We can find the conserved quantities if the Lagrangian is invariant to translation. For a translation, the transformation equations are:
where the function \(f_i\) is independent of the coordinates. Consider a translation in the \(q_1\) direction, where \(f_1=1\) and all other \(f_i=0\). Since time is not involved in the transformation, we also set \(g=0\), or consider only the invariance of the Lagrangian. If the Lagrangian is invariant under translations in the \(q_1\) direction, then the quantity:
is conserved. That is, we recover the observation that was made about cyclic coordinates: if a coordinate does not appear explicitly in the Lagrangian (thus the Lagrangian is invariant when that coordinate is transformed), the conjugate momentum for that coordinate is conserved.
More generally, for each direction in space that the Lagrangian is symmetric, momentum in that direction is a conserved quantity. In cartesian coordinates, this is the principle of conservation of momentum.