Write downthe Lagrangian L(xj, x 2 , i 1 , i 2 ) fortwoparticlesof equal masses. rnj = m2 = m, confined to the x axis and connected by a spring with potential energy U = !kx 2 • [Here x is the extension of the spring, x = (x 1 - x 2 - l), where l is the spring's unstretched length, and I assume that mass 1 remains to the right of mass 2 at all times.] Rewrite .C in terms of the new variables X = !(x 1 + x 2 ) (the CM position) and x (the extension), and write down the two Lagrange equations for X and x. (c) Solve for X (t) and x(t) and describe the motion.

The consumption of fossil fuels could be

Aparticle of mass m is projected with an initial velocity v0 in a direction making an angle α with the horizontal level ground as shown in the figure. the motion of the particle occurs under a uniform gravitational field g pointing downward. (a) write down the lagrangian of the system by using the cartesian coordinates (x, y). (b) is there any cyclic coordinate(s). if so, interpret it (them) physically. (c) find the euler-lagrange equations. find at least one constant of motion. (d) solve the differential equation in part (c) and obtain x and y coordinates of the projectile as a function of time. (e) construct the hamiltonian of the system, h, and write down the hamilton’s equations (canonical equations) of motion.

Which type of bond is found between the atoms of a molecule? a. metallic b. delocalized electron c. covalent d. ionic