The classical mechanics of non-conservative systems (1210.2745v2)

Abstract: Hamilton's principle of stationary action lies at the foundation of theoretical physics and is applied in many other disciplines from pure mathematics to economics. Despite its utility, Hamilton's principle has a subtle pitfall that often goes unnoticed in physics: it is formulated as a boundary value problem in time but is used to derive equations of motion that are solved with initial data. This subtlety can have undesirable effects. I present a formulation of Hamilton's principle that is compatible with initial value problems. Remarkably, this leads to a natural formulation for the Lagrangian and Hamiltonian dynamics of generic non-conservative systems, thereby filling a long-standing gap in classical mechanics. Thus dissipative effects, for example, can be studied with new tools that may have application in a variety of disciplines. The new formalism is demonstrated by two examples of non-conservative systems: an object moving in a fluid with viscous drag forces and a harmonic oscillator coupled to a dissipative environment.

The Classical Mechanics of Non-Conservative Systems

Chad R. Galley's paper presents a novel formulation of Hamilton's principle which aligns with initial value problems and extends the applicability of Lagrangian and Hamiltonian mechanics to non-conservative systems. Hamilton's principle, traditionally framed with boundary value problems in time, has limited capacity in addressing non-conservative interactions, where the dynamics must evolve from initial conditions.

Key Contributions

The research makes a pivotal observation regarding the incompatibility of Hamilton's boundary-focused principle with initial conditions typically encountered in deciding non-conservative systems' dynamics. Addressing this discrepancy leads to the development of a framework that successfully incorporates non-conservative systems into classical mechanics, where the dynamics involve energy dissipation among other features. This framework opens up new possibilities for analyzing dynamics that are inherently non-conservative, such as those systems experiencing dissipative forces.

Innovative Framework

Galley introduces a modification to the traditional variational method by proposing an approach that doubles the degrees of freedom within Hamilton's framework. This formalism ensures the equations of motion account for systems' dissipative behaviors and is characterized by a new action defined in terms of a doubled variable set. This employs dual trajectory paths, effectively incorporating initial conditions to unambiguously determine system evolution.

The paper stipulates an additional function, $K(q_a, \dot{q}_a, t)$, which provides a formal means of including non-conservative forces within the Lagrangian formalism. This function cannot typically manifest as the difference of a potential energy, setting it apart from conservative systems. Consequently, the governing dynamics become more general, seamlessly integrating the effects of non-conservative forces.

Application Highlights

1. Viscous Drag Forces: This formalism is applied to derive the dynamics of a particle experiencing viscous drag. Traditional methods cannot accommodate such dissipative force expressions directly within the Lagrangian mechanics framework. Galley's approach captures both linear and nonlinear drag scenarios while retaining consistency with physical observations, such as Stokes' law and large Reynolds number behaviors.
2. Coupled Harmonic Oscillators: The framework is further demonstrated using an oscillator coupled to a dissipative medium. Here, the effective dynamics of the primary oscillator, after integrating out the medium effects, account for causal behavior dictated by initial conditions, correcting a fundamental misalignment between conventional Hamiltonian mechanics and non-conservative settings.

Implications and Future Directions

The theoretical advancements presented imply significant ramifications in both classical mechanics and potential extensions to quantum mechanics, where similar non-conservative dynamics might be studied through a path integral quantization or the adapted Poisson bracket structure. Importantly, this admits potential applications in statistical physics, nonlinear dynamical systems, and numerical simulation methods, where dissipative forces play a critical role.

The approach provides a bridge to quantifying and numerically characterizing non-conservative systems more precisely, offering a basis for further methodological innovations in the paper of open systems. Furthermore, it suggests new directions for the exploration of symplectic geometry, owing to the modified structure introduced by doubled variables.

Overall, Galley's work fills a substantial gap in theoretical mechanics, aptly addressing long-standing limitations concerning dynamical systems' interactions beyond conservative forces. It offers a robust, mathematically consistent foundation for the continued exploration of complex systems within various physical and theoretical contexts.