Variational approach to nonholonomic and inequality-constrained mechanics

This paper presents a novel scalar action formulation, derived from the Schwinger-Keldysh formalism, that successfully describes non-holonomic and inequality-constrained mechanical systems by recovering Lagrange-d'Alembert dynamics through direct extremization, thereby enabling new analytical and computational approaches that bypass traditional equations of motion.

The Gist

Imagine you are trying to predict the path of a rolling ball, a spinning top, or a robot arm. For centuries, physicists have had a "magic recipe" called Hamilton's Principle to do this. Think of it like a GPS for the universe: it says that nature always chooses the path that minimizes a specific "score" (called the Action). If you know the starting point and the ending point, this recipe tells you exactly how the object moved in between.

The Problem: The "No-Slip" Rule
However, this magic recipe has a major blind spot. It works perfectly for things that just move freely or are tied to a fixed track (like a bead on a wire). But it fails miserably for things with non-holonomic constraints.

What's that? Imagine a car. A car can't just slide sideways; its wheels force it to move forward or backward. This is a rule based on velocity (how fast and in what direction it's going), not just position. Or think of a coin rolling on a table; it can't just teleport to a new spot without rolling. These rules are tricky. For a long time, physicists had to throw away the "magic recipe" for these systems and instead use messy, force-by-force calculations (like Newton's laws) to figure out what happens. It was like trying to solve a puzzle by counting every single piece individually instead of seeing the big picture.

The Solution: A Quantum Trick
In this paper, the authors, Rothkopf and Horowitz, found a way to bring the "magic recipe" back for these tricky systems. They didn't invent a new law of physics; instead, they borrowed a tool from Quantum Mechanics.

Here is the analogy:
Imagine you want to know the best route for a road trip.

• Old Way (Hamilton): You look at the map, pick a start and a finish, and ask, "What is the single best path?"
• The Problem: If your car has a rule like "You can only turn left," the old map doesn't work because it doesn't account for the direction you are currently facing.
• The New Way (The Authors' Approach): They use a technique called the Schwinger-Keldysh formalism. Imagine you don't just send one car on a trip. You send two identical cars at the same time:
  1. Car A (The Forward Car): Drives from the start to the finish.
  2. Car B (The Backward Car): Drives from the finish back to the start.

In the quantum world, these two cars interact. The authors realized that if you write a "score" (Action) that measures the difference between what Car A did and what Car B did, you can force them to agree on the rules of the road (like the non-slip wheels) without needing to know the exact forces pushing the wheels.

How It Works in Plain English

1. Double the Variables: They double the number of variables in their math. Instead of just tracking position $x$, they track $x_1$ and $x_2$.
2. The "Ghost" Constraint: They add a special "ghost" term to the score. This term acts like a referee. If the car tries to slide sideways (violating the rule), the referee penalizes the score heavily.
3. The Optimization: They then ask a computer to find the path where this score is minimized. Because of the way they set up the "two cars" and the "ghost referee," the computer naturally finds the correct path that obeys the rolling/sliding rules.

Why This Is a Big Deal

• It's Universal: Before this, you had to use different, messy math for every different type of constraint (rolling, sliding, hitting a wall). Now, they have one general formula that handles all of them.
• It Handles "Bumps": The method also works for things that hit hard surfaces (like a ball bouncing in a box). Instead of manually telling the computer "bounce here," the math automatically figures out the bounce as part of the path optimization.
• Robotics and AI: This is huge for robotics. Robots often have wheels or joints that can't move sideways. By using this new "Action" formula, engineers can use powerful machine learning tools (which love "score functions") to teach robots how to move more efficiently and naturally.

The Takeaway
The authors took a complex, abstract idea from quantum physics (the "two-car" time loop) and adapted it for everyday classical mechanics. They proved that even for systems with tricky, velocity-dependent rules (like rolling wheels), nature still follows a "least effort" path—you just have to look at it through the right lens (the doubled degrees of freedom) to see it.

They didn't just write down equations; they built a computer program that "optimizes" the path directly, skipping the messy middle steps of calculating forces. It's like giving a robot a goal and a set of rules, and letting it figure out the perfect movement on its own, without a human needing to calculate every push and pull.

Technical Summary

Here is a detailed technical summary of the paper "Variational approach to nonholonomic and inequality-constrained mechanics" by A. Rothkopf and W. A. Horowitz.

1. Problem Statement

Classical mechanics relies heavily on variational principles (Hamilton's principle), where the physical trajectory of a system extremizes an action functional $S = \int L \, dt$. While this framework works perfectly for holonomic systems (constraints depending only on position) and conservative forces, it fails for two major classes of problems:

• Non-holonomic systems: Systems subject to non-integrable velocity-dependent constraints (e.g., rolling wheels, spinning disks). Standard attempts to include these via Lagrange multipliers in Hamilton's principle yield incorrect equations of motion because the velocity dependence of the constraints is not handled consistently with the variation of the path.
• Inequality-constrained systems: Systems involving position inequalities (e.g., particles hitting hard walls) or sliding friction. These often lead to non-smooth trajectories (impacts) and dissipative forces that cannot be derived from a standard scalar action principle.

Currently, these systems are treated using the Lagrange-d'Alembert principle or Gauss's principle at the level of equations of motion, bypassing a global action formulation. This limits the application of powerful analytical tools (like Noether's theorem) and computational methods (like variational integrators and physics-informed neural networks) to these systems.

2. Methodology

The authors propose a new variational formulation based on the Schwinger-Keldysh-Galley (SKG) formalism, originally developed for non-equilibrium quantum field theory and rediscovered by Galley for classical non-conservative systems.

Core Mechanism: Doubled Degrees of Freedom
Instead of a single trajectory $q(t)$, the SKG formalism introduces a closed time contour with two branches:

1. Forward branch ($q_1$): Represents the system evolving forward in time.
2. Backward branch ($q_2$): Represents the system evolving backward.

The action is constructed as:
$$S_{SKG} = \int dt \left( L[q_1, \dot{q}_1] - L[q_2, \dot{q}_2] + \Lambda[q_1, \dot{q}_1, q_2, \dot{q}_2] \right)$$
where $\Lambda$ is an interaction term. The physical limit is recovered by defining sum and difference coordinates ($q_+ = (q_1+q_2)/2$, $q_- = q_1-q_2$) and enforcing $q_- = 0$ after variation.

Key Innovations:

• General Non-Holonomic Action: The authors introduce doubled Lagrange multipliers ($\lambda_1, \lambda_2$) as dynamical degrees of freedom. The generalized action includes terms that couple the constraint functions $g_a(q_+, \dot{q}_+)$ to the difference coordinates $q_-$.
  • Variation with respect to $\lambda_-$ recovers the constraint $g_a(q_+, \dot{q}_+) = 0$.
  • Variation with respect to $q_+$ yields the correct Lagrange-d'Alembert equations of motion, including the constraint forces (Chetaev forces).
  • Crucially, this formulation works for non-linear velocity constraints without requiring auxiliary variables or perturbative expansions.
• Inequality Constraints & Friction: The framework incorporates inequality constraints ($g(q) \leq 0$) and sliding friction by modeling normal forces and friction forces as interaction terms $\Lambda$ dependent on $q_-$ and $\dot{q}_+$.
  • Normal forces are modeled using regularized delta functions (Gaussian potentials) to handle impacts smoothly in the numerical discretization.
  • Friction forces are added as velocity-dependent dissipative terms.

Numerical Implementation:
To validate the theory, the authors bypass the derivation of equations of motion entirely. They:

1. Discretize the action using Summation-by-Parts (SBP) finite difference operators, which mimic integration by parts exactly in the discrete domain.
2. Treat the discretized action as a scalar cost function.
3. Use numerical optimization (Mathematica's FindMinimum) to find the trajectory that minimizes the action, directly solving for the critical point.

3. Key Contributions

1. First General Action for Non-Holonomic Systems: The paper provides an explicit, non-perturbative action functional for general non-holonomic systems (linear and non-linear velocity constraints) that reproduces the correct Lagrange-d'Alembert equations upon extremization.
2. Unified Framework for Inequalities and Friction: It extends the variational approach to systems with position inequality constraints (hard walls) and sliding friction, automatically handling non-smooth trajectories and impacts without manual intervention.
3. Bypassing Equations of Motion: The authors demonstrate that one can solve constrained mechanical problems by directly optimizing the action functional, avoiding the need to derive and solve complex ordinary differential equations (ODEs) first.
4. Conservation Laws: The framework preserves the structure required for Noether's theorem. The authors show how conserved charges (like energy) are encoded in the boundary terms of the SKG action, even in the presence of constraints and dissipation.

4. Results

The authors validated their approach on three canonical examples, comparing the results of direct action minimization against standard solutions (Newton's laws or Lagrange-d'Alembert ODEs):

• Rolling-Spinning Disk: A disk rolling and spinning on an incline with non-integrable velocity constraints.
  • Result: The numerical optimization of the SKG action reproduced the physical trajectories and Lagrange multipliers with high precision, matching the solution of the standard differential equations. Energy was conserved within numerical limits.
• Particle in a Hard-Walled Tumbler: A particle falling under gravity inside a circular container, experiencing elastic collisions.
  • Result: The method successfully generated the non-smooth trajectory (bouncing) without manually identifying impact points. As the regularization width ($\sigma$) of the wall potential decreased, the solution converged to the hard-wall limit.
• Sliding Particle with Friction: A particle sliding down an incline with Coulomb friction.
  • Result: The action-based approach accurately captured the dissipative dynamics and the transition between sticking and sliding, matching Newtonian solutions.

5. Significance and Impact

• Theoretical Unification: This work bridges the gap between classical mechanics and quantum field theory formalisms, showing that the "in-in" (Schwinger-Keldysh) formalism is the natural classical limit for initial value problems with constraints.
• Computational Advantages: By formulating constrained mechanics as a global optimization problem, the method opens the door for:
  • Variational Integrators: Developing symplectic-like numerical solvers that inherently respect constraints.
  • Machine Learning: Providing the necessary scalar cost function (action) to train Physics-Informed Neural Networks (PINNs) for robotics and control, where inductive bias is crucial.
  • Inverse Problems: Simplifying the inference of initial conditions or control forces to achieve a target state (inverse kinematics) in complex constrained systems.
• Broad Applicability: The framework is applicable to robotics, autonomous transport, soft robotics (contact mechanics), and granular materials, where non-holonomic and inequality constraints are ubiquitous.

In summary, Rothkopf and Horowitz have successfully constructed a general, explicit action principle for non-holonomic and inequality-constrained mechanics, transforming a class of problems previously solvable only via differential equations into a global variational problem.