what is the first part of the matrix? "Fill"

Derive using the Lagrangian and Lagrange's equation only. Please draw a diagram

Determine the general solution of the 1-dimensional Laplace equation on the cylinder coordinates and ball coordinates!

. Obtain the Euler – Lagrange equation for the extremals of the functional X2 | [y2 - yy'+y'2 ]dlx X1

Introduction to Classical Dynamics The Lagrangian Method Please I need a complete solution of this, thank you.

The Hamiltonian for the one dimensional quantum oscillator is 1 p² 1 Ĥ = 1² + ½ k²² = 12 + √ mw² ಠ2m 2m 2 where k = mw². 1) Define the operators ₁₁ and ₁₁ such that Ĥ = ½ħw (p² + ²). Define Ĥ2 as a function of 1 and p₁ such that Ĥ = hwĤ₂. - 2) Let us define the new operators â (1 + i₁) and ↠= ½(î₁ — ip₁). Express ₁ and p₁ as a function of â and â³. Knowing that [^^1,î₁] = i and [1, 1] = -i, calculate âât and â†â. Express Ĥ2 as a function of a and at. 3) Let us define Ñ such that Ĥ₂ = Ñ + ½. Knowing that Ĥ, Ĥ₂ and Ñ have the same eigenstates, what are their corresponding eigenvalues?

Problem 2 The relativistic Lagrangian for a particle of rest mass m moving along the x-axis in a potential V(x) is given by 2 L = -mc² 1 V(x) c2 (a) Derive the Euler-Lagrange equation of motion. (b) Show that it reduces to Newton's equation in the limit |*| << c. (c) Compute the Hamiltonian H of the system. Eliminate ȧ from the Hamiltonian by using the equation ƏL p = ax and write H = H(p, x) as a function of x and p only.

4. The Lagrangian of a relativistic particle is given by v2 L = -moc² 1- where mo is the rest mass of the particle, v is its velocity, and V is not velocity dependent. Find the generalized momentum and the Hamiltonian H. It may be shown that the relativistic kinetic energy T is -moc² (1-2) - V. Check that H = T + V.

This is question 8.8 in John R. Taylor's "Classical Mechanics" textbook by the way! (ISBN: 9781891389221)

please solve