By E. Poisson

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**Example text**

There is one EL equation for each generalized coordinate. 4. The rest of the recipe is concerned with solving the equations of motion. The methods for doing this are varied, and they depend on the particular situation, just as they do in the Newtonian formulation. Let us first verify that the recipe is compatible with Newton’s laws. Consider a particle moving in three-dimensional space and subjected to a potential V (x, y, z). As indicated, we use Cartesian coordinates to describe the motion of the particle.

For ease of notation set k = m/τ . 6. A particle of mass m is traveling in the x direction. At time t = 0 it is located at x = 0 and has a speed v0 . The particle is subjected to a frictional force which opposes the motion; its magnitude is equal to βv 2 , where v = v(t) is the particle’s speed at time t and β is a positive constant. (a) What is the speed of the particle as a function of time? (b) What is the position of the particle as a function of time? 7. A particle of mass m rests on top of a sphere of radius R.

B) Plot the orbit in the x-y plane. 99, and let φ range from 0 to 16π. What is happening to the major axis of the ellipse? 8 Additional problems 1. An inclined plane makes an angle α with the horizontal. A projectile is launched from point A at the bottom of the inclined plane. Its initial speed is v0 , and its initial velocity vector makes an angle β with the horizontal. The projectile eventually hits the inclined plane at point B. Air resistance is negligible. (a) Calculate the range R of the projectile, the distance between points A and B.