Losing Energy with Hamilton’s Principle of Least Action
Author: Chad Galley
Affiliation: Theoretical Astrophysics (TAPIR), California Institute of Technology, Pasadena, USA
Classical mechanics is the foundation of physics and of all students’ course work in engineering and the physical sciences. The bricks of this foundation were first laid by Galileo then by Newton and finally by the likes of D’Alembert, Hamilton, Lagrange, Poisson, and Jacobi in the 18th and 19th centuries. What resulted was a framework of physical laws and formalisms for virtually any problem one wishes to study in fluid mechanics , electromagnetism , statistical mechanics , and even quantum theory later on, to give just a few examples.
One important and pervasive formulation of classical mechanics is due to Hamilton, who showed that a physical system evolves to either minimize or maximize a quantity called the action which, loosely speaking, is the accumulation in time of the difference between the kinetic and potential energies . This important result, called Hamilton’s variational principle of stationary action or Hamilton’s principle for short, is the primary way to derive equations of motion for many systems, from the ubiquitous simple harmonic oscillator to supersymmetric string theories. Unfortunately, Hamilton’s principle has a well-known shortcoming: it generically cannot account for the irreversible effects of energy loss that are always present in any real world application, experiment, or problem. But why is that?
The answer has to do with the very formulation of Hamilton’s principle: “The physical configuration of a system is the one that evolves from the given state A at the initial time to the given state B at the final time such that the action is stationary.” This raises the question: how can one know the final state, especially when the system is losing energy? Isn’t the point to determine the final state from initial conditions? That’s how the real world works after all, through cause then effect. Remarkably, answering these questions correctly leads to a natural way to describe generic systems with a variational principle, even those that do not conserve energy .
The questions above are usually addressed, if at all, using a somewhat circular reasoning as follows. In practice, one applies Hamilton’s principle to derive equations of motion that are then solved with initial data. The fixed final state used in Hamilton’s principle is argued then as being the one associated with that specific solution. However, that specific final state is only determined after applying Hamilton’s principle to get the equations of motion in the first place. Perhaps this is a passable explanation but it doesn’t seem completely satisfactory because we usually do not have access to the environment that a system loses energy to so we cannot freely adjust the final states of those inaccessible degrees of freedom to accommodate the above explanation.
For these reasons and others, it is important to generalize Hamilton’s variational principle in a way that does not require fixing the final state of the system but is determined instead from the initial state only. The details of how this is achieved are reported in . The take-home result is that eliminating dependence on the final state requires a formal doubling of the degrees of freedom in the problem. These doubled variables are fictitious but their average values are the physical ones of interest whereas their difference does not contribute to the physical evolution of the system. Figure 1 one shows a cartoon of the usual Hamilton’s principle on the left and of Hamilton’s principle on the right generalized to accommodate for energy losses (or gains). The arrows in Figure 1 indicate the direction in time to integrate the Lagrangian of the system along that path.
Figure 1: Left: A cartoon of Hamilton’s principle. Dashed lines denote the virtual displacements and the solid line denotes the stationary path. Right: A cartoon of Hamilton’s principle compatible with initial data (i.e., the final state is not fixed). In both cartoons, the arrows on the paths indicate the integration direction for the line integral of the Lagrangian.
Doubling the variables in this formal way has an interesting natural consequence. Just as the potential V in classical mechanics is an arbitrary function for conservative systems, we now have the freedom to introduce an additional arbitrary function K that couples together the doubled variables. In many ways K is analogous to V in classical mechanics because K generates the forces and interactions that account for energy loss or gain in a similar way that V generates forces and interactions that conserve energy.
To summarize, the seemingly innocuous problem with specifying the final state in Hamilton’s principle leads to a generalization based solely on the initial state. Achieving this requires formally doubling the degrees of freedom that, in turn, allows for an extra arbitrary function K to be introduced that generically accounts for the dynamical forces and interactions that cause energy loss or gain in the system. This new variational principle may have broad applicability in a wide range of practical and theoretical problems across multiple disciplines.
 G. K. Batchelor, "An Introduction to Fluid Dynamics" (Cambridge University Press, Cambridge, England, 1967).
 J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1999), 3rd ed.
 K. Huang, Statistical Mechanics (Wiley, New York, 1963).
 H. Goldstein, Classical Mechanics (Addison-Wesley, Reading MA, 1980), 2nd ed.
 C. R. Galley, “Classical mechanics of nonconservative systems”, Physical Review Letters, 110, 174301 (2013). Abstract.