4. Hamiltonian equations of motion A particle of mass m moving in two dimensions has the...

4. Hamiltonian equations of motion A particle of mass m moving in two dimensions has the Hamiltonian H(x, y, px, py) = (px − cy) ^2 + (py + cx)^ 2 /2m where c is a constant. Find the equations of motion for x, y, px, py. Use these to write the equations of motion for the complex variables ζ = x + iy and ρ = px + ipy. Eliminate ρ to show that m¨ζ − 2ic ˙ζ = 0 . Find the general solution for ζ(t) and ρ(t). A particle has the initial position (x, y) = (0, 0) and initial momentum (px, py) = (mU, 0) at t = 0. Find its position, momentum and velocity as functions of t.

Expert Solution

This is the correct sequence of answers.

I have first computed the general equation of motion using the standard formula and then substituting these equations in other variables to compute Tao and rho and finally, I have computed position, momentum, and velocity.

genius_generous answered 2 years ago

write parametric equations in 2D for a particle moving in a clockwise motion along a circle...

write parametric equations in 2D for a particle moving in a clockwise motion along a circle radius 2 such that at t=0 , the particle is at point (-2,0) Please I need the answer with steps and explanation.. Clear handwriting please.. Thank you

A moving particle of mass M and impulse P breaks down in two fragments. One of...

A moving particle of mass M and impulse P breaks down in two fragments. One of these has a mass m1=1.00 MeV/c2 and an impulse p1 = 1.75Mev/c in the +x direction. The other one has a mass m2 = 1.50 MeV/c2 and an impulse p2 = 2.005 MeV/c in the +y direction. Find: a) The initial impulse (value and direction) b) Total energy of the initial particle c) The mass M of the initial particle d) The value of...

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Consider a particle of mass m moving in a two-dimensional harmonic oscillator potential : U(x,y)=...

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Consider a particle of mass m moving in a two-dimensional harmonic oscillator potential : U(x,y)= 1/2 mω^2 (x^2+y^2 ) a. Use separation of variables in Cartesian coordinates to solve the Schroedinger equation for this particle. b. Write down the normalized wavefunction and energy for the ground state of this particle. c. What is the energy and degeneracy of each of the lowest 5 energy levels of this particle? %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

One particle whose mass is defined by m is moving along a certain line in the...

One particle whose mass is defined by m is moving along a certain line in the region x>0 is subjected to a conservative net force given by the following equation: F(x)= F₀x eˣ^²/²ᴸ^² The constants F and L are both positive real quantities with appropriate physical dimensions. Find the formula for the corresponding U(x), with the given assumption that the limₓ↠∞U(x)=0. Secondly, we know that the particle is launched from an initial location x₀>>>L towards x=0. (the >>> mean that...

Problem 1: The energy E of a particle of mass m moving at speed v is...

Problem 1: The energy E of a particle of mass m moving at speed v is given by: E2 = m2 c4 + p2 c2 (1) p=γmv (2) 1 γ = 1−v2/c2 (3) This means that if something is at rest, it’s energy is mc2. We can define a kinetic energy to be the difference between the total energy of an object given by equation (1) and the rest energy mc2. What would be the kinetic energy of a baseball...

A particle with mass m begins its motion from the state of rest under the influence...

A particle with mass m begins its motion from the state of rest under the influence of an oscillating force F(t) = F0 sin(γ t). Find the particle’s velocity and the distance traveled at time t1, when the force reaches its first maximum, and at time t2, when the force becomes equal to zero.

Consider a particle moving in two spatial dimensions, subject to the following potential: V (x, y)...

Consider a particle moving in two spatial dimensions, subject to the following potential: V (x, y) = ( 0, 0 ≤ x ≤ L & 0 ≤ y ≤ H ∞, otherwise. (a) Write down the time-independent Schr¨odinger equation for this case, and motivate its form. (2) (b) Let k 2 = 2mE/~ 2 and rewrite this equation in a simpler form. (2) (c) Use the method of separation of variables and assume that ψ(x, y) = X(x)Y (y). Rewrite...

A particle with mass m and energy E is moving in one-dimension from right to legt....

A particle with mass m and energy E is moving in one-dimension from right to legt. It is incident on the step potential V(x)=0 for x<0 nd V(x)=V0 for x>0 where E>V0>0. Find the reflection coefficient R in terms of m,E and V0, and h-bar.

Particle A of mass m, initial velocity 20i (m/s) has a collision with a stationary particle...

Particle A of mass m, initial velocity 20i (m/s) has a collision with a stationary particle B of mass 2m. After collision, VA(final)=10i+5j (m/s). a) Find VB(final) if the system (particle A plus B) linear momentum is conserved (both i and j directions). What are the velocities of center of the system before and after collision?b) Find the system’s % KE lost due to the collision (m=20.0gram). c) If the collision time between A and B is 0.050 s, what...

(a) An elevator of mass m moving upward has two forces acting on it: the upward...

(a) An elevator of mass m moving upward has two forces acting on it: the upward force of tension in the cable and the downward force due to gravity. When the elevator is accelerating upward, which is greater, T or w? (b) When the elevator is moving at a constant velocity upward, which is greater, T or w? (c) When the elevator is moving upward, but the acceleration is downward, which is greater, T or w? (d) Let the elevator...

Question