Is there a canonical way of lifting a vector field $X$ on $M$ to its tangent bundle $TM$? Is there a canonical way of lifting vector field $X$ on $M$ to its tangent bundle $TM$?
I came up with this question while studying tangent bundle formalism of Lagrangian mechanics. In Lagrangian mechanics, a vector field on $X$ on $M$ may represent a continuous symmetry, and since the Lagrangian function is defined on $TM$, we want to lift $X$ to a vector field in $TM$. Is there a canonical way to do this?

Let $M$ be a differentiable manifold (in the context of lagrangian mechanics, you can see $M$ as the space of all possible positions of the system). Suppone that on $M$ lagrangian coordinates $q=(q^j)$ are defined. 
A vector field $X$ on $M$ can be written as
$$X = X^j(q) \frac{\partial}{\partial q^j}$$
where $X^j$ is the $j$-th component of $X$. 
Then the lifting of $X$ is the vector field on $TM$ defined by
$$X^T= X^j(q) \frac{\partial}{\partial q^j} + (\frac{d}{dt}X^j(q)) \frac{\partial}{\partial \dot{q^j}}. $$
