No Ponzi game condition and transversality condition are the same? Given the following non-stochastic planning problem with finite horizon, 
\begin{align}
&amp;\max_{\{k_{t+1}\}}\sum^T_{t=0}\beta^tU[f(k_t-k_{t+1})] \\
\text{s.t. } &amp; 0\leq k_{t+1}\leq f(k_t)\\
&amp; k_0 &gt;0 \text{ (given)}.
\end{align}
I found that in order to make the first order conditions necessary and sufficient I have to add the so called no Ponzi game condition, i.e. 
\begin{gather}
\lim_{T \rightarrow \infty} \frac{k_{T+1}}{R_{T+1}} \geq 0
\end{gather}

When written with the equal sign, this condition can be interpreted as the willingness of not keeping any capital at the end of life. And this is the same interpretation of the so called transversality condition. 

Thus, is it right to interpret the no Ponzi game condition as a finite horizon version of the transversality condition? If not, which is the difference between them?
Is it right to interpret the no Ponzi game condition as a finite
  horizon version of the transversality condition?


No. The "No-Ponzi-Game" or "solvency" condition is an external constraint imposed on the individual by the market/other participants. The individual would very much like to violate it.

The Transversality condition must be satisfied in order for the individual to maximize indeed its intertemporal utility. It is an optimization condition.

So they are conceptually very different aspects of the problem.

Finally the No-ponzi-game/solvency condition is not inherently of finite horizon -it extends to the infinite horizon also.