Use the Euler equation and variational calculus of on the Lagrange function (L(q,q’,t)) to arrive at...
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Use
the Euler equation and variational calculus of on the Lagrange
function (L(q,q’,t)) to arrive at...
5. Euler and Lagrange developed the calculus of variations. (a) Explain in a sentence what the ca...
5. Euler and Lagrange developed the calculus of variations. (a) Explain in a sentence what the calculus of variations is about. (b) Find the Euler-Lagrange equation (assuming y(0) 0 and y(1) 0 for minimizing the integral: (ul(t)(t)) dt. + (y(t))2 Recall that for the Lagrangian density function L(t, y, u), the Euler-Lagrange equation is as a function of t where u(t) = t). (c) How is the Euler-Lagrange equation in general related to the directional derivative of vector calculus? Why...
Please help with 2.4. It must be solved using the Euler-Lagrange equation. The answer is a...
Please help with 2.4. It must be solved using the Euler-Lagrange
equation. The answer is a right cylinder, but I'm not even sure how
to use the E-L equation to get that answer. Thank you!
2.4 Consider a solid of revolution of a given height. Determine the shape of the solid if it has the minimum moment of inertia about its axis 2.5 Consider the variational problem for variable end points. (a) Let
QUESTION 1 Consider the variational problem with Lagrangian function (1,3,2,3,») - ++y* - 328y. (1.1) Obtain...
QUESTION 1 Consider the variational problem with Lagrangian function (1,3,2,3,») - ++y* - 328y. (1.1) Obtain the Euler-Lagrange equations and solve them. (10) (1.2) For this system (a) Construct the Hamiltonian function from L (b) Obtain the Hamilton's equations. (e) Write down the Hamilton-Jacobi equation. (6) (4) (4) (24)
(a) Show that, if y satisfies the Euler-Lagrange equation associated with the integral 2. qy2) dx,...
(a) Show that, if y satisfies the Euler-Lagrange equation associated with the integral 2. qy2) dx, I = (6) where p() and q(x) are known functions, then I has the value 12 (b) Show that, if y satisfies the Euler-Lagrange equation associated with (6) and if z(x) is an arbitrary differentiable function for which z(x)z(r2) = 0, (7) 1 then (yyz)da= 0. + Hence show that by replacing y in (6) by the function (y + z), where the condition...
(4) Let f R -R be a strictly conve:r C2 function and let 0 a) Write the Euler-Lagrange equation...
(4) Let f R -R be a strictly conve:r C2 function and let 0 a) Write the Euler-Lagrange equation for the minimizer u.(x) of the following problem: minimize u subject to: u E A, where A- 0,REC1[0 , 1and u (0 a u(1)b) b) Assuming the minimizer u(a) is a C2 function, prove t is strictly convex
(4) Let f R -R be a strictly conve:r C2 function and let 0
a) Write the Euler-Lagrange equation for the minimizer u.(x)...
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...
Construct the Lagrange L function for a particle that moves on the surface of a parabolic...
Construct the Lagrange L function for a particle that moves on
the surface of a parabolic cylinder immersed in the gravity field.
Find the equation of motion
. W c
Consider the Lagrangian density ih Construct the equations of motion for the field from the Euler-Lagrange...
Consider the Lagrangian density ih Construct the equations of motion for the field from the Euler-Lagrange equations, and show that it leads to the Schrödinger equation in dt2m and its complex conjugate.
3. Find all critical points of dt dt with the constraint PP = 8 0 (c and boundary conditions x(0) - 0, x(1)- 3. Hint: Write the Euler Lagrange equation (there is no dependence on t), and then use the...
3. Find all critical points of dt dt with the constraint PP = 8 0 (c and boundary conditions x(0) - 0, x(1)- 3. Hint: Write the Euler Lagrange equation (there is no dependence on t), and then use the boundary conditions and the constraint to reach a system of 2 equations (with quadratic terms) of two unknown constants a, b Solve it by first finding a quadratic equation for a/b
3. Find all critical points of dt dt with...
Problem 1. For each of the following functions f (x,y,y'), use the Euler-Lagrange equations to derive...
Problem 1. For each of the following functions f (x,y,y'), use the Euler-Lagrange equations to derive a differential equation for the function y(x) that minimizes the functional Fy (x,y,y') dx. Do all calculations by hand. 1. f(x,y,y') = { (y')? – eXy 2. f (x, y, y') = 3y2 – ery 3. f (x,y, y') =y(1+(y)2) "? 4. f (x,y,y') =
5. Euler and Lagrange developed the calculus of variations. (a) Explain in a sentence what the calculus of variations is about. (b) Find the Euler-Lagrange equation (assuming y(0) 0 and y(1) 0 for minimizing the integral: (ul(t)(t)) dt. + (y(t))2 Recall that for the Lagrangian density function L(t, y, u), the Euler-Lagrange equation is as a function of t where u(t) = t). (c) How is the Euler-Lagrange equation in general related to the directional derivative of vector calculus? Why...
Please help with 2.4. It must be solved using the Euler-Lagrange
equation. The answer is a right cylinder, but I'm not even sure how
to use the E-L equation to get that answer. Thank you!
2.4 Consider a solid of revolution of a given height. Determine the shape of the solid if it has the minimum moment of inertia about its axis 2.5 Consider the variational problem for variable end points. (a) Let
QUESTION 1 Consider the variational problem with Lagrangian function (1,3,2,3,») - ++y* - 328y. (1.1) Obtain the Euler-Lagrange equations and solve them. (10) (1.2) For this system (a) Construct the Hamiltonian function from L (b) Obtain the Hamilton's equations. (e) Write down the Hamilton-Jacobi equation. (6) (4) (4) (24)
(a) Show that, if y satisfies the Euler-Lagrange equation associated with the integral 2. qy2) dx, I = (6) where p() and q(x) are known functions, then I has the value 12 (b) Show that, if y satisfies the Euler-Lagrange equation associated with (6) and if z(x) is an arbitrary differentiable function for which z(x)z(r2) = 0, (7) 1 then (yyz)da= 0. + Hence show that by replacing y in (6) by the function (y + z), where the condition...
(4) Let f R -R be a strictly conve:r C2 function and let 0 a) Write the Euler-Lagrange equation for the minimizer u.(x) of the following problem: minimize u subject to: u E A, where A- 0,REC1[0 , 1and u (0 a u(1)b) b) Assuming the minimizer u(a) is a C2 function, prove t is strictly convex
(4) Let f R -R be a strictly conve:r C2 function and let 0
a) Write the Euler-Lagrange equation for the minimizer u.(x)...
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...
Construct the Lagrange L function for a particle that moves on
the surface of a parabolic cylinder immersed in the gravity field.
Find the equation of motion
. W c
Consider the Lagrangian density ih Construct the equations of motion for the field from the Euler-Lagrange equations, and show that it leads to the Schrödinger equation in dt2m and its complex conjugate.
3. Find all critical points of dt dt with the constraint PP = 8 0 (c and boundary conditions x(0) - 0, x(1)- 3. Hint: Write the Euler Lagrange equation (there is no dependence on t), and then use the boundary conditions and the constraint to reach a system of 2 equations (with quadratic terms) of two unknown constants a, b Solve it by first finding a quadratic equation for a/b
3. Find all critical points of dt dt with...
Problem 1. For each of the following functions f (x,y,y'), use the Euler-Lagrange equations to derive a differential equation for the function y(x) that minimizes the functional Fy (x,y,y') dx. Do all calculations by hand. 1. f(x,y,y') = { (y')? – eXy 2. f (x, y, y') = 3y2 – ery 3. f (x,y, y') =y(1+(y)2) "? 4. f (x,y,y') =
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