# Approximating solutions to minima of the discrete Lagrangian

I have been stuck on this problem for a week and I'm not sure whether or not it is hard or I'm just missing something obvious.

General gist of the problem

I have a variational problem on a Riemannian manifold where I am approximating a Lagrangian with a discrete Lagrangian. I am looking for some sort of guarantee that critical points of the discrete Lagrangian can be approximated by curves of a certain level of differentiability such that they end up approximating a solution of the original problem. A reference would be great as I don't mind following this up myself but I don't know where to search.

My specific problem

I have a complete Riemannian manifold $$M$$ and I have a functional $$f : C^{2}_{v_0, v_1}([0,1], M) \rightarrow \mathbb{R}$$ defined over $$C^2$$ curves $$x : [0, 1] \rightarrow M$$ satisfying $$\dot{x}(0) = v_0 \in TM$$ and $$\dot{x}(1) = v_1 \in TM$$ and we now add a few additional constraints such as $$x(\frac{1}{3}) = x_1$$ and $$x(\frac{2}{3}) = x_2$$. The functional is then defined as$$f(x) = \int_0^1 \| \nabla_{\dot{x}(t)} \dot{x}(t) - V\|^2 dt$$

where $$V$$ is a bounded Lipschitz (in the sense there is a $$k$$ such that $$\||V(y)\| - \|V(x)\| | \leq k d(x,y)$$) smooth vector field. I now consider the discrete Lagrangian defined by the formula: $$f_n(x_n) = \dfrac{1}{N} \left(\sum_{i=1}^{N-1} \|N^2(l_i^+ + l_i^-) - V\|^2 \right)$$

Here $$x_i := x(\frac{i}{N})$$, and the covariant derivative is approximated using the functions $$l_i^+, l_i^-$$ where I define $$l_i^+$$ by $$\log_{x_i} x_{i+1}$$ and $$l_i^-$$ by $$\log_{x_i} x_{i-1}$$. Again, the $$x_i$$ are constrained under the same constraints by requiring that $$N l_0^+ = v_0$$ and $$N l_N^- = -v_1$$. We can set $$N$$ to be a multiple of $$3$$ so that we can set $$x_{\frac{N}{3}} = x_1$$ and $$x_{\frac{2N}{3}} = x_2$$.

If I find a critical point of $$f_n$$, call it $$y_n$$, can I find a curve $$y$$ that's $$C^2$$ which agrees with $$y_n$$ at $$t = \frac{i}{N}$$ such that $$\|f(y) - f_n(y_n)\| \rightarrow 0$$?

• This is a sort of interpolation problem. In flat space, good candidates are splines and Hermite intepolants. You might be able to use these locally. Mar 23 '13 at 19:55
• Thanks for the suggestion. I will try locally using minimisers of $\int_{t_0}^{t_1} \| x^{(2)} - V \|^2$ and piecing them together. Maybe I can piece them together in a way so that I don't significantly add to the functionals value across any split. Mar 24 '13 at 2:53