Lagrange Equations, Hamiltonian Principle and Calculus of Variations:

Curves that give the shortest distance between two points on a given surface are called the geodestics of the surface.

a) Using the calculus of variations, show that the geodesics of a spherical surface are great circles, i.e. circles whose centers lie at the center of the sphere.

b) Generalize the calculus of variations to a function of many independent variables, $q_i,\dot{q_i}$, such as the Lagrangian, $L(q_i,\dot{q_i})$, to show that Lagrange's equations:

$$\frac{d}{dt}\frac{\partial L}{\partial \dot{q_i}} - \frac{\partial L}{\partial q_i}$$

follow from Hamilton's principle (Goldstein, pg.36):

The motion of the system from time $t1$ to time $t2$ is such that the line integral:

$$I=\int_{t1}^{t2} L dt$$

where $L=T-V$, has a stationary value for the correct path of the motion. That is, out of all possible paths by which the system point could travel from its position at time $t1$ to its position at time $t2$, it will actually travel along that path for which the value of the integral is stationary. By the term ``stationary value'' for a line integral we mean that the integral along the given path has the same value to within first-order infinitesimals as that along all neighboring paths (i.e., those that differ from it by infinitesimal displacements). Note that $I$ is referred to as the action or action integral.