Trivial conserved Noether's current with second derivatives I'm considering a symmetry transformation on a Lagrangian
$$ \delta A = \int L(q +\delta q, \dot{q} + \delta \dot{q} , \ddot{q} + \delta \ddot{q}) dt $$
the general variation takes the form 
$$ \delta A = \int \frac{ \partial L}{\partial q} \delta q + \frac{\partial L}{\partial \dot{q}} \delta \dot{q} + \frac{\partial L}{\partial \ddot{q}} \delta \ddot{q} dt $$ 
Now, the second term inside the integral is normally handled as:
$$ \int \frac{\partial L}{\partial \dot{q}} \delta \dot{q} = \frac{\partial L}{\partial \dot{q}} \delta q - \int \frac{\partial}{\partial t}( \frac{\partial L}{\partial \dot{q}} ) \delta q $$
the third terms requires some more work, I'm having it as:
$$ \int \frac{\partial L}{\partial \ddot{q}} \delta \ddot{q} = \frac{\partial L}{\partial \ddot{q}} \delta \dot{q} - \frac{\partial}{\partial t}( \frac{\partial L}{\partial \ddot{q}} ) \delta q + \int \frac{\partial^2}{\partial^2 t}( \frac{\partial L}{\partial \ddot{q}} ) \delta q $$
So my variation (that in the case of symmetry must be zero up to boundary terms is)
$$ \delta A = \int \Big \{ \frac{\partial L}{\partial q} - \frac{\partial}{\partial t}( \frac{\partial L}{\partial \dot{q}} ) + \frac{\partial^2}{\partial^2 t}( \frac{\partial L}{\partial \ddot{q}} ) \Big \} \delta q dt
 + \Big \{ \frac{\partial L}{\partial \dot{q}} - \frac{\partial}{\partial t}( \frac{\partial L}{\partial \ddot{q}} ) \Big \} \delta q  +
 \frac{\partial L}{\partial \ddot{q}} \delta \dot{q}
$$ 
Now, I'm taking both boundary terms to be conserved currents:
$$ \frac{\partial L}{\partial \dot{q}} - \frac{\partial}{\partial t}( \frac{\partial L}{\partial \ddot{q}} ) $$
and 
$$ \frac{\partial L}{\partial \ddot{q}} $$
But if the second is a conserved current, then its derivative is zero, and the conserved current becomes trivially identical to the first-order case
What the error in my derivation?
 A: Comments to the question (v2):


*

*Let there be given a Lagrangian 
$$\tag{1} L(q,v,a,t), \qquad v^i~:=~\dot{q}^i,\qquad a^i~:=~\dot{v}^i,\qquad \jmath^i~:=~\dot{a}^i, $$ 
that depends on up to second time derivative. 

*Let 
$$\tag{2} \delta q^i~=~\varepsilon Y^i(q,v,a,t) ,$$ 
be a (global, vertical) quasi-symmetry of the Lagrangian, i.e. there exists a function $f(q,v,a,\jmath,t)$ such that 
$$\tag{3} \delta L ~=~ \varepsilon \frac{df}{dt}.$$
Here $\varepsilon$ is an infinitesimal constant parameter.

*Noether's (first) theorem states that a single quasi-symmetry (3) corresponds to a single on-shell conservation law$^1$
$$\tag{4}  \frac{dQ}{dt}~\approx~0. $$

*The corresponding (full) Noether charge is in this case
$$\tag{5}  Q~:=~\left(\frac{\partial L}{\partial v^i} - \frac{d}{dt}\frac{\partial L}{\partial a^i}\right)Y^i +\frac{\partial L}{\partial a^i}\frac{dY^i}{dt} - f. $$

*The error in OP's derivation seems to be that it is basically non-existing.
--
$^1$ Here the $\approx$ symbol denotes equality modulo Euler-Lagrange (EL) equations. Note that in order to have well-defined EL eqs. it is necessary to impose appropriate boundary conditions.
