Variational reduction of homogenous Lagrangian systems

This paper establishes a variational reduction procedure for Lagrangian systems with scaling symmetries, enabling trajectory reconstruction via quadratures and characterizing critical points through scaling-analogous Lagrange-Poincaré equations, while also investigating their relationship with the Herglotz variational principle.

The Gist

Imagine you are trying to navigate a massive, complex maze. The maze represents a physical system (like a swinging pendulum or a planet orbiting a star), and the path you take is the "trajectory" of that system. Usually, figuring out the exact path requires solving very difficult math problems that involve tracking every single detail of the system's position and speed at every moment.

This paper is about a clever shortcut. The authors show that if your maze has a special kind of "scaling symmetry"—meaning the maze looks the same whether you zoom in or zoom out—you can solve a much simpler, smaller version of the problem first. Once you solve the small version, you can easily "reconstruct" the full, complex path without doing all the heavy lifting again.

Here is a breakdown of their ideas using everyday analogies:

1. The "Zoom" Symmetry (Scaling)

Most physical systems are described by a "Lagrangian," which is essentially a mathematical recipe that tells you how the system moves.

• Standard Symmetry: Imagine a maze where if you rotate it 90 degrees, it looks exactly the same. You can ignore the rotation and just solve for the shape.
• Scaling Symmetry (This Paper): Imagine a maze where if you zoom in or out (change the scale), the rules of the maze stay the same, just the size changes. The authors focus on systems where the "recipe" for movement scales up or down linearly. Think of a fractal pattern: a small piece looks like the whole thing.

2. The Shortcut: Reduction

The authors ask: Can we throw away the "zoom" information, solve the problem on the "shape" alone, and then put the "zoom" back in later?

• The Old Way: You try to calculate the path of a particle moving on a giant, expanding balloon. You have to track its position on the balloon and how fast the balloon is inflating simultaneously.
• The New Way (Reduction): You strip away the inflation part. You solve the path of the particle on a fixed balloon (the "reduced" system). This is much easier.
• The Catch: The "reduced" system isn't just a simpler version of the original; it lives on a slightly different mathematical structure (a "line bundle"). Think of it as solving the puzzle on a flat map, but knowing that the map can stretch or shrink.

3. Reconstructing the Full Path

Once you have the solution to the simple, reduced problem, how do you get back to the real, complex world?

• The authors provide a "reconstruction recipe." It's like having a blueprint for a house (the reduced solution) and a separate instruction manual on how to scale that house up or down (the quadrature).
• You take the blueprint, apply the scaling instructions, and boom—you have the full trajectory of the original system. The math shows this final step only requires a simple integration (a "quadrature"), which is like adding up a list of numbers rather than solving a complex differential equation.

4. The "Scaling-Lagrange-Poincaré" Equations

In physics, there are famous equations (Euler-Lagrange) that tell you how things move. When you reduce a system with standard symmetries (like rotation), you get a specific set of equations called "Lagrange-Poincaré equations."

• The authors discovered a new set of equations specifically for these "zoom" symmetries. They call them Scaling-Lagrange-Poincaré equations.
• These are the "rules of the road" for the reduced system. If you follow these rules, you are guaranteed to find the correct path for the reduced problem, which you can then expand back to the real world.

5. The "Herglotz" Detour

The paper also checks if this new method is related to another famous mathematical tool called the Herglotz principle (which deals with systems where energy isn't conserved, like a car losing fuel).

• The Finding: They found that, surprisingly, these two methods are not the same. You cannot simply swap one for the other. The "zoom" reduction works differently than the "energy-loss" (Herglotz) method. It's like finding that a shortcut through a forest doesn't lead to the same destination as a shortcut through a tunnel, even if they look similar on a map.

Summary

In simple terms, this paper proves that for physical systems that behave the same at different sizes (scaling symmetry):

1. You can simplify the math by ignoring the size changes.
2. You solve the simplified problem using a new set of specific rules (the Scaling-Lagrange-Poincaré equations).
3. You can then easily rebuild the full, complex motion from that simple solution.

It's a powerful tool for mathematicians and physicists to break down complex, "self-similar" problems into manageable chunks, solve the chunk, and then scale the answer back up to reality.

Technical Summary

Technical Summary: Variational Reduction of Homogeneous Lagrangian Systems

Problem Statement
The paper addresses the variational reduction of Lagrangian systems defined by homogeneous Lagrangian functions (specifically, those homogeneous of degree 1 with respect to a scaling symmetry). While the reduction of symplectic Hamiltonian systems with scaling symmetries is well-established—resulting in contact structures and allowing reconstruction of original trajectories up to one quadrature—the variational formulation for the Lagrangian side remained less developed. The central question is whether a reduced variational principle exists for homogeneous Lagrangian systems such that:

1. Its critical points (reduced trajectories) allow for the reconstruction of the critical points of the original Hamilton principle.
2. The reconstruction can be achieved up to quadratures.
3. The critical points of the reduced principle can be characterized by a system of ordinary differential equations (ODEs) analogous to the Lagrange-Poincaré equations found in standard symmetry reduction.

Additionally, the authors investigate the relationship between these reduced homogeneous systems and the Herglotz variational principle (which governs action-dependent Lagrangians and contact Hamiltonian systems), specifically asking if the sets of critical points for these two frameworks coincide.

Methodology
The authors employ a geometric approach within the $C^\infty$ category, utilizing differential geometry, Lie group actions, and variational calculus.

1. Standard Symmetries Review: The paper begins by reviewing variational reduction for systems with standard symmetries (Lie group actions). In the abelian case with a flat connection, the reduction occurs on the associated bundle $(Q \times \mathfrak{g})/G$. The reconstruction involves solving an equation for the group element, solvable up to quadratures.
2. Scaling Symmetries Setup: The authors define a homogeneous Lagrangian system $(Q, L)$ where $L$ is homogeneous of degree 1 with respect to a principal action $\psi: \mathbb{R}^+ \times Q \to Q$. Since $\mathbb{R}^+$ is contractible, the principal bundle $\pi: Q \to Q/\mathbb{R}^+$ is trivial, allowing for the existence of a global scaling function $f: Q \to \mathbb{R}^+$.
3. Construction of Reduced Action:
   • A flat principal connection $\varpi_f = d \ln f$ is defined.
   • The Atiyah diffeomorphism $\alpha_f$ maps the tangent bundle $TQ$ to $T(Q/\mathbb{R}^+) \times \mathbb{R}$.
   • A reduced Lagrangian $\ell$ is defined on $T(Q/\mathbb{R}^+) \times \mathbb{R}$ via the relation $L = \ell \circ \alpha_f \circ p \cdot (f \circ \tau_Q)$.
   • Crucially, the original action functional $A(\gamma) = \int L(\dot{\gamma}) dt$ is shown to be proportional to a new functional $\check{A}(x, y) = \int \exp(\int_0^t y(s) ds) \ell(\dot{x}, y) dt$, where $x = \pi(\gamma)$ and $y = d \ln f(\dot{\gamma})$.
4. Variational Analysis: The authors define "interrelated" curves and variations. They establish that a curve $\gamma$ is a critical point of $A$ if and only if the associated pair $(x, y)$ is a critical point of $\check{A}$ under specific boundary conditions (vanishing variations at endpoints for $x$, and zero integral for $y$).
5. Derivation of Equations: By applying the calculus of variations to $\check{A}$, the authors derive the scaling-Lagrange-Poincaré equations, a system of ODEs governing the reduced dynamics.
6. Comparison with Herglotz: The paper analytically compares the scaling-Lagrange-Poincaré equations with the equations derived from the Herglotz principle for action-dependent Lagrangians.

Key Contributions and Results

• Existence of Reduced Variational Principle: The paper proves that for a homogeneous Lagrangian system with a principal $\mathbb{R}^+$-action, a reduced variational principle exists on the space $(Q/\mathbb{R}^+) \times \mathbb{R}$.
• Reconstruction Theorem: It is demonstrated that trajectories of the original system can be reconstructed from the critical points of the reduced principle up to a single quadrature. The reconstruction formula is given by:
  $$\gamma(t) = \psi \left( e^{\int_0^t y(s) ds}, (\pi, f)^{-1}(x(t), \varsigma) \right)$$
  where $\varsigma$ is a constant determined by initial conditions.
• Scaling-Lagrange-Poincaré Equations: The critical points of the reduced action are characterized by the following system of ODEs (in local coordinates or using covariant derivatives):
  • Horizontal Equation:
    $$-\frac{D}{Dt} \frac{\partial \ell}{\partial \dot{x}} - y \frac{\partial \ell}{\partial \dot{x}} + \frac{\partial \ell}{\partial x} = 0$$
  • Vertical Equation:
    $$-\frac{d}{dt} \frac{\partial \ell}{\partial y} - y \frac{\partial \ell}{\partial y} + \ell = 0$$
    These equations are structurally similar to, but distinct from, the standard Lagrange-Poincaré equations.
• Examples: The theory is illustrated with examples including kinetic Lagrangians on manifolds with homotheties, Jacobi metrics, and the harmonic oscillator.
• Non-Equivalence with Herglotz Principle: A significant negative result is established: the set of critical points of the reduced variational principle for a homogeneous Lagrangian does not generally coincide with the set of critical points of the Herglotz variational principle for any action-dependent Lagrangian, and vice versa. The authors provide counterexamples showing that the dynamical structures are fundamentally different, despite both being related to contact geometry in broader contexts.

Significance
The paper extends the classical theory of variational reduction (previously established for standard Lie group symmetries) to the context of scaling symmetries. It provides a rigorous variational framework for homogeneous Lagrangian systems, offering a method to reduce the dimensionality of the problem while preserving the ability to reconstruct the full dynamics.

The authors explicitly state that their work clarifies the variational structure of these systems and distinguishes them from action-dependent systems governed by the Herglotz principle. The paper is dedicated to the memory of H. Cendra, a colleague known for his work in reduction theory, and aims to contribute to the understanding of how scaling symmetries interact with variational principles. The work is presented as a foundational step, with future research directions identified as extending these results to general homogeneity (involving $\mathbb{R}^\times$) and pre-symplectic systems, as well as developing discrete analogues.