4. Consider a harmonic oscillator in two dimensions described by the Lagrangianтmw?L22where (r, ø) are the polar coordinates, m > 0 is the mass, and w > 0 the oscillationfrequency of the oscillator.(a) Find the canonical momenta p, and pø, and show that the Hamiltonian is given bymw?H(r, 0, Pr; Po) =2m2mr?2(b) Show that p is a constant of motion.(c) In the following we will perform a variable transformation (r, 6, p,, Pa) → (Q,,Qo, Pr, Pgdefined byQ, = 7.lp,P,Pg = PeQo = 20,%3D%3D2r2where l is an arbitrary constant length that just ensures that Q, has also the dimensionof length. Show that this transformation is canonical.(d) The Hamiltonian becomes in the new variables2P?mw?l2H(Q..Qo» Pr, Pa) = Q,P; +mlQ,-Qr.2mlShow that the fact that energy E is conserved and H = E allows us to introduce a newHamiltonian HK and a new energy EK such that HKwith%3DEk is equivalent to H = E,P2P?Hg(Qr.Qá, Pr, Pa) =|2m2mQ? Q,Determine the constant a and the new energy EK as a function of m,w, E.(e) In question 4 (d) just above you will find that one always has EK < 0 and a > 0. Givea physical argument why this must indeed be the case.

Since you have posted a question with multiple subpart we will solve first three sub-parts for you.…

Answered: 4. Consider a harmonic oscillator in…

Similar questions

2. One end of a long rod is fixed at the origin, and the rod rotates about that point in the horizontal plane with angular frequency w. A bead of mass m is constrained to slide without friction along the rod (assume the rod is long enough that you don't have to worry about it sliding off the end). Let r be the radial position of the bead. (a) Find the Lagrangian and equations of motion of the system. (b) Identify a symmetry of the system and find the corresponding conserved quantity. Verify directly from the equations of motion that it is conserved. Show from the equations of motion that the kinetic energy of the bead is not conserved, and explain why. r m Figure 2: Question 2
In the following vibration system, disk 2 rolls on a non-slip part with uniform mass distribution and radius R. The mass of disk 2 and parts 1 and 3 is m. If x1 and x2 are the absolute displacement of part 1 and hinge A, respectively,A) Get the differential equations of motion using Lagrange equations. (I ̅_Disk=1/2 mr^2)B) Write what kind of couplings there are.
3. A bead of mass m is threaded on a frictionless wire that is bent into a helix with cylin- drical polar coordinates (p, ø, z) satisfying z The z axis points vertically up and gravity vertically down. co and p = R, with c and R constants (a) as a function of and its conjugate momentumpo. (b) Using as your generalized coordinate, write down the Hamiltonian H Write down the Hamilton's equations and solve for ø and hence z.
A particle of mass m described by one generalized coordinate q movesunder the influence of a potential V(q) and a damping force −2mγq˙ proportional to its velocity with the Lagrangian L = e2γt(1/2 * mq˙2 − V (q)) which gives the desired equation of motion. (a) Consider the following generating function: F = eγtqP - QP.Obtain the canonical transformation from (q,p) to (Q,P) and the transformed Hamiltonian K(Q,P,t). (b) Let V (q) = (1/2)mω2q2 be a harmonic potential with a natural frequency ω and note that the transformed Hamiltonian yields a constant of motion. Obtain the solution Q(t) for the damped oscillator in the under damped case γ < ω by solving Hamilton's equations in the transformed coordinates. Then, write down the solution q(t) using the canonical coordinates obtained in part (a).
Part b
Let G(u, v) = (3u + v, u - 2v). Use the Jacobian to determine the area of G(R) for: (a) R = [0, 3] x [0, 5] (b) R = [2, 5] x [1, 7]
Quartic oscillations Consider a point particle of mass m (e.g., marble whose radius is insignificant com- pared to any other length in the system) located at the equilibrium points of a curve whose shape is described by the quartic function: x4 y(x) = A ¹ Bx² + B² B²), (1) Where x represents the distance along the horizontal axis and y the height in the vertical direction. The direction of Earth's constant gravitational field in this system of coordinates is g = −gŷ, with ŷ a unit vector along the y direction. This is just a precise way to say with math that gravity points downwards and greater values of y point upwards. A, B > 0. (a) Find the local extrema of y(x). Which ones are minima and which ones are maxima? (b) Sketch the function y(x). (c) What are the units of A and B? Provide the answer either in terms of L(ength) or in SI units. (d) If we put the point particle at any of the stationary points found in (a) and we displace it by a small quantity³. Which stationary locations…
My system is a pendulum attached to moving horizontal mass m_1 and the pendulum m_2 that is shifted by X_o from origin. I have the lagrangian of my system what would be my equations of motions in terms of small angle approximation and what’s is their frequency?
A massless spring with equilibrium length d and spring constant k connects two particles. The system is flat and horizontal, yet it may spin and vibrate (ccompress\stretch).1- Determine the system's Lagrangian.2- Determine the system's Hamiltonian.3- Calculate Hamilton's equations of motion. It should be noted that the generalized momenta can be omitted. -It is worth noting that as the mass spins, it begins to expand. Hint: make your coordinate system's origin the center of the unstretched spring. also In generalized coordinates of (r_i) and (theta_i) , express your equations.
Q1
A particle of mass m described by one generalized coordinate q movesunder the influence of a potential V(q) and a damping force −2mγq˙ proportional to its velocity. Show that the following Lagrangian gives the desired equation of motion: L = e2γt(1/2 * mq˙2 − V (q))