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Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of

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Page 1: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 2: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 3: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 4: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 5: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 6: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 7: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 8: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 9: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 10: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 11: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 12: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 13: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 14: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 15: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 16: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 17: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 18: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 19: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 20: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 21: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 22: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 23: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 24: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 25: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 26: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 27: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 28: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 29: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 30: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 31: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 32: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 33: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 34: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 35: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 36: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 37: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 38: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 39: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 40: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 41: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 42: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 43: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 44: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 45: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 46: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 47: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 48: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 49: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 50: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 51: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 52: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 53: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 54: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 55: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 56: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 57: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 58: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 59: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 60: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 61: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 62: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
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Page 64: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
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Page 66: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 67: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 68: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
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Page 83: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
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Page 85: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 86: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 87: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 88: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
Page 89: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
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Page 91: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
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Page 95: Claude Bernard University Lyon 1math.univ-lyon1.fr/~clarke/ClarkeMemoir.pdf · the calculus of variations: to minimize the integral functional At (x (t), ei(t)) dt over a class of
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