My question:
This is just differentiation and I did it. I got $$T'(\epsilon)[L(T(\epsilon),x(T(\epsilon);\epsilon),\dot x(T(\epsilon);\epsilon))]+\int_0^{T(\epsilon)}L_x(t,x(t,\epsilon),\dot x(t,\epsilon))\frac{\partial x(t,\epsilon)}{\partial \epsilon}dt+\int_0^{T(\epsilon)}L_{\dot x}(t,x(t,\epsilon),\dot x(t,\epsilon))\frac{\partial \dot x(t,\epsilon)}{\partial \epsilon}dt$$ at $\epsilon=0$.
Since the result should be represented by $\tau$, $\xi$ and $\eta$.
So I just naively think the result is $$\tau L(T^*, x^*(T^*),\dot x^*(T^*))+\int_0^{T^*}L_x(t,x^*(t),\dot x^*(t)) \eta dt+\int_0^{T^*}L_{\dot x}(t,x^*(t),\dot x^*(t) )\frac{\partial \dot x(t,\epsilon)}{\partial \epsilon}dt|_{\epsilon =0}$$ But in my result there is no $\xi$! Where am I wrong? Can anyone help me with this?
Thanks for your help!