MatFys. Week 2, Nov , 2005, revised Nov. 23

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1 MtFys Week 2, Nov , 2005, revised Nov. 23 Lectures This week s lectures will be bsed on Ch.3 of the text book, VIA. Mondy Nov. 21 The fundmentls of the clculus of vritions in Eucliden spce nd its relevnce for mechnics will be explined. The discussion of the concept of differentibility of functionls will be bit more detiled thn in the book. The importnce of invrince of the principle of lest ction under coordinte chnges will be stressed nd discussed with regrd to lter developments. Wednesdy Nov. 23 The min emphsis will be on the Legendre trnsformtion nd its use in obtining the Hmiltonin form of the equtions of motion in Eucliden spce. A few pplictions of these equtions will be discussed, illustrting the effectiveness of the formlism of Hmiltonin mechnics, which is further developed lter in the book, but outside the scope of the present course. Tutorils Mondy Nov. 21 Solve the following exercises: Exercises 7 nd 10 in Ch.1 of the notes. Exercise 1A Let F : Ω R 3 be C vector field nd γ : [, b] R 3 smooth curve nd ϕ : [c, d] [, b] C 1 -function with ϕ (t) 0 for ll t [c, d], i.e. either ϕ > 0 or ϕ < 0 by continuity. Let γ = γ ϕ. 1

2 ) Show tht F dx = F dx if ϕ > 0, γ γ F dx = F dx if ϕ < 0. γ γ b) Show tht if nd only if γ F dx = 0 for ll closed curves γ F dx = F dx γ 1 γ 2 for ll curves γ 1 : [, b] R 3 nd γ 2 : [c, d] R 3 such tht γ 1 () = γ 2 (c) nd γ 1 (b) = γ 2 (d). Exercise 2A Show tht the vector fields F (x, y) = ( y, x), (x, y) R 2, nd G(x, y) = ( ) y x2 + y, x, (x, y) R 2 \ {(0, 0)}, 2 x2 + y 2 re not conservtive. Hint. Evlute the integrls of F nd G long suitble closed curve. (See lso VIA p.29.) Exercise 3A Consider prticle in one dimension with potentil energy U(x) = x 4, i.e. its eqution of motion is mẍ = U (x) = 4x 3. ) Show tht the energy E = 1 2 mẋ2 x 4 is conserved. (See the Theorem on p.16 in VIA on energy conservtion for one prticle in one dimension.) 2

3 b) Use ) to show tht for solution x(t), t [t 1, t 2 ], the time t 2 t 1 it tkes to move from point x 1 to point x 2 is given by t 2 t 1 = x2 x 1 dx, 2 m x4 ) ssuming tht x 1 < x 2 nd tht ẋ 0 on [t 1, t 2 ]. c) Use b) to show tht, for E > 0, the prticle excpes to infinity in finite time, i.e. the solution is not globlly defined. Exercise 4A Consider system of N prticles intercting by the grvittionl force, i.e. the force F i cting on prticle i is given by formuls ( ) in Ch.1 of the notes. ) Show tht F i = U x i, where the function U, the grvittionl potentil, is defined by for x i x j, if i j. U(x 1,..., x N ) = i<j Gm i m j x i x j b) Use Newton s second lw (1.14) to show tht the totl energy E = i 1 2 m iẋ 2 i + U(x 1,..., x N ) is conserved, i.e. it is constnt function of time for ny solution (x 1 (t),..., x N (t)) to the equtions of motion. Wednesdy Nov 23 Solve the following exercises: Exercise 5A ) Let f : [, b] R n be continuous function nd define Φ(x) = 3 f(t) x(t)dt,

4 where x : [, b] R n is curve. Show tht Φ is differentible, in the sense of VIA p. 56, nd equls its differentil. Determine the sttionry points (extremls) of Φ. b) Show tht, if Φ is differentible functionl, then its differentil is uniquely defined. Exercise 6A Consider the energy functionl E(x) = m 2 ẋ 2 dt, defined on (piecewise) smooth curves x : [, b] R 3. ) Write down the Euler-Lgrnge equtions. Show tht there exists exctly one solution such tht x() = x 1 nd x(b) = x 2 for ny given x 1, x 2 R 3. Show tht this solution is minimum point of E on the spce of curves from x 1 to x 2. Hint. For the lst question it my be usefull to compute the vlue of E for the difference between n rbitrry curve nd the unique solution. b) Write down the expression for E in sphericl coordintes (r, θ, ϕ), given by (x, y, z) = r(cos θ cos ϕ, cos θ sin ϕ, sin θ), for r > 0, π/2 θ π/2, 0 ϕ 2π. Determine the corresponding Euler-Lgrnge equtions nd show tht stright lines through the origin, with liner prmetristion, re solutions. Exercise 7A Consider the length functionl L(x) = ẋ dt, defined on (piecewise) smooth curves x : [, b] R 2. ) Write down the corresponding Euler-Lgrnge equtions nd show tht stright lines, with rbitrry prmetristion, re solutions. Argue tht these re the only solutions, nd show tht they re minimum points of L on the spce of curves with fixed end points x 1 nd x 2. 4

5 Hint. The lst question my be nswered by showing tht for ny smooth curve x from x 1 to x 2 the function f(t) = L(x [,t] ) increses more rpidly tht the function g(t) = x(t) x 1, for t [, b]. To compre the derivtives f nd g one cn mke use of Cuchy-Schwrz. b) Write down the expression for L in polr coordintes (r, θ), given by (x, y) = r(cos θ, sin θ), r > 0, 0 θ 2π. Determine the corresponding Euler-Lgrnge equtions nd show tht stright lines through the origin, with rbitrry prmetristion, re solutions. Exercise 8A Let the functionl Φ be given by Φ(x) = L(x, ẋ)dt, where the Lgrnge function L does not depend explicitly on t. Show tht, if x is solution to the corresponding Euler-Lgrnge equtions, then E(t) = ẋ L (x(t), ẋ(t)) L(x(t), ẋ(t)) ẋ is constnt function of t [, b]. We sy tht E is conserved quntity. Exercise 9A Let the functionl Φ be defined by 1 + ẋ 2 Φ(x) = dt, x where x : [, b] R + is positive C 1 -function. ) Show tht the corresponding Euler-Lgrnge eqution is xẍ + ẋ = 0. b) To solve this eqution, use first the result of the previous exercise to show tht ny solution x fulfills x 2 + (xẋ) 2 = constnt. 5

6 Then proceed to find ll solutions nd verify tht for given x 1, x 2 > 0 there is exctly one solution x such tht x() = x 1 nd x(b) = x 2. Hint. It my be usefull to rewrite the conservtion eqution in terms of y = x 2. The correct form of the solutions is x(t) = t 2 + Bt + C, where B, C re constnts. Bergfinnur Durhuus 6