
C. The speed increases as x increases, but is independent of y.
CT14-4. Which of these problems could be solved using the calculus of
variations with constraints?
1. Find the shortest path between two points on the surface of a cylinder.
2. Find the surface with fixed area that encloses the maximum volume.
3. Find the path between two points that minimizes the time for a
particle to slide (without friction) between the points.
4. Find the path between two points that minimizes the area of the
surface of revolution (formed by rotating the path around an axis).
A) None
D) Exactly three
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B) Only one
E) All four
C) Exactly two
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Week 14 page 3 of 5
C. 1 and 2 can be solved using CoV with constraints. 3 and 4 are also
calculus of variations problems, but can be formulated without constraint
functions.
y
CT14-5. To solve Dido’s problem, what is
the function that must be maximized?
A) J  y  
B) J  y  
C) J  y  
 y  
a

1  y2 dx
a
x
a
  y  y  dx
2
a
a
 y
a
2

  1  y  dx
2
D) J  y  

a

1  y2  y dx
a
A. The integral of y is the area under the curve. The second term is the
Lagrange multiplier which enforces the fixed perimeter constraint.
CT14-6. What is the Lagrangian of a particle
(mass m) attached to a spring (spring
constant k)?
1
1
L
x
,
x
,
t

kx

mx


A)
2
2
B) L  x , x , t   mx  kx
1
1
L
x
,
x
,
t

mx

kx


C)
2
2
D) L  x , x , t   mx
m
2
2
2
2
2
2
2
 kx 2
1
L
x
,
x
,
t

mx


E)
2
2
1
 kx 2
2
E. The Lagrangian is the kinetic energy minus the potential energy.
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Week 14 page 4 of 5
CT14-7. What is the Lagrangian of a pendulum (mass m,
length l)? Assume the potential energy is zero when  is zero.
1
L

,

,
t

m


A)

  mg 1  cos 
2
1 2 2
L

,

,
t

m   mg 1  cos 
B)
2
1 2
L

,

,
t

m  mg 1  cos 
C)
2
1 2
L

,

,
t

m  mg 1  cos 
D)
2






2 2
m
A. A and B have the correct kinetic energy term, while A and D have the
correct potential energy term (the correct sign).
CT14-8. Which of these constraints is holonomic?
1.
2.
3.
4.
A particle is constrained to slide on the inside of a sphere.
A cylinder rolls down an inclined plane.
Two moving particles are connected by a rod of length l.
A moving car is constrained to obey the speed limit.
A) None
D) Exactly three
B) Only one
E) All four
C) Exactly two
D. Only 4 is nonholonomic, because it involves the velocity of the object.
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Week 14 page 5 of 5
CT14-9. For which of these systems could you use
Lagange’s equations of motion?
1. A double pendulum: a pendulum (mass m, length l) has
a second pendulum (mass m, length l) connected to its
bob.
2. A projectile moves in two dimensions with gravity and
air resistance.
3. A bead slides without friction on a circular, rotating
wire.
A) 1 only
D) 1 and 2
B) 2 only
E) 1 and 3
C) 3 only
E. 2 must be excluded because the problem has air
resistance. 1 and 3 are both good candidates for Lagrange’s
equations.
CT14-10. What would be a good choice of generalized
coordinates for the double pendulum? (Assume the pendula
are constrained to move in the x-y plane.)
A) the Cartesian coordinates of the bob positions: x1, y1
(first bob) and x2, y2 (second bob)
B) the Cartesian coordinates of the first bob: x1, y1 and the
Cartesian coordinates of the second bob, treating the
first bob as the origin: x’2, y’2
C) the angles made between each pendulum rod and the
vertical: 1, (first bob) 2 (second bob)
D) the angles made between a line drawn from each pendulum bob to the
pivot and the vertical: 1, (first bob) 2 (second bob)
C. These angles will make the problem easiest to solve.
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