5. Consider the functional Jlſ(t), Þ(t)] = " (Hğ,p,t) - à p) dt of the 2n...
7. Consider the functional J(y) = SỐ[r(t)? + g(t)y?] dt. Find the Hamiltonian H(t, y, p) and find Hamilton's equations for the problem.
5*. Consider all sequences (ai,. .., an) such that a, are nonnegative integers and a ai+ 2. Let P, n and Rn be the number of such sequences which start from 0, 1 and 2 respectively. (a) Compute P, Qn, Rn by writing down all such sequences for n 1,2,3. (b) Prove that P, Qn Rn satisfy the recurrence relations: (c) Translate the above equations into linear equations for the generating functions for P, Qn, Rn (d) Solve these equations...
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
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,...
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...
5. Consider the signal x (t) = cos (2n . 500) + cos (2n . 1 500). Its spectrum X1c" consists of a pair of spectral lines at positive and negative frequencies. Use the MATLAB command fft to find and plot the signal's spectrum using various values of N.
Differential equations question. dp/dt = 0.3 (1-p/10) (p/10-2)p 1. (5 points) Consider the given population model, where P(t) is the population at time t A. For what values of P is the population in equilibrium? B. For what values of P is it increasing? C. For what values is it decreasing? : (i-T-YE -2) p dt120 her
Problem 6. Consider the n independent trails in Problem 5. Let On be the probability that there is no three consecutive successes in n trails. (1). Show that limn+cQn = 0 (2). Show that Qn = (1 - pQn-1 + p(1 - pQn-2 + p (1 - p)Qn-3 for n 3 (Hint: condition on the first failure). Problem 5. Suppose we do n independent trails that each has a probability P E (0,1) to result in success. Let Pn be...
Question 1: (5 marks) Consider a two-species model for populations Ni and N2 follows as N1 (a -bN1 cN2) dt N2 (d - eN2 - Ni) dt (a) What kind of interaction does this system of equations represent? (b) Show that the equations can be simplified to dn1 an n1 (1 d7 dn2 Bn2 (1n2-n1). dT mT into the system of equations and picking by substituting N = kn\, N2 = ln2 and t appropriate constants k, l and m...
Please answer ALL parts OF LETTER F and show work/explain LETTER F 5. Consider the following concentration table 4 NH3(g) 0.30 mol 702(g) 0.80 mol 〈=> 2N,04(g) + 6H20(g) I (initial moles) C (change) E (equilibrium moles) a) What are the values for each of the v;? b) Show that Σ v, Xi-0. c) Fill in the concentration table in terms of 5, the extent of reaction. d) According to your concentration table what are the minimum and maximum values...