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In the last decades a peculiar family of solutions of the Circular RestrictedThree Body Problem has been used to explain the temporary captures of smallbodies and spacecrafts by a planet of the Solar System. These solutions, whichtransit close to the Lagrangian points $L_1,L_2$ of the CRTBP, have beenclassified using the values of approximate local integrals and of the Jacobiconstant. The use for small bodies of the Solar System requires to consider ahierarchical extension of the model, from the CRTBP to the the full $N$planetary problem. The Elliptic Restricted Three Body, which is the firstnatural extension of the CRTBP, represents already a challenge, since globalfirst integrals such as the Jacobi constant are not known for this problem. Inthis paper we extend the classification of the transits occurring close to theLagrangian points $L_1,L_2$ of the ERTBP using a combination of the Floquettheory and Birkhoff normalizations. Provided that certain non-resonanceconditions are satisfied, we conjugate the Hamiltonian of the problem to anintegrable normal form Hamiltonian with remainder, which is used to defineapproximate local first integrals and to classify the transits of orbitsthrough a neighbourhood of the Lagrange equilibria according to the values ofthese integrals. We provide numerical demonstrations for the Earth-Moon ERTBP.

Transits close to the Lagrangian solutions $L_1,L_2$ in the Elliptic Restricted Three-body Problem