Systematic Nondimensionalization via Scaling Symmetries

• Systematic nondimensionalization via scaling symmetries is a formal methodology that removes redundant dimensional structure by exploiting scaling invariance in dynamical systems.
• It reduces parameter count and clarifies empirical content by reformulating equations in terms of invariant, dimensionless observables, thereby simplifying complex models.
• The approach applies across Lagrangian, Hamiltonian, ODE, and PDE frameworks using algorithmic workflows that generalize the Buckingham π-theorem for diverse scientific applications.

Systematic nondimensionalization via scaling symmetries is the formal methodology of eliminating surplus dimensional structure from dynamical systems by exploiting their intrinsic scaling (dynamical similarity) symmetries. This procedure allows the reformulation of the original equations in terms of invariants under scaling, thus enabling a representation where only physically meaningful, dimensionless observables remain. Such reformulations clarify the empirical content, reduce parameter count, yield reduced-order equations, and provide direct insights into phenomena such as singularity resolution and the arrow of time. This paradigm encompasses Lagrangian, Hamiltonian, ODE, and PDE contexts, applies to both variational and non-variational scaling symmetries, and extends to algorithmic reductions in high-dimensional models.

1. Formal Structure of Dynamical Similarity and Scaling Symmetry

A dynamical similarity (scaling symmetry) is defined by the existence of a one-parameter group of transformations acting on the phase space (and possibly on couplings and time) such that the system's action rescales homogeneously: $$\mathcal{M}^* S = c S + \text{(constant)}, \quad c > 0$$ For a canonical Hamiltonian system $(M, \omega, H)$, the scaling vector field $D$ satisfies: $L_D \omega = \omega, \quad L_D H = \Lambda H$ with $L_D$ the Lie derivative along $D$. The transformed variables evolve as $q \rightarrow \lambda^n q$, $p \rightarrow \lambda^{1-n} p$, $H \rightarrow \lambda^{-1} H$, such that the physically observable combinations (e.g., $q\cdot p$, the action) rescale consistently. If the original system lacks an apparent degree-one scaling symmetry, couplings $a_i$ can be promoted to dynamical variables, lifting to an extended phase space $\hat M = M \times \mathbb{R}^{2k}$, with a corresponding lifted scaling symmetry $\hat D$ satisfying $L_{\hat D}\hat\omega = \hat\omega$, $L_{\hat D}\hat H = \hat H$ (Bravetti et al., 2022, Gryb et al., 2021, Sloan, 2021).

2. Construction of Dimensionless Invariants and Reduced Variables

The key step is the explicit identification of scale-carrying variables and the algorithmic construction of dimensionless invariants. For each variable $X$ of scaling weight $w$ under $D$, and scale function $\rho$, define

$X_{\mathrm{inv}} := X ~ \rho^{-w}$

This invariantization is extended to all phase-space coordinates, time, and parameters. The minimal sufficient set $\mathcal{A} = \{a^j\}$ of invariants is selected so that:

• The closed reduced dynamics can be written solely in terms of $\mathcal{A}$ and an internal clock,
• $\mathcal{A}$ provides autonomy, sufficiency, and necessity for empirical predictions.

Tables of scaling weights and generator actions (Lie derivatives) systematize this selection process across model classes (Gryb et al., 2021, Bludman et al., 2010, Tanburn et al., 15 Dec 2025, Sloan, 2021).

3. Reduced (Contact/Kirillov) Hamiltonian and Nondimensional Equations

Reduction of a symplectic Hamiltonian system by its scaling symmetry yields a $(2n-1)$-dimensional contact Hamiltonian system characterized by a contact form $\eta$ and a contact Hamiltonian $\mathscr{H}$ (Bravetti et al., 2022, Bravetti et al., 2023): $$(C,\eta,\mathscr H), \quad \eta = \frac{i_D \omega}{\rho}\Big|_{\rho=1}, \quad \mathscr{H} = -\frac{H}{\rho}\Big|_{\rho=1}$$ The equations of motion on $C$ (the contact manifold) are: $$\dot{q}^a = \frac{\partial \mathscr{H}}{\partial p_a}, \;\; \dot{p}_a = -\frac{\partial \mathscr{H}}{\partial q^a} - p_a \frac{\partial \mathscr{H}}{\partial S}, \;\; \dot{S} = p_a \frac{\partial \mathscr{H}}{\partial p_a} - \Lambda \mathscr{H}$$ The result is an autonomous, frictional (contact or Herglotz) system in terms of dimensionless quantities, with the eliminated scale appearing only through dissipation-like or monotonizing effects (e.g., the emergence of an arrow of time). The Kirillov reduction further endows the reduced system with a homogeneous Jacobi bracket structure, fully encoding the dynamics of the observables (Bravetti et al., 2023).

4. Algorithmic Workflow for Systematic Nondimensionalization

For ODE and PDE systems, especially rational first-order systems, the scaling symmetry and its invariant algebra can be determined entirely formally (Tanburn et al., 15 Dec 2025). The generic algorithm proceeds as follows:

1. Write the system in explicit variable and parameter form.
2. Compute the exponent matrix $K$ encoding the monomial structure of all rational functions in the system.
3. Calculate the maximal scaling torus by finding the kernel (using Hermite normal form) to identify the independent scaling actions.
4. Construct all dimensionless invariants as monomials invariant under the maximal scaling.
5. Substitute to express the dynamics in terms of these invariants, resulting in a minimal parameter and variable set.
6. For initial conditions or user-supplied invariants, extend the symmetry analysis by appending extra rows/columns to $K$ as needed.

This approach generalizes the Buckingham $\pi$-theorem to arbitrary-scale systems, encompassing high-dimensional and complex network models (Tanburn et al., 15 Dec 2025).

5. Classical and Contemporary Applications

Hamiltonian/Lagrangian Paradigms:

• Kepler Problem: Reduction of the two-body problem yields dimensionless equations in variables invariant under $r \rightarrow \lambda r$, $t \rightarrow \lambda^{3/2} t$; the resulting dimensionless system exactly captures all empirical content (including Kepler’s 3rd law) (Gryb et al., 2021, Bravetti et al., 2022, Sloan, 2021).
• Central-force and Virial Laws: Inverse-power-law interactions are reduced to parameter-free, first-order equations between scale invariants, yielding the generalized virial theorem (Bludman et al., 2010).
• Hydrostatic Stellar Structure: Lane–Emden polytropes arise from scaling-invariant nondimensionalization, with all mass-radius scaling, core radii, and structure recoverable from autonomous ODEs between invariants such as $u_n$, $v_n$ (Bludman et al., 2011, Bludman et al., 2012).
• Cosmological Dynamics (FLRW): Volume-eliminating dynamical similarity yields dimensionless Friedmann–Klein–Gordon equations that remain regular at the singularity and encode the arrow of time through measure focusing (Gryb et al., 2021, Bravetti et al., 2022, Sloan, 2021).

Biological/Epidemiological ODE/PDE Systems:

• Michaelis–Menten Kinetics: The algorithmic approach recovers the minimal set of dimensionless variables ($u=s/s_0$, $v=c/e_0$, etc.) and classic ODE reductions (Tanburn et al., 15 Dec 2025).
• Epidemic SI(R)S Models: High-dimensional parameter redundancies are removed via scaling symmetry analysis, reducing to the two-parameter extended Hethcote model; bifurcation and stability criteria then follow without rederivation (Nill, 2023).

A summary table of key structural features:

Model Class	Symmetry Eliminated	Invariant Manifold
Hamiltonian	Overall scale (e.g. length)	Contact manifold $C$
ODE (rational)	Independent unit rescalings	Torus quotient $\mathbb{T}^r$
Epidemic/biochem.	Time, population, rates	Dimensionless phase space

6. Advanced Theoretical Developments

Recent research has shown that the contact reduction process for scaling symmetry always exists if couplings are promoted to dynamical variables, thus universalizing the methodology even in the absence of manifest scaling symmetry (Bravetti et al., 2022). Multiple commuting symmetries (scaling plus Lie group) can be reduced in any order, yielding equivalent Kirillov Hamiltonian or Jacobi structures on the quotient (Bravetti et al., 2023). The approach extends to PDEs, continuum field theories, and systems with mixed symmetries, and provides a framework for direct implementation via integer-matrix algorithms (Hermite, Smith forms) (Tanburn et al., 15 Dec 2025).

7. Physical and Conceptual Implications

Systematic nondimensionalization via scaling symmetries not only reduces computational complexity and clarifies parameter dependence, but also elucidates deep structural features:

• Elimination of Unobservable Degrees of Freedom: All physical predictions are expressed purely in terms of dimensionless invariants; overall scale is rendered surplus to empirical content.
• Resolution of Singularities: Dimensionless reduced systems remain smooth where the original, dimensioned system is singular (e.g., cosmological big-bang becomes a type-2 Janus point) (Gryb et al., 2021).
• Emergence of the Arrow of Time: Contact measure flows exhibit dissipative focusing, providing a rigorous realization of entropy growth and time asymmetry—without appealing to a probabilistic past-hypothesis (Gryb et al., 2021, Bravetti et al., 2022).
• Parameter Reduction: All constant factors with the same scaling weight become redundant; only invariants constructed from their ratios remain in the reduced model (Tanburn et al., 15 Dec 2025, Nill, 2023).

A plausible implication is that systematic scaling reduction forms the rigorous backbone for any dimensionless theory of physical phenomena and provides a universal tool for extracting the minimal, genuinely predictive structure from complex models. The general methodology is now algorithmic and widely applicable, with active research into broader symmetry types and applications across mathematics, physics, and quantitative biology (Tanburn et al., 15 Dec 2025, Bravetti et al., 2022).