This paper has the goal of searching for natural trajectories that can be used for a particle or a spacecraft coming from a region of the space far from Jupiter system to be captured into this system by making close approaches with the Galilean satellites of the planet. The opposite situation is also possible and escape trajectories can also be found. This type of maneuver is called "Swing-By" and it is usual in astrodynamics. It was used in many space missions to reduce the fuel consumption by gaining or loosing energy from the gravity of a celestial body. Several famous examples are the Voyager, Cassini, Galileo and other missions. The idea of the present research is to study this type of maneuver using the Galilean satellites of Jupiter, to search for trajectories that change the two-body energy (particle or spacecraft)-(Jupiter) from positive to negative (a capture trajectory) or from negative to positive (an escape trajectory). Those trajectories can be used for a spacecraft going or leaving the planet Jupiter or to explain how particles can be captured or escape from Jupiter system by close approaches with the Galilean satellites. Initial conditions are varied to cover the whole possible alternatives and then small regions of captures and escapes are identified. After that, a study is made to see the accuracy of the Tisserand's method when applied to those close approach trajectories.Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)Instituto Nacional de Pesquisas Espaciais (INPE), BR-12201 São José dos Campos, BrazilUniversidade Estadual Paulista, FEG UNESP, Guaratinguetá, BrazilCNPq: 473387/2012-3CNPq: 304700/2009-6FAPESP: 2011/09310-7FAPESP: 2011/08171-3FAPESP: 2011/13101-4FAPESP: 2012/21023-6