Download Calculus of Variations with Applications to Physics & by Robert Weinstock PDF

By Robert Weinstock

E-book primarily divided into components. Chapters 1-4 contain heritage fabric, simple theorems and isoperimetric difficulties. Chapters 5-12 are dedicated to purposes, geometrical optics, particle dynamics, the idea of elasticity, electrostatics, quantum mechanics and different subject matters. routines in every one bankruptcy.

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Example text

Let n0 ∈ {1, 2, . 93) and the following hold: ⎧ there exists a function β ∈ C[0, 1] ∩ C 2 (0, 1) ⎪ ⎪ ⎪ ⎪ with β(t) ρn0 for t ∈ [0, 1] and with ⎪ ⎪ ⎨ q(t)f t, β(t) + β (t) 0 for t ∈ (0, 1) and ⎪ 1 1 ⎪ ⎪ q(t)f 2n0 +1 , β(t) + β (t) 0 for t ∈ 0, 2n0 +1 and ⎪ ⎪ ⎪ ⎩ q(t)f 1 − 1 , β(t) + β (t) 0 for t ∈ 1 − 1 , 1 . 95) 2 α(t) for t ∈ [0, 1]. 90). Suppose the following condition is satisfied: ⎧ let n ∈ {n0 , n0 + 1, . } and associated with each n we ⎪ ⎪ ⎪ ⎪ ⎨ have a constant ρn such that {ρn } is a decreasing sequence with limn→∞ ρn = 0 and there exists a ⎪ 1 ⎪ constant k0 > 0 such that for 2n+1 t 1 ⎪ ⎪ ⎩ and 0 < y ρn we have q(t)f (t, y) k0 .

Now since yn0 (0) = yn0 (1) = ρn0 there exists either (i) t1 , t2 ∈ (0, 1), t2 < t1 with αn0 (t) yn0 (t) M for t ∈ [0, t2 ), yn0 (t2 ) = M and yn0 (t) > M on (t2 , t1 ) with yn0 (t1 ) = 0; or (ii) t3 , t4 ∈ (0, 1), t4 < t3 with αn0 (t) yn0 (t) M for t ∈ (t3 , 1], yn0 (t3 ) = M and yn0 (t) > M on (t4 , t3 ) with yn0 (t4 ) = 0. We can assume without loss of generality that either t1 12 or t4 12 . Suppose t1 12 . P. Agarwal and D. 98), gn0 t, yn0 (t) = gn0 (t, M) + r M − yn0 (t) max f 1 , M , f (t, M) .

23. Let n0 ∈ {1, 2, . 74) hold. P. Agarwal and D. e. 109). 62) has a solution y ∈ C[0, 1] ∩ C 2(0, 1) with y(t) α(t) for t ∈ [0, 1]. P ROOF. For n = n0 , n0 + 1, . . 9. Without loss of generality assume ρn0 mint ∈[ 1 , 2 ] α(t). Fix n ∈ {n0 , n0 + 1, . }. Let tn ∈ [0, 13 ] and 3 3 sn ∈ [ 23 , 1] be such that α(tn ) = α(sn ) = ρn and α(t) ρn for t ∈ [0, tn ] ∪ [sn , 1]. Define αn (t) = if t ∈ [0, tn ] ∪ [sn , 1], if t ∈ (tn , sn ). 117)). 112) that for any constants r1 > 0, r2 > r1 , ∃hr1 ,r2 ∈ L1q [0, 1] with |f (t, u)| hr1 ,r2 (t) for t ∈ (0, 1) and r1 u r2 ).

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