I am having some trouble reconciling two definitions for the Lie derivative.
Let $X$ be a vector field on a smooth manifold, $M$, and let $\varphi_t(x)$ be the local flow through the point $x \in M$. Then, the usual formulation of the Lie derivative of a vector field, $Y$, at a point, $a \in M$, is
$$ \mathcal{L}_X Y (a) := \lim_{t \rightarrow 0} \frac{(D\varphi_{-t})(Y(\varphi_t(a))) \, - \, Y(a)}{t} $$
However, in the lecture notes I am using, the Lie derivative is defined as
$$ \mathcal{L}_X Y (a) := \left[\frac{\partial}{\partial{t}}((\varphi_t)_*Y)(a)\right]_{t=0} $$
When I unroll the definitions in the second formulation, I get a slightly different result to the usual definition of the Lie derivative (see below).
Where am I going wrong:
The second definition is given by
\begin{align} & \quad \,\, \left[\frac{\partial}{\partial{t}}((\varphi_t)_* Y)(a)\right]_{t=0} \\ &= \lim_{t \rightarrow 0} \frac{((\varphi_{t})_* Y)(a) \, - \, ((\varphi_{0})_*Y)(a) }{t} \\ &= \lim_{t \rightarrow 0} \frac{((\varphi_{t})_* Y)(a) \, - \, Y(a)}{t} \end{align}
Hence, to reconcile this with the usual definition, we want to show $((\varphi_{t})_* Y)_a = (D\varphi_{-t})Y_{\varphi_t(a)}$ (where, for ease of notation, we have moved the point at which we are evaluating the vector field to the subscript).
But, for $f \in C^\infty{M}$
\begin{align} & \,\, \quad ((\varphi_{t})_* Y)_a(f) \\ &= ((D\varphi_t) Y_{\varphi_{t}^{-1}(a)})(f) \\ &= ((D\varphi_t) Y_{\varphi_{-t}(a)})(f) \\ &= Y_{\varphi_{-t}(a)}(f \circ \varphi_t) \end{align}
whereas
\begin{align} & \,\, \quad ((D\varphi_{-t})Y_{\varphi_t(a)})(f) \\ &= Y_{\varphi_t(a)}(f \circ \varphi_{-t}) \end{align}
This seems to show that the two definitions for the Lie derivative agree only up to sign. Where am I going wrong?
The first definition is the correct one, the second one measure the derivative through the flow $t \mapsto \varphi_{-t}$ which is just the orginal one with the sign changed.