![7. Consider the functional J(y) = SỐ[r(t)? + g(t)y?] dt. Find the Hamiltonian H(t, y, p) and find Hamiltons equations for th](http://img.homeworklib.com/questions/23fbab80-a7d9-11ea-a754-ffc22cfcebdc.png?x-oss-process=image/resize,w_560)
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![Soi given Jy) = S(ocelyt + acery] de. let us find the Hamiltons the alem of H(ty,p] + Hamilton :- a staus that the developem](http://img.homeworklib.com/questions/cc341370-ad29-11ea-83d6-75903d1e6bf8.png?x-oss-process=image/resize,w_560)
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7. Consider the functional J(y) = SỐ[r(t)? + g(t)y?] dt. Find the Hamiltonian H(t, y, p)...
5. Consider the functional Jlſ(t), Þ(t)] = " (Hğ,p,t) - à p) dt of the 2n...
5. Consider the functional Jlſ(t), Þ(t)] = " (Hğ,p,t) - à p) dt of the 2n independent functions qı(t),...,qn(t), pi(t),..., p(n(t). Show that the extremals of J satisfy Hamilton's equations with Hamiltonian H.
1. A particle of mass m moves in three dimensions, and has position r(t)-(x(t), y(t), z(t)) at ti...
Mechanics. Need help with c) and d)
1. A particle of mass m moves in three dimensions, and has position r(t)-(x(t), y(t), z(t)) at time t. The particle has potential energy V(x, y, 2) so that its Lagrangian is given by where i d/dt, dy/dt, dz/dt (a) Writing q(q2.93)-(r, y, z) and denoting by p (p,P2, ps) their associated canonical momenta, show that the Hamiltonian is given by (show it from first principles rather than using the energy) H(q,p)H(g1, 92,9q3,...
6. (a) The signal y(t) is defined as follows: ' y(t) = r(r)dT Suppose that (t)...
6. (a) The signal y(t) is defined as follows: ' y(t) = r(r)dT Suppose that (t) tu(t). First sketch r(t). Then determine and sketch y(t). (b) Consider the signal p(t) Σ δ(t-2n). First sketch p(t). Then calculate the following integral: 27 p(T)dT -3
Problem 7. Consider the following system of linear differential equations: = ax + y, dt =...
Problem 7. Consider the following system of linear differential equations: = ax + y, dt = ay Let J = 0 Verify that exp(Jt) is the solutions. J = ja 1] a
Consider the problem minimize 1[r(-)] = 2 / r,(t)2 dt subject to the conditions r(0) - r(T)0 and ...
Consider the problem minimize 1[r(-)] = 2 / r,(t)2 dt subject to the conditions r(0) - r(T)0 and the constraint 0 r(t)2 dt 1. = Suppose that r : [0, π] R is a C2 function that! solves the above Let y : [0, π] R be any other C2 function such that y(0) Define problem a(s): (r(t) + sy(t))2 dt and a(s) a. Explain why a(0) 1 and i'(0) 0. b. Show that i'(0)= | z'(t) y' (t) dt-X...
Differentiating the Hamiltonian. Starting with H(t, q, Р) %3D р .qlt, q, P) — L(t, 9,...
Differentiating the Hamiltonian. Starting with H(t, q, Р) %3D р .qlt, q, P) — L(t, 9, q(t, q, P)) Рiа: (t, q. P)| — L(t, q, q(t, q. p)) Li=1 show that dH (4.163) дt dt along extremals
Differentiating the Hamiltonian. Starting with H(t, q, Р) %3D р .qlt, q, P) — L(t, 9, q(t, q, P)) Рiа: (t, q. P)| — L(t, q, q(t, q. p)) Li=1 show that dH (4.163) дt dt along extremals
2. Consider a mass m moving in R3 without friction. It is fasten tightly at one...
2. Consider a mass m moving in R3 without friction. It is fasten tightly at one end of a string with length 1 and can swing in any direction. In fact, it moves on a sphere, a subspace of R3 1 0 φ g 2.1 Use the spherical coordinates (1,0,) to derive the Lagrangian L(0,0,0,0) = T-U, namely the difference of kinetic energy T and potential energy U. (Note r = 1 is fixed.) 2.2 Calculate the Euler-Lagrange equations, namely...
3. A particle of mass m moves in one dimension, and has position r(t) at time t. The particle has...
Mechanics.
3. A particle of mass m moves in one dimension, and has position r(t) at time t. The particle has potential energy V(x) and its relativistic Lagrangian is given by where mo is the rest mass of the particle and c is the speed of light (a) Writing qr and denoting by p its associated canonical momenta, show that the Hamiltonian is given by (show it from first principles rather than using the energy mzc2 6 marks (b) Write...
As described in class, the Poisson Bracket [F, G] between two functions Fand G of the generalized positions q, and momenta pi is defined as: Consider a system with Hamiltonian H-P2/2m-Vr = (P, 2+py 2...
As described in class, the Poisson Bracket [F, G] between two functions Fand G of the generalized positions q, and momenta pi is defined as: Consider a system with Hamiltonian H-P2/2m-Vr = (P, 2+py 2+pz2y2m)-y(x"2 + y"2 + z ^2)-U2 where yis a constant. a) Evaluate [Lz, H] and interpret the result in two ways i.e. what it says about L, and what it says about H b) Using the Poisson Bracket and the given Hamiltonian, find the value of...
Consider the Mundel-Fleming small open economy model: Y=C(Y-T)+1(1) + G Y = F(K,L) (M/P) L(r+z® Y)...
Consider the Mundel-Fleming small open economy model: Y=C(Y-T)+1(1) + G Y = F(K,L) (M/P) L(r+z® Y) Goods Money C = 50+0.8(Y- T) M 3000 I = 200-20r r*=5 NX = 200-508 P = 3 G=T= 150 L(Y, r) Y - 30r 1- find the IS* equation (hint : y as a function of e) 2- find the LM* equation (hint, also relates y and maybe e) 3-draw the IS-LM curve I y 4- find the equilibrium interest rate (trick question!)...
5. Consider the functional Jlſ(t), Þ(t)] = " (Hğ,p,t) - à p) dt of the 2n independent functions qı(t),...,qn(t), pi(t),..., p(n(t). Show that the extremals of J satisfy Hamilton's equations with Hamiltonian H.
Mechanics. Need help with c) and d)
1. A particle of mass m moves in three dimensions, and has position r(t)-(x(t), y(t), z(t)) at time t. The particle has potential energy V(x, y, 2) so that its Lagrangian is given by where i d/dt, dy/dt, dz/dt (a) Writing q(q2.93)-(r, y, z) and denoting by p (p,P2, ps) their associated canonical momenta, show that the Hamiltonian is given by (show it from first principles rather than using the energy) H(q,p)H(g1, 92,9q3,...
6. (a) The signal y(t) is defined as follows: ' y(t) = r(r)dT Suppose that (t) tu(t). First sketch r(t). Then determine and sketch y(t). (b) Consider the signal p(t) Σ δ(t-2n). First sketch p(t). Then calculate the following integral: 27 p(T)dT -3
Problem 7. Consider the following system of linear differential equations: = ax + y, dt = ay Let J = 0 Verify that exp(Jt) is the solutions. J = ja 1] a
Consider the problem minimize 1[r(-)] = 2 / r,(t)2 dt subject to the conditions r(0) - r(T)0 and the constraint 0 r(t)2 dt 1. = Suppose that r : [0, π] R is a C2 function that! solves the above Let y : [0, π] R be any other C2 function such that y(0) Define problem a(s): (r(t) + sy(t))2 dt and a(s) a. Explain why a(0) 1 and i'(0) 0. b. Show that i'(0)= | z'(t) y' (t) dt-X...
Differentiating the Hamiltonian. Starting with H(t, q, Р) %3D р .qlt, q, P) — L(t, 9, q(t, q, P)) Рiа: (t, q. P)| — L(t, q, q(t, q. p)) Li=1 show that dH (4.163) дt dt along extremals
Differentiating the Hamiltonian. Starting with H(t, q, Р) %3D р .qlt, q, P) — L(t, 9, q(t, q, P)) Рiа: (t, q. P)| — L(t, q, q(t, q. p)) Li=1 show that dH (4.163) дt dt along extremals
2. Consider a mass m moving in R3 without friction. It is fasten tightly at one end of a string with length 1 and can swing in any direction. In fact, it moves on a sphere, a subspace of R3 1 0 φ g 2.1 Use the spherical coordinates (1,0,) to derive the Lagrangian L(0,0,0,0) = T-U, namely the difference of kinetic energy T and potential energy U. (Note r = 1 is fixed.) 2.2 Calculate the Euler-Lagrange equations, namely...
Mechanics.
3. A particle of mass m moves in one dimension, and has position r(t) at time t. The particle has potential energy V(x) and its relativistic Lagrangian is given by where mo is the rest mass of the particle and c is the speed of light (a) Writing qr and denoting by p its associated canonical momenta, show that the Hamiltonian is given by (show it from first principles rather than using the energy mzc2 6 marks (b) Write...
As described in class, the Poisson Bracket [F, G] between two functions Fand G of the generalized positions q, and momenta pi is defined as: Consider a system with Hamiltonian H-P2/2m-Vr = (P, 2+py 2+pz2y2m)-y(x"2 + y"2 + z ^2)-U2 where yis a constant. a) Evaluate [Lz, H] and interpret the result in two ways i.e. what it says about L, and what it says about H b) Using the Poisson Bracket and the given Hamiltonian, find the value of...
Consider the Mundel-Fleming small open economy model: Y=C(Y-T)+1(1) + G Y = F(K,L) (M/P) L(r+z® Y) Goods Money C = 50+0.8(Y- T) M 3000 I = 200-20r r*=5 NX = 200-508 P = 3 G=T= 150 L(Y, r) Y - 30r 1- find the IS* equation (hint : y as a function of e) 2- find the LM* equation (hint, also relates y and maybe e) 3-draw the IS-LM curve I y 4- find the equilibrium interest rate (trick question!)...
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