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2. Consider a mass m moving in R3 without friction. It is fasten tightly at one...
1. Show that the Lagrangians L(t,q, y) and Īct, 4, ) = L(1,4,0) + f/10, 9)...
1. Show that the Lagrangians L(t,q, y) and Īct, 4, ) = L(1,4,0) + f/10, 9) yield the same Euler-Lagrange equations. Here q e R and f(t,q) is an arbitrary function. 2 Lagrangian mechanics In mechanics, the space where the motion of a system lies is called the configuration space, which is usually an n-dimensional manifold Q. Motion of a system is defined as a curve q : R + Qon Q. Conventionally, we use a rather than 1 to...
3. A particle of mass m moves in one dimension, and has position r(t) at time t. The particle has...
Mechanics.
3. A particle of mass m moves in one dimension, and has position r(t) at time t. The particle has potential energy V(x) and its relativistic Lagrangian is given by where mo is the rest mass of the particle and c is the speed of light (a) Writing qr and denoting by p its associated canonical momenta, show that the Hamiltonian is given by (show it from first principles rather than using the energy mzc2 6 marks (b) Write...
1. A particle of mass m moves in three dimensions, and has position r(t)-(x(t), y(t), z(t)) at ti...
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,...
Question 3 3. Consider a plane pendulum consisting of a mass m suspended by a massless...
Question 3
3. Consider a plane pendulum consisting of a mass m suspended by a massless string of length I. Suppose that that time t-0 the pendulum is put into motion and the length of the string is shortened at a constant rate ot-a (ie. L(t)= Lo-at). Use the angle of the pendulum φ as your generalized coordinate. (a) (2 points) Obtain the Lagrangian and Hamiltonian for this system (b) (0.5 points) Is H conserved? How can you tell? (c)...
5. Consider the following time-dependent Lagrangian for a system with one degree of freedom , (10)...
5. Consider the following time-dependent Lagrangian for a system with one degree of freedom , (10) where 8, m and k are fixed real constants greater than zero. (total 10 points) (a) Write down the Euler-Lagrange equation of motion for this system, and interpret the resulting equation in terms of a known physical system. (1 point) (b) Find Hamiltonian via Legendre transformation. (1 point) (c) Show that q(t) and the corresponding canonical momentum p(t) can be found as follows for...
3. The Hamiltonian of a particle of mass m and charge q in a static magnetic field may be written 2 where πί Pi-qAi(x). We shall assume that the magnetic field B is uniform, so that AiEijkBjxk is a s...
3. The Hamiltonian of a particle of mass m and charge q in a static magnetic field may be written 2 where πί Pi-qAi(x). We shall assume that the magnetic field B is uniform, so that AiEijkBjxk is a suitable choice. (a) Find Hamilton's equation of motion for the particle. (Hint: To simplify the algebra, use the chain rule to write9and similarly for p) 8H UT, 0z,, and similarly for Sp use the chain rule to write oz (b) Show...
2. (35 points) A pendulum consists of a point mass (m) attached to the end of a spring (massless ...
2. (35 points) A pendulum consists of a point mass (m) attached to the end of a spring (massless spring, equilibrium length-Lo and spring constant- k). The other end of the spring is attached to the ceiling. Initially the spring is un-sketched but is making an angle θ° with the vertical, the mass is released from rest, see figure below. Let the instantaneous length of the spring be r. Let the acceleration due to gravity be g celing (a) (10...
1. Consider a spin-0 particle of mass m and charge q moving in a symmetric three-dimensional harm...
1. Consider a spin-0 particle of mass m and charge q moving in a symmetric three-dimensional harmonic oscillator potential with natural frequency W.Att-0 an external magnetic field is turned on which is uniform in space but oscillates with temporal frequency W as follows. E(t)-Bo sin(at) At time t>0, the perturbation is turned off. Assuming that the system starts off at t-0 in the ground state, apply time-dependent perturbation theory to estimate the probability that the system ends up in an...
2. Microcanonical ensemble: One-dimensional chain. (24 pts.) Consider a one-dimensional chain consisting of N segments as illus- trated in Figure 1. Let the length of each segment be a when the...
2. Microcanonical ensemble: One-dimensional chain. (24 pts.) Consider a one-dimensional chain consisting of N segments as illus- trated in Figure 1. Let the length of each segment be a when the long dimension of the segment is parallel to the chain and 0 when the long dimension is normal to the chain direction. Each segment has just two non-degenerate states: long dimension parallel to the chain or perpen- dicular to the chain. Now consider a macrostate of the chain in...
1. Consider a mass M moving near a flat surface (which we may take to be...
1. Consider a mass M moving near a flat surface (which we may take to be 0 in the presence of the gravitational acceleration g 9.8 m/s2. (a) Show using the Wilson Sommerfeld Quantization rule that the amplitude of bounces To and the system energy are quantized. For this purpose, use: / pio It may be useful to review the example of the harmonic oscillator where we used p md/dt and q-r. In the case of this question, one full...
1. Show that the Lagrangians L(t,q, y) and Īct, 4, ) = L(1,4,0) + f/10, 9) yield the same Euler-Lagrange equations. Here q e R and f(t,q) is an arbitrary function. 2 Lagrangian mechanics In mechanics, the space where the motion of a system lies is called the configuration space, which is usually an n-dimensional manifold Q. Motion of a system is defined as a curve q : R + Qon Q. Conventionally, we use a rather than 1 to...
Mechanics.
3. A particle of mass m moves in one dimension, and has position r(t) at time t. The particle has potential energy V(x) and its relativistic Lagrangian is given by where mo is the rest mass of the particle and c is the speed of light (a) Writing qr and denoting by p its associated canonical momenta, show that the Hamiltonian is given by (show it from first principles rather than using the energy mzc2 6 marks (b) Write...
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,...
Question 3
3. Consider a plane pendulum consisting of a mass m suspended by a massless string of length I. Suppose that that time t-0 the pendulum is put into motion and the length of the string is shortened at a constant rate ot-a (ie. L(t)= Lo-at). Use the angle of the pendulum φ as your generalized coordinate. (a) (2 points) Obtain the Lagrangian and Hamiltonian for this system (b) (0.5 points) Is H conserved? How can you tell? (c)...
5. Consider the following time-dependent Lagrangian for a system with one degree of freedom , (10) where 8, m and k are fixed real constants greater than zero. (total 10 points) (a) Write down the Euler-Lagrange equation of motion for this system, and interpret the resulting equation in terms of a known physical system. (1 point) (b) Find Hamiltonian via Legendre transformation. (1 point) (c) Show that q(t) and the corresponding canonical momentum p(t) can be found as follows for...
3. The Hamiltonian of a particle of mass m and charge q in a static magnetic field may be written 2 where πί Pi-qAi(x). We shall assume that the magnetic field B is uniform, so that AiEijkBjxk is a suitable choice. (a) Find Hamilton's equation of motion for the particle. (Hint: To simplify the algebra, use the chain rule to write9and similarly for p) 8H UT, 0z,, and similarly for Sp use the chain rule to write oz (b) Show...
2. (35 points) A pendulum consists of a point mass (m) attached to the end of a spring (massless spring, equilibrium length-Lo and spring constant- k). The other end of the spring is attached to the ceiling. Initially the spring is un-sketched but is making an angle θ° with the vertical, the mass is released from rest, see figure below. Let the instantaneous length of the spring be r. Let the acceleration due to gravity be g celing (a) (10...
1. Consider a spin-0 particle of mass m and charge q moving in a symmetric three-dimensional harmonic oscillator potential with natural frequency W.Att-0 an external magnetic field is turned on which is uniform in space but oscillates with temporal frequency W as follows. E(t)-Bo sin(at) At time t>0, the perturbation is turned off. Assuming that the system starts off at t-0 in the ground state, apply time-dependent perturbation theory to estimate the probability that the system ends up in an...
2. Microcanonical ensemble: One-dimensional chain. (24 pts.) Consider a one-dimensional chain consisting of N segments as illus- trated in Figure 1. Let the length of each segment be a when the long dimension of the segment is parallel to the chain and 0 when the long dimension is normal to the chain direction. Each segment has just two non-degenerate states: long dimension parallel to the chain or perpen- dicular to the chain. Now consider a macrostate of the chain in...
1. Consider a mass M moving near a flat surface (which we may take to be 0 in the presence of the gravitational acceleration g 9.8 m/s2. (a) Show using the Wilson Sommerfeld Quantization rule that the amplitude of bounces To and the system energy are quantized. For this purpose, use: / pio It may be useful to review the example of the harmonic oscillator where we used p md/dt and q-r. In the case of this question, one full...
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