From electromagnetism to mechanics: The role of action in Poincare's Memoir of 1905

We make a systematic and extensive study of the role of action and its invariance in the logic of the Palermo Memoir. This sheds a new light on the Memoir and on the mathematical contribution of Henri Poincare to relativistic mechanics. After a brief account of the history of the Least Action Principle in the late 19th century, we examine the origin of Poincare's Hamiltonian electromagnetic action. We exhibit how the invariance of the action (whatever it is, electromagnetic or not) is related by him to the velocity dependence of an electron lagrangian, provided the electron is stable and characterized by its position variables. With the group condition l = 1 (whose origin is also discussed), this invariance ensures the covariance of the Lorentz equation of the dynamics independently of electron models (which play the role of example or counter-example). Finally, the concept of action is the tool which had enabled Poincare to transfer to mechanics the symmetry he had just discovered in electromagnetism, and to obtain (before Max Planck) the relativistic lagrangian. We discuss the reasons why it did not lead him to the discovery of the general mass-energy equivalence.