Variational Calculus on Sub-Riemannian Manifolds O. C˘alin and V. Mangione Dedicated to the Memory of Grigorios TSAGAS (1935-2003), President of Balkan Society of Geometers (1997-2003) Abstract We provide invariant formulas for the Euler-Lagrange equation associated to sub-Riemannian geodesics. They use the concept of curvature and horizontal connection introduced and studied in the paper. Mathematics Subject Classification: 37J60, 53B21, 70H03 Key words: curvature, geodesics, connection 1 Introduction The geodesic is a concept which comes from Riemannian geometry. It is the curve with the minimum energy E =  1 0 1 2 | ˙ c(s)| 2 ds between two given points. At least two kind of constraints can be considered to act on the curve: holonomic and non- holonomic. A holonomic constraint is when the energy is perturbed by a potential U (c) and the energy becomes E =  1 0 ( 1 2 | ˙ c| 2 + U (c)) ds. The equation geodesic in this case is ∇ ˙ c ˙ c = −U ′ (c). The other kind of constraints are the nonholonomic ones (see [1], [8], [9]). These are constraints on the velocity of the curve. The energy to be minimized is E = =  1 0  1 2 | ˙ c| 2 +ω(˙ c)  ds. The paper deals with a presentation of the variational calculus for the case when ω is a 1-form of type (1.1) such that (1.3) does not vanish. It is said that these kind of sub-Riemannian manifolds are of step 2. They are also called Heisenberg manifolds (see [2]). In general a sub-Riemannian manifold is said to be of step k if k − 1 iterations need for the brackets of X j in order to span the whole tangent space. In section 5 we shall deal with examples of sub-Riemannian manifolds of superior type. The idea of the paper is to consider the solutions of the Euler-Lagrange system as geodesics in a certain connection with certain perturbation given by the curvature tensor defined in section 2. Section 3 shows that the classical Hamilton-Jacobi equation still holds if the gradient is modified into a horizontal gradient. The relationship * Balkan Journal of Geometry and Its Applications, Vol.8, No.1, 2003, pp. 21-32. c  Balkan Society of Geometers, Geometry Balkan Press 2003.