Homework Answers
According to Chegg's guidelines here I have done the first question only.but here I am for help you.that's why if you need 1.2 also.then comment me.i will help you surely.
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QUESTION 1 Consider the variational problem with Lagrangian function (1,3,2,3,») - ++y* - 328y. (1.1) Obtain...
QUESTION 1 Consader the variational problem with Lagrangan function Lis. )+ky-)-+ Obtaun the Ealer-Tagrange equatio...
QUESTION 1 Consader the variational problem with Lagrangan function Lis. )+ky-)-+ Obtaun the Ealer-Tagrange equations (Do not solve them) (1 1) (10) (1 2) For this system (a) Construct the Hamiltonian funeton from L (6) (b) Derive the Hamalton's equations (4) (e) Wnte down the Hamalton-Jacobs equation (4) (24]
QUESTION 1 Consader the variational problem with Lagrangan function Lis. )+ky-)-+ Obtaun the Ealer-Tagrange equations (Do not solve them) (1 1) (10) (1 2) For this system (a) Construct the Hamiltonian...
3. The Lagrangian for a relativistic particle of (rest) mass m is L=-me²/1- (A² - Elmo...
3. The Lagrangian for a relativistic particle of (rest) mass m is L=-me²/1- (A² - Elmo (The corresponding action S = ( L dt is simply the length of the particle's path through space-time.) (a) Show that in the nonrelativistic limit (v << c) the result is the correct nonrelativistic kinetic energy, plus a constant corresponding to the particle's rest energy. (Hint. Use the binomial expansion: for small 2, (1 + 2) = 1 +a +a(-1) + a(a-1)(-2) 13 +...
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...
Solve the following problems: Problem 1: masses&springs Two masses mand m2 connected by a spring of...
Solve the following problems:
Problem 1: masses&springs Two masses mand m2 connected by a spring of elastic constant k slide on a frictionless inclined plane under the effect of gravity. Let a be the angle between the the x axis and the inclined plane, r the distance between the two masses, l the position of the first mass with respect to the top of the plane (see figure). Considering the top of the plane to be the zero for potential...
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...
2. Consider a mass m moving in R3 without friction. It is fasten tightly at one...
2. Consider a mass m moving in R3 without friction. It is fasten tightly at one end of a string with length 1 and can swing in any direction. In fact, it moves on a sphere, a subspace of R3 1 0 φ g 2.1 Use the spherical coordinates (1,0,) to derive the Lagrangian L(0,0,0,0) = T-U, namely the difference of kinetic energy T and potential energy U. (Note r = 1 is fixed.) 2.2 Calculate the Euler-Lagrange equations, namely...
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)...
1 Consider a cylinder of mass M and radius a rolling down a half-cylinder of radius R as shown in...
question (c), (d), (e), (f) please. Thanks.
1 Consider a cylinder of mass M and radius a rolling down a half-cylinder of radius R as shown in the diagram (a) Construct two equations for the constraints: i rolling without slipping (using the two angles and θ), and ii) staying in contact (using a, R and the distance between the axes of the cylinders r). (b) Construct the Lagrangian of the system in terms of θ1, θ2 and r and two...
question (c), (d), (e), (f) please. Thanks. 1 Consider a cylinder of mass M and radius...
question (c), (d), (e), (f) please. Thanks.
1 Consider a cylinder of mass M and radius a rolling down a half-cylinder of radius R as shown in the diagram (a) Construct two equations for the constraints: i rolling without slipping (using the two angles and θ), and ii) staying in contact (using a, R and the distance between the axes of the cylinders r). (b) Construct the Lagrangian of the system in terms of θ1, θ2 and r and two...
Question 4: Consider a general utility function U(xi, x2). Let's now solve for the optimal bundle...
Question 4: Consider a general utility function U(xi, x2). Let's now solve for the optimal bundle generally using the Lagrangian Method. 1. Write down the objective function and constraint in math 2. Set up the Lagrangian Equation 3. Fnd the first derivatives. 4, Find the first order conditions, what's the interpretation for λ? 5. Rearrange them to get the tangency condition.
QUESTION 1 Consader the variational problem with Lagrangan function Lis. )+ky-)-+ Obtaun the Ealer-Tagrange equations (Do not solve them) (1 1) (10) (1 2) For this system (a) Construct the Hamiltonian funeton from L (6) (b) Derive the Hamalton's equations (4) (e) Wnte down the Hamalton-Jacobs equation (4) (24]
QUESTION 1 Consader the variational problem with Lagrangan function Lis. )+ky-)-+ Obtaun the Ealer-Tagrange equations (Do not solve them) (1 1) (10) (1 2) For this system (a) Construct the Hamiltonian...
3. The Lagrangian for a relativistic particle of (rest) mass m is L=-me²/1- (A² - Elmo (The corresponding action S = ( L dt is simply the length of the particle's path through space-time.) (a) Show that in the nonrelativistic limit (v << c) the result is the correct nonrelativistic kinetic energy, plus a constant corresponding to the particle's rest energy. (Hint. Use the binomial expansion: for small 2, (1 + 2) = 1 +a +a(-1) + a(a-1)(-2) 13 +...
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...
Solve the following problems:
Problem 1: masses&springs Two masses mand m2 connected by a spring of elastic constant k slide on a frictionless inclined plane under the effect of gravity. Let a be the angle between the the x axis and the inclined plane, r the distance between the two masses, l the position of the first mass with respect to the top of the plane (see figure). Considering the top of the plane to be the zero for potential...
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...
2. Consider a mass m moving in R3 without friction. It is fasten tightly at one end of a string with length 1 and can swing in any direction. In fact, it moves on a sphere, a subspace of R3 1 0 φ g 2.1 Use the spherical coordinates (1,0,) to derive the Lagrangian L(0,0,0,0) = T-U, namely the difference of kinetic energy T and potential energy U. (Note r = 1 is fixed.) 2.2 Calculate the Euler-Lagrange equations, namely...
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)...
question (c), (d), (e), (f) please. Thanks.
1 Consider a cylinder of mass M and radius a rolling down a half-cylinder of radius R as shown in the diagram (a) Construct two equations for the constraints: i rolling without slipping (using the two angles and θ), and ii) staying in contact (using a, R and the distance between the axes of the cylinders r). (b) Construct the Lagrangian of the system in terms of θ1, θ2 and r and two...
question (c), (d), (e), (f) please. Thanks.
1 Consider a cylinder of mass M and radius a rolling down a half-cylinder of radius R as shown in the diagram (a) Construct two equations for the constraints: i rolling without slipping (using the two angles and θ), and ii) staying in contact (using a, R and the distance between the axes of the cylinders r). (b) Construct the Lagrangian of the system in terms of θ1, θ2 and r and two...
Question 4: Consider a general utility function U(xi, x2). Let's now solve for the optimal bundle generally using the Lagrangian Method. 1. Write down the objective function and constraint in math 2. Set up the Lagrangian Equation 3. Fnd the first derivatives. 4, Find the first order conditions, what's the interpretation for λ? 5. Rearrange them to get the tangency condition.
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