Show that the function S = (mω/2)(q2 + α2) cot ωt − mωqα csc ωt is a solution of the Hamilton-Jacobi for Hamilton’s principal function for the linear harmonic oscillator with H =(1/2m)(p2 + m2ω2q2). Show that this function generates a correct solution to the motion of the harmonic oscillator.

Show that the function S = (mω/2)(q2 + α2) cot ωt − mωqα csc ωt is a solution of the Hamilton-Jacobi for Hamilton’s principal function for the linear harmonic oscillator with H =(1/2m)(p2 + m2ω2q2). Show that this function generates a correct solution to the motion of the harmonic oscillator.

Related questions

Q: (3) Consider the Hamiltonian given as H = +²+Fâ , where each term has a clear physical meaning. 2m 2…

Q: Find the Lagrangian and Lagrange's equations for a simple pendulum if the cord is replaced by a…

A: Spring constant of spring = k If unstretched the spring length is r0.The potential energy of the…

Q: A particle of mass m moving in one dimension is subject to the potential (x) = V₁²²2e-², a>0. a)…

A: We will first plot the graph of of given function qualitatively. Then we will do derivative test…

Q: which the Lagrangian is I = mc² (1-√√1-B²)-kx² where ß = == a) Obtain the Lagrange equation of…

A: Required to find the equation of motion.

Q: The time-evolution of a physical system with one coordinate q is described by the La- grangian 1 L +…

A: The Hamiltonian of the system is given by Where L = Lagrangian of the system p = conjugate…

Q: Consider the following unattached spring system. m₁ = 1, C₁ = 14, m₂ = 7 Write the stiffness matrix…

A: “As per the policy we are allowed to answer only 3 subparts at a time, I am providing the same.…

Q: 8. The Lagrangian of a system is given as L = mx7+qAx,- qd, where the symbols have the usual…

Q: Show that for a particle in a box of length L and infinite potential outside the box, the function…

Q: Consider a block of mass m on the end of a massless spring of spring constant k and equilibrium…

A: Since you have posted a question that has more than three subparts, we will solve the first three…

Q: Consider a mass m confined to the x axis and subject to a force Fx = kx where k > 0. (a) Write down…

A: If the particle in the given problem has energy E>0, then it can cross the origin. If it has…

Q: : The Hamiltonian for the one-dimensional simple harmonic oscillator is: mw? 1 ÎĤ =- + 2m Use the…

Q: Question one: Two masses m, M are joined with a light string of length L over a pulley of radius R…

A: Let center of disk be the level of zero potential energy and y1: height of m below the center of…

Q: Employing the power expansion to the solution of the equation of motion, show that for a…

A: Belongs to quantum dynamics and time development in quantum mechanics.

Q: Q = log (sin p) P = q cot p,

Q: 4.5 In aerodynamics and fluid mechanics, the functions u(x,y) and v(x,y) in f(z) = u(x,y) + i v(x,…

Q: If the Lagrange's function is given by m (4+ ġ142 + 93) – V(q1,92). 2 Prove that 2 (а) Н — Зт (Pi +…

A: Part a: The canonical momentum corresponding to q1 is given by: Similarly, the canonical momentum…

Q: Find the gradient and the Laplacian at V(x19, 2) = √(x-1)^-+ (13-b)+(2-c)

Q: A simple pendulum of mass m and length L is shown in the figure below. Take the reference of…

A: Part (a) The given pendulum is constrained to move in the XY plane only. Hence z coordinate of the…

Q: Q2: A pendulum bob of mast m is suspended by a string of length / from a car of mass M which moves…

A: To determine: (a) The Lagrangian function The mass of the pendulum is m, the length of the pendulum…

Q: Let a two-degree-of-freedom system be described by the Hamiltonian = 1/ (p² + p ²) + V(x, y) and…

A: Given Hamilton : And the potential energy V is a homogeneous function of degree -2 for all

Q: The time-evolution of a physical system with one coordinate q is described by the La- grangian L = ?…

A: **as per our company guidelines we are supposed to answer only first 3 sub-parts. Kindly repost…

Q: 1. For an one-dimensional harmonic oscillator, the Hamiltonian is given as p2 +kx?. 1 H 2m (1a)…

Q: 2.32 Consider a harmonic oscillator of mass m undergoing harmonic motion in two dimensions x and y.…

Q: A particle of mass m moves in the one-dimensional potential U(x) = U, Se-x/a Sketch U(x). Identify…

A: Here, we use the maxima/minima conditions to get the required.

Q: 17. Consider a particle in a one- dimensional simple harmonic oscillator, subject to perturbing…

A: Given x=h2mω(a+a+) where, h=h2π And from the rule of lowering and raising operator. a|n> =…

Q: Show that the Young's modulus Y, modulus of rigidity n and Poisson's ratio o are related by the…

A: In elasticity, Young's modulus=Modulus of rigidity=Poisson's ratio=

Q: A wire of the shape of y=ax² is rotating around its vertical axis with an angular velocity wo (see…

Q: Show that the minimum energy of a simple harmonic oscillator is hω/2. What is the minimum energy in…

A: The mean value of x2av is the mean square devation ∆x2By substituting the mean square deviation…

Q: Consider a mechanical system with one degree of freedom x, conjugate momentum p evolving under the…

A: a) For a given system defined by Hamiltonian H(x,p); Microcanonical ensemble is defined as follows…

Q: Prove the triple product identity Ax(B×C)= B(A·C)-C(A·B).

Q: For a particle confined on a ring (with periodic boundaries) the appropriate wavefunction and…

A: Given: Hamiltonian operator = H^ = -ℏ22Id2dϕ2ψml(ϕ) = eimlϕ2π1/2

Q: The Hamiltonian of a three-level system is represented by the matrix 22 Vo 2V + 1 22 H = 3V where Vo…

A: Hey dear this is related to perturbation theory have a look at solution

Q: In the following vibration system, disk 2 rolls on a non-slip part with uniform mass distribution…

A: Given data, Mass of each component = m Spring constant of the spring = k

Q: Laplace transforms find broad applications in the modelling of oscillators for energy harvesting…

A: Since you have posted multiple questions, we will provide the solution only to the first question as…

Q: Please provide a written answer

Q: Using Lagrange's equation, obtain the equation of motion for a particle in plane polar coordinates.

A: A particle in a plane has a two-dimensional motion with two coordinates, x and y. Let the potential…

Q: Demonstrate that the Hamiltonian operator for a particle experiencing a harmonic potential V (x) =…

A: The Hamiltonian for a particle in a harmonic potential is given by the sum of the kinetic and…

Q: Consider the following scalar and vectors: Ã = 41+ 3) + 7k T = 3x+8yz + 2zx 1. Calculate for the…

A: REQUIRED LAPLACIAN. DIVERGENCE.

Q: 3. Consider a single particle of mass m in spherical coordinates, with the kinetic energy lees so…

Q: Using what are canonical transformations ? generating function F=1m 009² cót @. Discuss the problem…

Question

Show that the function

S = (mω/2)(q² + α²) cot ωt − mωqα csc ωt
is a solution of the Hamilton-Jacobi for Hamilton’s principal function for the
linear harmonic oscillator with

H =(1/2m)(p² + m²ω²q²).

Show that this function generates a correct solution to the motion of the
harmonic oscillator.

Expert Solution