![Problem 1-1: Noethers theorem tells us that whenever a system exhibits a continuous symmetry, there is an associated conserved quantity in the system. Find the associated conserved quantity for space translations. Hint: Assume a general system Lagrangian of the form L(q,4, t) and start with the Euler-Lagrange equation of motion.]](http://img.homeworklib.com/questions/418d6a50-3f30-11eb-bbb3-639e8de4be7a.png?x-oss-process=image/resize,w_560)
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Problem 1-1: Noether's theorem tells us that whenever a system exhibits a continuous symmetry, there is...
Recall how we saw in class that if we add a total time derivative of a...
Recall how we saw in class that if we add a total time derivative of a function g(g; t) on configuration space to the Lagrangian then the equations of motion were unaffected, i.e. the new Lagrangian gives the same Euler-Lagrange equations as the original Lagrangian L. The reason this worked was that the action only shifted by a boundary term, which doesn't affect the extremization procedure We have also seen that if q is a cyclic coordinate, meaning aL/aq0, then...
4. The dynamics of a simple harmonic oscillator is described by a Lagrangian (a) Show that the La...
Question (b) and (c) ONLY.
Thanks.
4. The dynamics of a simple harmonic oscillator is described by a Lagrangian (a) Show that the Lagrangian changes with a full derivative, L' = L +X, consequently the action is invariant under the two-parameter transformation yA sin(t w) +Bcos(t )+x (b) Find the two independent constants of the motion associated with an infinitesimal version of the above transformation, and identify their physical meaning. (c) Use the results of (b) to write down the...
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...
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...
-US Help 1 System Announcements Anton, Calculus! Early Transcendentals, lle Start Time: 10:47 PM / Remaining:...
-US Help 1 System Announcements Anton, Calculus! Early Transcendentals, lle Start Time: 10:47 PM / Remaining: 79 min. ES Question 2 Find T() and N(t) at the given point. x = e' cost, y = e' sint, z = e'; t = 0 Enter the vector i as $7, the vector jas 7, and the vector k as T(0) = Edit N(0) = Edit US Anton, Calculus: Early Transcendentals, 11e Help | System Announcements tart Time: 10:47 PM / Remaining:...
Use the following information To help you solve the following questions. Show all work for thumbs...
Use the following information
To help you solve the following questions. Show all
work for thumbs up.
3.1 Rotations and Angular-Momentum Commutation Relations 159 We are particularly interested in an infinitesimal form of Ry: (3.1.4) where terms of order & and higher are ignored. Likewise, we have R0= ° :- R(E) = 1 (3.1.5) and (3.1.5b) - E01 which may be read from (3.1.4) by cyclic permutations of x, y, zthat is, x y , y → 2,2 → x....
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra...
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra be the spacing between the inner and outer conductors. (a) Let the radii of the two conductors be only slightly different, so that d << ra. Show that the result derived in Example 24.4 (Section 24.1) for the capacitance of a cylindrical capacitor then reduces to Eq. (24.2), the equation for the capacitance of a parallel-plate capacitor, with A being the surface area of...
Chapter overview 1. Reasons for international trade Resources reasons Economic reasons Other reasons 2. Difference between...
Chapter overview 1. Reasons for international trade Resources reasons Economic reasons Other reasons 2. Difference between international trade and domestic trade More complex context More difficult and risky Higher management skills required 3. Basic concept s relating to international trade Visible trade & invisible trade Favorable trade & unfavorable trade General trade system & special trade system Volume of international trade & quantum of international trade Commodity composition of international trade Geographical composition of international trade Degree / ratio of...
Recall how we saw in class that if we add a total time derivative of a function g(g; t) on configuration space to the Lagrangian then the equations of motion were unaffected, i.e. the new Lagrangian gives the same Euler-Lagrange equations as the original Lagrangian L. The reason this worked was that the action only shifted by a boundary term, which doesn't affect the extremization procedure We have also seen that if q is a cyclic coordinate, meaning aL/aq0, then...
Question (b) and (c) ONLY.
Thanks.
4. The dynamics of a simple harmonic oscillator is described by a Lagrangian (a) Show that the Lagrangian changes with a full derivative, L' = L +X, consequently the action is invariant under the two-parameter transformation yA sin(t w) +Bcos(t )+x (b) Find the two independent constants of the motion associated with an infinitesimal version of the above transformation, and identify their physical meaning. (c) Use the results of (b) to write down the...
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...
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...
-US Help 1 System Announcements Anton, Calculus! Early Transcendentals, lle Start Time: 10:47 PM / Remaining: 79 min. ES Question 2 Find T() and N(t) at the given point. x = e' cost, y = e' sint, z = e'; t = 0 Enter the vector i as $7, the vector jas 7, and the vector k as T(0) = Edit N(0) = Edit US Anton, Calculus: Early Transcendentals, 11e Help | System Announcements tart Time: 10:47 PM / Remaining:...
Use the following information
To help you solve the following questions. Show all
work for thumbs up.
3.1 Rotations and Angular-Momentum Commutation Relations 159 We are particularly interested in an infinitesimal form of Ry: (3.1.4) where terms of order & and higher are ignored. Likewise, we have R0= ° :- R(E) = 1 (3.1.5) and (3.1.5b) - E01 which may be read from (3.1.4) by cyclic permutations of x, y, zthat is, x y , y → 2,2 → x....
Consider a cylindrical capacitor like that shown in Fig. 24.6. Let d = rb − ra be the spacing between the inner and outer conductors. (a) Let the radii of the two conductors be only slightly different, so that d << ra. Show that the result derived in Example 24.4 (Section 24.1) for the capacitance of a cylindrical capacitor then reduces to Eq. (24.2), the equation for the capacitance of a parallel-plate capacitor, with A being the surface area of...
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