Is Langrangian Formalism Adequately Describing Energy Conservation?

In most physical theories, total energy is conserved. For example, when the kinetic energy of a particle decreases, the potential energy increases accordingly. For some physical systems, energy is not conserved. For example, if we consider a particle moving with friction, the energy of the particle itself is not conserved: it is transformed into thermal energy of the surrounding medium. For simple systems, energy is easy to define. For more complex physical systems, such a definition is not easy. To describe energy of generic systems, physicists came up with a general notion of energy based on the Lagrangian formalism – a minimal-action representation of physical theories which is now ubiquitous. For many physical theories, this notion leads to physically meaningful definitions of energy. In this paper, we show that there are also examples when the Lagrangian-motivated notion of energy is not physically meaningful at all – e.g., according to this definition, all dynamical systems are energyconserving.