family of metrics with same geodsics For every bi-invariant metric on a lie group we know geodesics  are flow of left invariant vector fields, so this question naturally arise:
are there family of metrics on manifolds that have same geodesics?
 A: There are examples of metrics with the same geodesics on one manifold. You may find a discussion about them in  Can one recover a metric from geodesics?
The short summery of the examples from the answer in the following:  for any two metrics $g_1$ on $M_1$ and $g_2$ on $M_2$ the product metrics $g=g_1+ g_2$ and $\bar g= 2g_1 + g_2$  on $M_1 + M_2$ have the same geodesics considered as PARAMETERIZED curves. 
If you consder geodesics as  curves  without preferable  parameterisation on them,  a local example of two 2-dim metrics  that have the same geodesics is due to Dini 1869 and is (for arbitrary functions $X(x)$ and $Y(y)$ such that the next formulas do correspond to metrics)    $$g= (X(x)- Y(y))(dx^2 + dy^2) \ \  and  \ \ \bar g= \left(\tfrac{1}{Y(y)}- \tfrac{1}{X(x)}\right)\left(\tfrac{dx^2}{X(x)} + \tfrac{dy^2}{Y(y)}\right).$$
The example can be generalized for any dimension (Levi-Civita 1896), and the formulas of Dini and Levi-Civita  are actually local normal forms of metrics having the same geodesics (see the discussion in http://arxiv.org/abs/1301.2492,  where also metrics of indefinite signature are discussed). 
If you are interested in metrics having the same geodesics with a biinvariant metric, then only the first example is possible. More precisly, if a bi-invariant  Riemannian metric $g$
  has the same geodesics considered as unparameterised curves with a metric $\bar g$,
 then parameterised geodesics of these two metrics are also the same. Moreover, 
 the second metric is also bi-invariant and moreover the metric $g$   in the appropriate coordinate system has  the form      $g_0+ g_1 +...+g_k$ such that $g_0$ is $dx_1^2 +...+ dx^2_{k_0}$ and the metric $\bar g$ in the same coordinate system has the form 
$g'_0 + \lambda_1 g_1 + ...+ \lambda_kg_k$, where $\lambda_i$ are constants and $g'_0$
is $\sum_{i,j=1}^{k_0} C_{ij} dx_idx_j$, where $C_{ij}$ is a constant matrix.    
This  follows from the result of Sinjukov (Dokl. Akad. Nauk SSSR (N.S.) 98, (1954) 21--23) ,  see the discussion in  Geodesic transformations of the complex projective plane
See also the question/answer in  A property of bi-invarient Finsler metrics on SU(N) which is also related
