Solving The Jacobi Equation
Let and complete this to get an orthonormal basis at . Parallel transport it to get a basis all along . This gives an orthonormal basis with . The Jacobi field can be written in co-ordinates in terms of this basis as and thus
and the Jacobi equation can be rewritten as a system
for each . This way we get a linear ordinary differential equation (ODE). Since this ODE has smooth coefficients we have that solutions exist for all and are unique, given and, for all .
Read more about this topic: Jacobi Field
Famous quotes containing the words solving the, solving, jacobi and/or equation:
“More than a decade after our fellow citizens began bedding down on the sidewalks, their problems continue to seem so intractable that we have begun to do psychologically what government has been incapable of doing programmatically. We bring the numbers down—not by solving the problem, but by deciding it’s their own damn fault.”
—Anna Quindlen (b. 1952)
“Will women find themselves in the same position they have always been? Or do we see liberation as solving the conditions of women in our society?... If we continue to shy away from this problem we will not be able to solve it after independence. But if we can say that our first priority is the emancipation of women, we will become free as members of an oppressed community.”
—Ruth Mompati (b. 1925)
“During the long ages of class rule, which are just beginning to cease, only one form of sovereignty has been assigned to all men—that, namely, over all women. Upon these feeble and inferior companions all men were permitted to avenge the indignities they suffered from so many men to whom they were forced to submit.”
—Mary Putnam Jacobi (1842–1906)
“A nation fights well in proportion to the amount of men and materials it has. And the other equation is that the individual soldier in that army is a more effective soldier the poorer his standard of living has been in the past.”
—Norman Mailer (b. 1923)