Second variation formula - Revision history

Vipul: 2 revisions

2008-05-18T20:08:47Z

2 revisions

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Vipul at 11:59, 5 August 2007

2007-08-05T11:59:26Z

← Older revision		Revision as of 11:59, 5 August 2007
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	<math>\frac{1}{2} \frac{\partial^2 E(\overline{\alpha} (u^1,u^2))}{\partial u^1 \partial u^2} (0) = - \sum_t <W_t, \Delta_t DW/dt> - \int_0^1 <W_2, D^2 W_1/dt^2 + R(W,V)V> dt</math>		<math>\frac{1}{2} \frac{\partial^2 E(\overline{\alpha} (u^1,u^2))}{\partial u^1 \partial u^2} (0) = - \sum_t <W_t, \Delta_t DW/dt> - \int_0^1 <W_2, D^2 W_1/dt^2 + R(W,V)V> dt</math>

			==Facts==

			Here are two important and apparently remarkable things about the second variation formula:

			* The dependence of the right side on the 2-parameter variation is only ''through'' the [[variation vector field]]
			* Although the left side is symmetric, the right side does not ''a priori'' appear symmetric

Vipul at 11:57, 5 August 2007

2007-08-05T11:57:04Z

New page

==Statement==

===Notation===

Let <math>M</math> be a [[Riemannian manifold]] and <math>p,q \in M</math>. Let <math>\Omega = \Omega(M;p,q)</math> be the [[path space]] of <math>M</math>, viz the space of [[piecewise smooth path]]s from <math>p</math> to <math>q</math>.

Let the points of non-smoothness by <math>0=t_0 < t_1 < t_2 \ldots < t_k</math>.
Let <math>\alpha</math> be a [[2-parameter variation]] of <math>\omega</math>, and <math>W_1</math> and <math>W_2</math> be its [[variation vector field]]s along the two parameters. Let <math>V_t = d\omega/dt</math> wherever <math>\omega</math> is differentiable. For each <math>t_i</math> where <math>\omega</math> is not smooth, let <math>\Delta_{t_i}V = V_{t_i^+} - V_{t_i^-}</math>.

Denote by <math>\overline{\alpha}</math> the map sending <math>u \in U</math> (here <math>U</math> is the open set in <math>\R^2</math> over which the variation is defined) to the curve <math>t \mapsto \alpha(u_1,u_2,t)</math>.

===The formula===

<math>\frac{1}{2} \frac{\partial^2 E(\overline{\alpha} (u^1,u^2))}{\partial u^1 \partial u^2} (0) = - \sum_t <W_t, \Delta_t DW/dt> - \int_0^1 <W_2, D^2 W_1/dt^2 + R(W,V)V> dt</math>