Projections in Polar coordinate system I really understand what projections in Cartesian coordinate system, I can imagine this, but I absolutely do not understand projection in polar system. For example, I have a speed, $U$, and I must find projections $U_r ; U_{\phi}$ in polar system $(r,\phi)$
Google didn't help me.
 A: Projecting can be understood like decomposing a vector into the sum of vectors whose direction match that of the coordinate system versors. In polar coordinates the versors are: $\hat \rho$ oriented along the radius pointing outwards of the origin; and $\hat \phi$ which is a vector tangential to the circle formed by the counter clockwise rotation of $\rho$ in the point where $\rho$ is pointing.
Below, a vector in black is decomposed into its projections on $\hat \rho$ in red and $\hat \phi$ in blue.

A: This answer assumes you want to find the projections of a vector onto the polar basis at some point away from the origin. This is equivalent to rmhleo's advice and differs from that given by Kyle in the comments to the question which address a different (and simpler) problem.

The notion of projection is the same. 

Given an arbitrary vector $\vec{v}$, the components of $\vec{v}$ in terms of the coordinates $\{a,b,c\}$ ($v_a$, $v_b$, and $v_c$) are the amounts by which $\vec{v}$ points in the direction of the unit vectors associated with each coordinate: $\{\vec{e}_a,\vec{e}_b,\vec{e}_c\}$. 
These can be computed with the inner (or dot) product $v_a = \vec{v} \cdot \vec{e}_a = |v|\,|e_a| \cos\theta_{va}$ (where $\theta_{ea}$ is the angle between $\vec{v}$ and $\vec{e}_a$). 

Of course if two vectors $\vec{u}$ and $\vec{v}$ are already known in terms of a common set of coordinates (say $\{i,j,k\}$) then we can also write the dot product as $$\vec{u} \cdot \vec{v} = \sum_{c \in \{i,j,k\}} u_c  v_c \,.$$
Now comes the stuff that depends on the coordinate system you are using. 
While the unit vectors in Cartesian coordinates are constant (that is $\vec{e}_x$ always points in the same direction), the unit vectors for polar coordinates depend on where they are evaluated. The radial unit vector $\vec{e}_r$ always points directly away from the origin, and the polar unit vector always points tangent to the circle of constant $r$ in the direction of increasing $\phi$, which means we can find the polar unit vectors at point $\vec{p} = (r,\phi)$ in terms of the Cartesian unit vectors as
$$
\begin{align*}
\vec{e}_r(\vec{p}) &= +\vec{e}_x \cos \phi + \vec{e}_y \sin \phi \\
\vec{e}_\phi(\vec{p}) &= -\vec{e}_x \sin \phi + \vec{e}_y \cos \phi \,.
\end{align*}
$$ 
From this we can deduce the general projection formula in terms of the point $\vec{p}$ where it is evaluated.
$$
\begin{align*}
\vec{v}_r(\vec{p})
&= \vec{v} \cdot \vec{e}_r(\vec{p}) \\
&= v_x \cos\phi + v_y \sin\phi \\
\vec{r}_\phi(\vec{p}) 
&= \vec{v} \cdot \vec{e}_\phi(\vec{p}) \\
&= -\vec{v}_x \sin\phi + \vec{v}_y \cos\phi \,.
\end{align*}
$$
