Department of Mathematics
Monterey, California 93943
1
,Contents
1 Functions of n Variables 1
1.1 Unconstrained Minimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Constrained Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Examples, Notation 10
2.1 Notation & Conventions ............................................................................................. 13
2.2 Shortest Distances ....................................................................................................... 14
3 First Results 21
3.1 Two Important Auxiliary Formulas: .......................................................................... 22
3.2 Two Important Auxiliary Formulas in the General Case.......................................... 26
4 Variable End-Point Problems 36
4.1 The General Problem .................................................................................................. 38
4.2 Appendix....................................................................................................................... 41
5 Higher Dimensional Problems and Another Proof of the Second Euler
Equation 46
5.1 Variational Problems with Constraints ...................................................................... 47
5.1.1 Isoparametric Problems ..................................................................................... 47
5.1.2 Point Constraints ................................................................................................ 51
6 Integrals Involving More Than One Independent Variable 59
7 Examples of Numerical Techniques 63
7.1 Indirect Methods .......................................................................................................... 63
7.1.1 Fixed End Points................................................................................................ 63
7.1.2 Variable End Points ........................................................................................... 71
7.2 Direct Methods ............................................................................................................ 74
8 The Rayleigh-Ritz Method 82
8.1 Euler’s Method of Finite Differences .......................................................................... 84
9 Hamilton’s Principle 90
10 Degrees of Freedom - Generalized Coordinates 97
11 Integrals Involving Higher Derivatives 104
12 Piecewise Smooth Arcs and Additional Results 110
13 Field Theory Jacobi’s Neccesary Condition and Sufficiency 116
i
,List of Figures
1 Neighborhood S of X0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Neighborhood S of X0 and a particular direction H . . . . . . . . . . . . . . 2
3 Two dimensional neighborhood of X0 showing tangent at that point . . . . . 5
4 The constraint φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
5 The surface of revolution for the soap example ........................................................ 11
6 Brachistochrone problem ............................................................................................. 12
7 An arc connecting X1 and X2 ................................................................................................................................. 15
8 Admissible function η vanishing at end points (bottom) and various admissible
functions (top) .............................................................................................................. 15
9 Families of arcs y0 + νη .............................................................................................. 17
10 Line segment of variable length with endpoints on the curves C, D ........................ 22
11 Curves described by endpoints of the family y(x, b) ................................................. 27
12 Cycloid .......................................................................................................................... 29
13 A particle falling from point 1 to point 2 .................................................................. 29
14 Cycloid .......................................................................................................................... 32
15 Curves C, D described by the endpoints of segment y34 ............................................................. 33
16 Shortest arc from a fixed point 1 to a curve N. G is the evolute ........................... 36
17 Path of quickest descent, y12, from point 1 to the curve N ...................................... 40
18 Intersection of a plane with a sphere......................................................................... 56
19 Domain R with outward normal making an angle ν with x axis............................. 61
20 Solution of example given by (14) .............................................................................. 71
21 The exact solution (solid line) is compared with φ0 (dash dot), y1 (dot) and
y2 (dash) ....................................................................................................................... 85
22 Piecewise linear function.............................................................................................. 86
23 The exact solution (solid line) is compared with y1 (dot), y2 (dash dot), y3
(dash) and y4 (dot) ...................................................................................................... 88
24 Paths made by the vectors R and R + δR ................................................................ 90
25 Unit vectors er, eθ, and eλ ......................................................................................................................................... 94
26 A simple pendulum .................................................................................................... 99
27 A compound pendulum ............................................................................................. 100
28 Two nearby points 3,4 on the minimizing arc ......................................................... 112
29 Line segment of variable length with endpoints on the curves C, D ...................... 116
30 Shortest arc from a fixed point 1 to a curve N. G is the evolute ......................... 118
31 Line segment of variable length with endpoints on the curves C, D ...................... 120
32 Conjugate point at the right end of an extremal arc .............................................. 121
33 Line segment of variable length with endpoints on the curves C, D ...................... 123
34 The path of quickest descent from point 1 to a cuve N .......................................... 127
ii
, Credits
Much of the material in these notes was taken from the following texts:
1. Bliss - Calculus of Variations, Carus monograph - Open Court Publishing Co. - 1924
2. Gelfand & Fomin - Calculus of Variations - Prentice Hall 1963
3. Forray - Variational Calculus - McGraw Hill 1968
4. Weinstock - Calculus of Variations - Dover 1974
5. J. D. Logan - Applied Mathematics, Second Edition -John Wiley 1997
The figures are plotted by Lt. Thomas A. Hamrick, USN and Lt. Gerald N. Miranda,
USN using Matlab. They also revamped the numerical examples chapter to include Matlab
software and problems for the reader.
iii
Monterey, California 93943
1
,Contents
1 Functions of n Variables 1
1.1 Unconstrained Minimum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Constrained Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Examples, Notation 10
2.1 Notation & Conventions ............................................................................................. 13
2.2 Shortest Distances ....................................................................................................... 14
3 First Results 21
3.1 Two Important Auxiliary Formulas: .......................................................................... 22
3.2 Two Important Auxiliary Formulas in the General Case.......................................... 26
4 Variable End-Point Problems 36
4.1 The General Problem .................................................................................................. 38
4.2 Appendix....................................................................................................................... 41
5 Higher Dimensional Problems and Another Proof of the Second Euler
Equation 46
5.1 Variational Problems with Constraints ...................................................................... 47
5.1.1 Isoparametric Problems ..................................................................................... 47
5.1.2 Point Constraints ................................................................................................ 51
6 Integrals Involving More Than One Independent Variable 59
7 Examples of Numerical Techniques 63
7.1 Indirect Methods .......................................................................................................... 63
7.1.1 Fixed End Points................................................................................................ 63
7.1.2 Variable End Points ........................................................................................... 71
7.2 Direct Methods ............................................................................................................ 74
8 The Rayleigh-Ritz Method 82
8.1 Euler’s Method of Finite Differences .......................................................................... 84
9 Hamilton’s Principle 90
10 Degrees of Freedom - Generalized Coordinates 97
11 Integrals Involving Higher Derivatives 104
12 Piecewise Smooth Arcs and Additional Results 110
13 Field Theory Jacobi’s Neccesary Condition and Sufficiency 116
i
,List of Figures
1 Neighborhood S of X0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Neighborhood S of X0 and a particular direction H . . . . . . . . . . . . . . 2
3 Two dimensional neighborhood of X0 showing tangent at that point . . . . . 5
4 The constraint φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
5 The surface of revolution for the soap example ........................................................ 11
6 Brachistochrone problem ............................................................................................. 12
7 An arc connecting X1 and X2 ................................................................................................................................. 15
8 Admissible function η vanishing at end points (bottom) and various admissible
functions (top) .............................................................................................................. 15
9 Families of arcs y0 + νη .............................................................................................. 17
10 Line segment of variable length with endpoints on the curves C, D ........................ 22
11 Curves described by endpoints of the family y(x, b) ................................................. 27
12 Cycloid .......................................................................................................................... 29
13 A particle falling from point 1 to point 2 .................................................................. 29
14 Cycloid .......................................................................................................................... 32
15 Curves C, D described by the endpoints of segment y34 ............................................................. 33
16 Shortest arc from a fixed point 1 to a curve N. G is the evolute ........................... 36
17 Path of quickest descent, y12, from point 1 to the curve N ...................................... 40
18 Intersection of a plane with a sphere......................................................................... 56
19 Domain R with outward normal making an angle ν with x axis............................. 61
20 Solution of example given by (14) .............................................................................. 71
21 The exact solution (solid line) is compared with φ0 (dash dot), y1 (dot) and
y2 (dash) ....................................................................................................................... 85
22 Piecewise linear function.............................................................................................. 86
23 The exact solution (solid line) is compared with y1 (dot), y2 (dash dot), y3
(dash) and y4 (dot) ...................................................................................................... 88
24 Paths made by the vectors R and R + δR ................................................................ 90
25 Unit vectors er, eθ, and eλ ......................................................................................................................................... 94
26 A simple pendulum .................................................................................................... 99
27 A compound pendulum ............................................................................................. 100
28 Two nearby points 3,4 on the minimizing arc ......................................................... 112
29 Line segment of variable length with endpoints on the curves C, D ...................... 116
30 Shortest arc from a fixed point 1 to a curve N. G is the evolute ......................... 118
31 Line segment of variable length with endpoints on the curves C, D ...................... 120
32 Conjugate point at the right end of an extremal arc .............................................. 121
33 Line segment of variable length with endpoints on the curves C, D ...................... 123
34 The path of quickest descent from point 1 to a cuve N .......................................... 127
ii
, Credits
Much of the material in these notes was taken from the following texts:
1. Bliss - Calculus of Variations, Carus monograph - Open Court Publishing Co. - 1924
2. Gelfand & Fomin - Calculus of Variations - Prentice Hall 1963
3. Forray - Variational Calculus - McGraw Hill 1968
4. Weinstock - Calculus of Variations - Dover 1974
5. J. D. Logan - Applied Mathematics, Second Edition -John Wiley 1997
The figures are plotted by Lt. Thomas A. Hamrick, USN and Lt. Gerald N. Miranda,
USN using Matlab. They also revamped the numerical examples chapter to include Matlab
software and problems for the reader.
iii
