Consider a particle of mass m with kinetic energy T = mx² moving in one dimension in a potential V(x). Use the Euler-Lagrange equations to find the equation of motion.
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- Express the Lagrangian for a free particle moving in a plane in a plane polar coordinates. From this proves that, in terms of radial and tangential components, the acceleration inpolar coordinates isa = (¨r − rθ˙2) er + (rθ¨ + 2 r˙ θ˙) eθ(where er and eθ are unit vectors in the positive radial and tangential directions).Two particles, each of mass m, are connected by a light inflexible string of length l. The string passes through a small smooth hole in the centre of a smooth horizontal table, so that one particle is below the table and the other can move on the surface of the table. Take the origin of the (plane) polar coordinates to be the hole, and describe the height of the lower particle by the coordinate z, measured downwards from the table surface. Here, the total force acting on the mass which is on the table is -T r^ (r hat). Why?See the provided image of the Michelson-Morley experiment and the following calculations for tA→C→A and tA→B→A. What would v/c have to be to make tA→C→A 1 percent larger than tA→B→A?
- determine the general solution to thegiven differential equation (D2 − 2D + 2)2(D2 − 1)y = 0.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.Problem 3: A mass m is thrown from the origin att = 0 with initial three-momentum po in the y direction. If it is subject to a constant force F, in the x direction, find its velocity v as a function of t, and by integrating v find its trajectory. Check that in the non-relativistic limit the trajectory is the expected parabola. Hint: The relationship F = P is still true in relativistic mechanics, but now p = ymv instead of p = mv. To find the non-relativistic limit, treat c as a very large quantity and use the Taylor approximation (1+ x)" = 1 + nx when a is small.