I am starting to learn Classical Mechanics and while studying Noether's Theorem, stumbled upon the following question:
Let $(q_1,q_2)$ be the Lagrangian coordinates in $\mathbb{R}^2$ of two particles moving on a line, of equal masses $m_1=m_2=m$ respectively. Consider the following Lagrangian
$L(q,\dot{q}):=\frac{1}{2}m\dot{q_1}^2+\frac{1}{2}m\dot{q_2}^2-exp\{q_2-q_1\}$,
or the corresponding equivalent Hamiltonian
$H(q_1,q_2,p_1,p_2):=\frac{1}{2m}p_1^2+\frac{1}{2m}p_2^2+exp\{q_2-q_1\}$.
(1) Find a one parameter group of diffeomorphisms $\phi_s$ acting as a symmetry of $L$ and find an associated conserved quantity using Noether's Theorem.
(2) Compute explicitly the general solution $q_1(t),q_2(t)$ of the equations of motion.
My reference being Arnold's "Mathematical Methods of Classical Mechanics" where the version of Noether's Theorem states that, for a smooth manifold $M$, if the Lagrangian system $(M,L)$ admits the one parameter group of diffeomorphisms, then the Lagrangian system of equations corresponding to $L$ has a first integral $I:TM\rightarrow \mathbb{R}$.
I have a feeling that it might be useful to change the coordinates $q_1,q_2$ to $x=\frac{q_1+q_2}{2}$ and $y=q_1-q_2$ (and the corresponding momenta), but then I am kind of stuck and can use any suggestions, insights and ways to tackle the problem. Thanks in advance.
Your idea is fine: $$q_1 \mapsto q_1 + s, \ \ q_2 \mapsto q_2 + s$$ is a symmetry.
If, as suggested, we write the equations of motion with respect to the coordinates $$ x = \tfrac 1 {\sqrt 2} (q_1 + q_2), \ \ y = \tfrac 1 {\sqrt 2} (q_1 - q_2),$$
then the Lagrangian becomes
$$ L = \tfrac 1 2 \dot x^2 + \tfrac 1 2 \dot y^2 + \exp(\sqrt 2 y),$$
which is manifestly invariant under the transformation $$ x \mapsto x + \sqrt{2} s, \ \ y \mapsto y.$$
and the conserved quantity is simply $$ p_x = \frac{\partial L}{\partial \dot x} = \dot x = \tfrac 1 {\sqrt 2} (q_1 + q_2).$$