A Unifying Action Principle for Classical Mechanical Systems (2409.11063v1)

Abstract: The modern theory of classical mechanics, developed by Lagrange, Hamilton and Noether, attempts to cast all of classical motion in the form of an optimization problem, based on an energy functional called the classical action. The most important advantage of this formalism is the ability to manifestly incorporate and exploit symmetries and conservation laws. This reformulation succeeded for unconstrained and holonomic systems that at most obey position equality constraints. Non-holonomic systems, which obey velocity dependent constraints or position inequality constraints, are abundant in nature and of central relevance for science, engineering and industry. All attempts so far to solve non-holonomic dynamics as a classical action optimization problem have failed. Here we utilize the classical limit of a quantum field theory action principle to construct a novel classical action for non-holonomic systems. We therefore put to rest the 190 year old question of whether classical mechanics is variational, answering in the affirmative. We illustrate and validate our approach by solving three canonical model problems by direct numerical optimization of our new action. The formalism developed in this work significantly extends the reach of action principles to a large class of relevant mechanical systems, opening new avenues for their analysis and control both analytically and numerically.

• The paper extends classical variational principles to non-holonomic systems using a novel Schwinger-Keldysh approach.
• It reformulates the action as an initial value problem by doubling degrees of freedom to simplify constraint enforcement.
• Numerical validations on canonical models confirm its robust performance and alignment with established classical laws.

A Unifying Action Principle for Classical Mechanical Systems

In the paper of classical mechanics, the formalism provided by Lagrange, Hamilton, and Noether has facilitated the transformation of physical principles into an optimization problem, characterized by the classical action, an energy functional. While effective for unconstrained and holonomic systems, which only obey position equality constraints, this formalism has remained elusive for non-holonomic systems that adhere to velocity-dependent or position inequality constraints. The work of Rothkopf and Horowitz presents a significant extension to this classical action principle, successfully encompassing these non-holonomic systems by leveraging insights from quantum field theory.

Non-Holonomic Constraints

Non-holonomic systems, prevalent in sectors such as robotics and autonomous transport, and critical for understanding contact forces in material processing, pose inherent difficulties due to their velocity-dependent or inequality constraints. Traditional methods, reliant on Newton's second law or the Lagrange-d’Alembert principle, necessitated direct force calculation and could not benefit from the variational treatment useful for holonomic systems.

Classical Schwinger-Keldysh Approach

Rothkopf and Horowitz propose a novel classical action for non-holonomic systems, derived from the classical limit of the Schwinger-Keldysh action principle known in quantum field theory. This approach rewrites the traditional variational problem to an initial value problem, accommodating the causality of physical systems efficiently. By doubling the degrees of freedom, the proposed action avoids the pitfalls of Hamilton's principle, which necessitates final state information, thus making it more suitable for systems characterized by initial conditions.

Reformulation of the Action for Non-Holonomic Systems

The paper extends the classical Schwinger-Keldysh formalism by incorporating Lagrange multipliers to enforce velocity-dependent constraints, revealing a general action: $$S_{\rm cSK}=\int dt \, [L[{\bf q}_1,\dot {\bf q}_1] - L[{\bf q}_2,\dot {\bf q}_2] + \lambda_-^a g^a({\bf q}_+,\dot {\bf q}_+) - \lambda_+^a q_-^i \frac{\partial g^a}{\partial \dot q^i}]$$ Here, $q_+ = ({\bf q}_1 + {\bf q}_2)/2$ and $q_- = {\bf q}_1 - {\bf q}_2$. The inclusion of non-holonomic constraints through this method ensures correct system trajectories, reflecting the Lagrange-d'Alembert adjustments for velocity constraints.

Numerical Optimization and Validation

The authors validate their approach through numerical simulations on canonical model problems such as the rolling-spinning disc and the particle under gravity in a tumbler. Using summation-by-parts (SBP) finite difference operators for discretization, they demonstrate how their reformulation correctly predicts system behavior. The numerical results for these test cases show excellent agreement with the traditional Lagrange-d’Alembert principle and Newton's laws, proving the robustness of the method.

Implications and Future Applications

The unifying action principle outlined in this work extends the versatility of action-based analysis to systems with non-holonomic constraints, offering significant practical and theoretical benefits. This development allows the use of variational methods in areas where detailed force calculations were previously necessary, broadening the application scope in engineering and physical sciences.

This paper’s implications reach beyond merely addressing a gap in classical mechanics. By providing a robust framework for variational treatment of non-holonomic systems, myriad applications—ranging from robotics control systems to the analysis of contact dynamics in material sciences—could potentially benefit. Furthermore, the work opens avenues for the incorporation of more complex interactions and constraints, considering both smooth and non-smooth system behaviors.

In summary, Rothkopf and Horowitz’s formulation marks a substantial progression in the treatment of classical mechanical systems. By extending the variational principles to include a broader class of constraints, this work paves the way for innovative analytical and numerical techniques, optimizing the analysis and control of a significant range of mechanical systems.