A particle of mass m moves in a central force field with potential V(r). The La- grangian in terms of spherical polar coordinates (r, 0, 4) is m (2 +rô² +r² sin² 04? ) – V(r). 1. Find the momenta (p, Po, Po) conjugate to (r,0,p). 2. Find the Hamiltonian H(r,0,4,,Pr, Pe, Pg). 3. Write down the explicit Hamiltonian's equations of motion. ?
Q: What is the Hamiltonian of a system? Show that the kinetic energy operator commutes with the linear…
A: The momentum operator in 1D is defined as px=-ihδδx the Kinetic energy operator is Tx=-h22mδ2δx2…
Q: In class, we developed the one-dimensional particle-in-a-box model and showed that the wavefunction…
A:
Q: For each of the following vector fields F, decide whether it is conservative or not by computing the…
A:
Q: Shortest Line in 3D Prove that the shortest path between two points in 5. three dimensions is a…
A: The distance between the two points in the three dimensions can be given as follow:
Q: Demonstrate that in an electromagnetic field, the gauge transformation transfers the L to an…
A: I wrote an answer in a Page:
Q: A ball of mass m, slides over a ếy sliding inclined plane of mass M and angle a. Denote by X, the…
A: The expression for the degree of freedom is given as, F=3N-K Here, N is the number of particles,…
Q: Verify that the Hamiltonian equation H(x, p, t) = T + V = p2/2m + (k/2) (x − v0t)2 leads to the same…
A: The Hamilton’s equations of motion are ∂H∂p=x˙, and ∂H∂x=-p˙ From Newton's second law p˙=mx¨
Q: Go back to question 6 but this time assume uk=0.2. a) How much time elapses before the block…
A: This question belongs to friction.
Q: Using the del operator in tesian coordinates to derive the operator in the cylindrical
A:
Q: Show that if the Hamiltonian and a quantity F are constants of motion, then af/at is also a…
A:
Q: A small metal ball of mass m and negative charge −q0 is released from rest at the point (0,0,d) and…
A: The magnitude of electric field is given as,
Q: A heavy particle is projected from a point at the foot of a flying plane, inclined at an angle 45°…
A:
Q: Q1: (1 0 0 The Hamiltonian is represented by the matrix H = 0 0 2 0 0 0 2 Where o is positive real…
A:
Q: For the motion of a projectile write the Lagrangian and the Hamiltonian functions in polar…
A: The Lagrangian of a system is defined as:L=T-V………(1)where,T is the Kinetic energy.V is the Potential…
Q: What do you know about the difference between Newtonian, Lagrangian and Hamiltonian in describing…
A: There some basic deffierence between Newtonian lagrangian and Hamiltonian mechanics. Below some…
Q: Verify that each of the following force fields is conservative. Then find, for each, a scalar…
A:
Q: Find the equations of motion for the Lagrangian L = eγt(mq˙2/2 − kq2/2) How would you describe the…
A:
Q: Consider a particle of spin s = 3/2. (a) Find the matrices representing the operators S,,S,,Ŝ,,S and…
A:
Q: Q.6) Read the quest ions carefully and choose the correct option: (i) The Hamiltonian funct ion in…
A: In the given question, We have to discuss about Hamiltonian function.
Q: A particle of mass m is subject to an attractive central force F(r) p3 For which values of the…
A: Given: A particle of mass m and attractive central force, Fr=-kr3
Q: L=T-V = 1²2 8² +mg | Cos Write down the Lagrange equation for a single generalised coordinate q.…
A: We have a Lagrange given by L=T-V=ml2θ˙2/2+mglcosθ which is for the case of a simple pendulum of…
Q: A hemispherical bowl of radius a is held fixed with its rim upwards and horizontal. A particle of…
A: Lagrangian is nothing but the difference between kinetic energy and the potential energy of an…
Q: Find the equations of motion for the Lagrangian L = eγt(mq˙2/2 − kq2/2) How would you describe the…
A:
Q: What is hamiltonian equation
A: Given data: Hamiltonian equation.
Q: Q2: 0 0 The Hamiltonian is represented by the matrix H = u 0 0 1 Where u is positive real number. 1…
A:
Q: Use the Euler's Lagrange equationto prove clearly that the shortest distance between two points in…
A:
Q: What is Hamiltonian cycle?
A: What is hamiltonian cycle.
Q: Can both members of a third-law force pair appear on the same free-body diagram? Explain why or why…
A: Force is a vector quantity that is defined as the rate of change of an object's momentum. The force…
Q: In Poincare transformation if scalar field is invariant under translation, then prove that generator…
A: In this question we have to answer related to Poincare Transformation.Please give positive feedback…
Q: Show that the position operator (Ê = x) and the hamiltonian operator (Ĥ : -(n2/2m)d²/dx² + V (x))…
A: Proof for Position operator being hermitian x^ψψ=∫-∞∞xψ*ψdx=∫-∞∞xψ*ψdx=∫-∞∞ψ*xψdx=ψx^ψ
Q: Show that the total energy eigenfunctions ψ210(r, θ, φ) and ψ211(r, θ, φ) are orthogonal. Do you…
A:
Q: 1) Give the Lagrange function and the Lagrangian equation(s) of motion of the second kind for the…
A:
Q: Show that the following operators are Hermitian, a) Momentum operator b) Hamiltonian operator c) i…
A:
Q: What do you mean by Harmonic ascillator. Prove that the Hamiltonian tor Harmonic oscillator can be…
A: A harmonic oscillator, Hamiltonian of harmonic oscillator, H = ?
Q: A hemispherical bowl of radius a is held fixed with its rim upwards and horizontal. A particle of…
A: Lagrangian of a system is the difference between kinetic energy and potential energy of that system.…
Q: Consider a particle of mass m acted upon by a central potential (r is distance and a>0 is a…
A:
Q: Given the Hamiltonian: H=p²/2m+ V(x) What is the value of the commutator [H,x]?
A:
Q: Yoyo The figure below shows a crude model of a yoyo. A massless string 4. suspended vertically from…
A: The total kinetic energy of the body is
Q: At a certain point of a body, the components of the cauchy stress tensor are given by [2 5 37 [0] =…
A: introduction: Cauchy stress vector tn at point P on the plane in terms of normal and shear…
Q: Using the property of Re z and Im z to show that the hyperbola x2 – y? = 1 can be written as: z2 +…
A:
Q: Q1: 1. The Hamiltonian is represented by the matrix H = @ 0 2 0 Where o is positive real number. 0 2…
A:
Trending now
This is a popular solution!
Step by step
Solved in 3 steps with 3 images
- Problem 3 The Lagrangian of a particle in spherical coordinates is given by 1 L(r, 0, 6; 1, 0, 0) = ¸m[r² + r²ġ² + r² (sin 0)² ¿²] – V(r) . Write down the momenta conjugate to each generalised coordinate. Write down the Hamiltonian and Hamil- ton's equations.1. Consider the 2D motion of a particle of mass u in a central force field with potential V(r). a) Find the r, o polar-coordinate expression of the Lagrangian for this system and write down the corresponding Euler-Lagrange e.o.m.s. b) Note that the angular variable o is cyclic. What is the physical interpretation of the correspond- ing integral of motion? (For the definitions of the italicized terms see this link.) c) Solve for o in terms of this integral of motion and substitute the result into the Euler-Lagrange equation for r. Show that the result can be arranged to look like a purely 1D e.o.m. of the form dVef(r) (1) dr Identify in the process the explicit expression for Vef(r), which will depend among other things on the integral of motion. d) Take now k V (r) = with k > 0 to be an attractive electrostatic/gravitational-type potential. Sketch the profile of the corresponding effective potential function Vef(r). Find the equilibrium solution for the correspond- ing e.o.m. (1). What…A particle of mass m is projected upward with a velocity vo at an angle a to the horizontal in the uniform gravitational field of earth. Ignore air resistance and take the potential energy U at y=0 as 0. Using the cartesian coordinate system answer the following questions: a. find the Lagrangian in terms of x and y and identiy cyclic coordinates. b. find the conjugate momenta, identify them and discuss which are conserved and why c. using the lagrange's equations, find the x and y components of the velocity as the functions of time.
- Which of the following statements is false? I. The reduced mass of a two-particle system is always less than m₁ and less than m2. II. When we solve a two-particle system (whose potential-energy V is a function of only the relative coordinates of the two particles) by dealing with two separate one-particle systems, V is part of the Hamiltonian operator of the fictitious particle with mass equal to the reduced mass. III. For a one-particle problem with V = br², where b is a positive constant, the stationary-state wave functions have the form = ƒ (r) Y (0, ¢) O II OI ||| None of the given choices. II and IIIQ.n.3 A central force is defined to be a force that points radially, and whose magnitude depends on only r. That is, F(r) = F(r) `r. Show that a central force is a conservative force, by explicitly showing that Vx F = 0 Q.n.4 Consider two particles of masses ml and m2. Let m1 be confined to move on a circle of O plane, centered at x = y = 0. Let m2 be confined to move on a circle of radius radius a in the z = b in the z = c plane, centered at x = y = 0. A light (massless) spring of spring constant k is attached between the two particles. a) Find the Lagrangian for the system. Q.n.5 Oral VivaFor each of the following vector fields F , decide whether it is conservative or not by computing the appropriate first order partial derivatives. Type in a potential function f (that is, Vf = F) with f(0,0) = 0. If it is not conservative, type N. A. F (x, y) = (4x – 2y) i+ (-2x + 12y) j f (x, y) = B. F (x, y) = 2yi+ 3xj f (x, y) = C. F (x, y) = (2 sin y) i + (–4y + 2x cos y) j f (x, y) = Note: Your answers should be either expressions of x and y (e.g. "3xy + 2y"), or the letter "N"
- Consider an ideal gas system composed of Krypton atoms at a temperature of 302.15 K and 116.64 kPa in pressure. The atomic mass of Krypton is 83.8 Da. Given that the total volume of the system is 1 Liter: a. Calculate the total Hamiltonian of the system. b. Calculate the relative fluctuation (coefficient of variation) in the HamiltonianProblem 6 Consider the following Lagrangian describing the two-dimensional motion of a particle of mass m in an inertial system (1, 2), L = m² (1² + ±²) - m² (x² - w² (x² + a x²) - bx₁ x2, 2 where a and b are constant numbers. (i) Write down the Euler-Lagrange equations for the system. (ii) For generic values of a and b, what are the conserved quantities of the system and what are the corresponding Noether symmetries? Consider next the case a = 1 and b = 0. Discuss if there are any additional symmetries and find the corresponding conserved quantities. (iii) Write down the Hamiltonian of the system and the corresponding Hamilton equations for generic values of a and b. (iv) The time evolution of a generic physical observable O(x, p) is described by the equation where Ӧ = {O,H}, {A, B} = Σ 2 ДА ӘВ дхп дрп ДА ӘВ дрп дхп JB ) n=1 is the Poisson bracket of A with B. x := (x1, x2) and p = (P1, P2) denote the position vector and the momentum of the particle, respectively. Using this result,…420 for a 2D 21 (;2 + 0 r² ) + Suppose that you have the Lagrangian L system in plane polar coordinates (r, 0). Determine the Hamiltonian.
- A Hamiltonian is given in matrix form as ħ (wo W1 A₁ = 7/7 (@₂₁ 000) 2 a. What are the energy eigenvalues? What are the energy eigenvectors? b.Two particles, each of mass m, are connected by a light inflexible string of length l. The string passes through a small smooth hole in the centre of a smooth horizontal table, so that one particle is below the table and the other can move on the surface of the table. Take the origin of the (plane) polar coordinates to be the hole, and describe the height of the lower particle by the coordinate z, measured downwards from the table surface. i. sketch all forces acting on each mass ii. explain how we get the following equation for the total energyA particle of mass m in a gravitational field slides on the inside of a smooth parabola of revolution whose axis is vertical. Using the distance from the axis r, and the azimuthal angle φ as generalized coordinates, find the following. a) The Lagrangian of the system. b) The generalized momenta and the corresponding Hamiltonian c) The equation of motion for the coordinate r. d) If dφ/dt = 0, show that the particle can execute small oscillations about the lowest point of the paraboloid and find the frequency of these oscilllations.