Non-Conservative Potential Term

• Non-conservative potential terms are defined as functionals that represent forces not derived from a scalar potential, accounting for irreversible effects like dissipation.
• They are integrated into variational and Hamiltonian frameworks via a doubling of degrees of freedom to model time-asymmetric dynamics and path-dependent work.
• Applications span classical mechanics, quantum field theory, and data-driven models, providing new insights into symmetry breaking and energy non-conservation.

A non-conservative potential term is a mathematical object or functional introduced into the dynamical equations of classical, quantum, or stochastic systems to capture effects that do not derive from a standard potential energy function. Unlike conservative systems, where all forces can be written as gradients of a scalar potential and energy is conserved, non-conservative systems include both dissipative and non-potential-generated forces, leading to time-asymmetric evolution, energy loss, or other irreversible processes. Non-conservative potential terms formalize these effects in a way that permits rigorous analysis, computation, and generalization of methods otherwise restricted to conservative settings.

1. Fundamental Definition and Types

Non-conservative potential terms generalize the concept of a potential to cases where the classical force, drift, or dynamical generator cannot be written purely as a negative gradient. In deterministic dynamics, the force field $F(\mathbf{x})$ is non-conservative if there does not exist a scalar function $U(\mathbf{x})$ such that $F = -\nabla U$. Such non-conservative contributions can be:

• Dissipative (e.g., friction, viscous drag, radiation reaction): these extract energy from the system.
• Non-dissipative but non-conservative (“curl” or “circulatory” forces): these generate path-dependent work without necessarily dissipating energy directly (Yavari et al., 22 May 2025, Kycia, 13 Jul 2025).
• Non-conservative products: PDE source terms arising from non-smooth domain geometry or non-divergence-form operators (Colombo et al., 2021).

In action formalism, non-conservative terms are often implemented as extra functionals—denoted $K$ or $\mathcal{K}$—depending on both the state variables and (crucially) their “doubled” or history-reversed copies to consistently encode non-reversible effects (Galley, 2012, Galley et al., 2014, Saha et al., 1 Sep 2025).

2. Formal Construction in Variational and Hamiltonian Frameworks

To systematically analyze non-conservative systems while preserving the strengths of Lagrangian and Hamiltonian mechanics, a doubling of degrees of freedom is employed. Given coordinates $q_1(t),\, q_2(t)$ (or fields $\phi_1,\, \phi_2$), the total action is

$$S[q_1, q_2] = \int_{t_i}^{t_f} dt \left(L(q_1, \dot{q}_1) - L(q_2, \dot{q}_2) + K(q_1, q_2, \dot{q}_1, \dot{q}_2) \right).$$

Here, $K$ is the non-conservative potential term, antisymmetric under the exchange of indices. Variation and imposition of the “equality condition” at the final time (i.e., $q_1(t_f) = q_2(t_f)$) yield equations of motion for the physical variable $q$ (upon setting $q_1 = q_2 = q$), where the non-conservative force is generated by the derivatives of $K$ with respect to the “difference” variable:

$$\frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} = \left[\frac{\partial K}{\partial q_{-,i}} - \frac{d}{dt}\frac{\partial K}{\partial \dot{q}_{-,i}} \right]_{PL}$$

with $q_{-} = q_1 - q_2$ and $PL$ denoting the physical limit $q_{-} \to 0$ (Galley, 2012, Galley et al., 2014, Saha et al., 1 Sep 2025).

In field theories, this methodology permits explicit construction of dissipative models (e.g., a damped scalar field) while maintaining a well-defined canonical structure and causality (Saha et al., 1 Sep 2025).

3. Path-Dependence and Decomposition: Geometric Structure

Non-conservative potential terms fundamentally relate to the geometric properties of the force field in phase or configuration space. When recast as differential forms, the work form $\omega = g(F, \cdot)$ is analyzed via the Helmholtz and Hodge decompositions:

• Conservative component: The exact part—$\omega = df$—corresponds to conservative forces.
• Antiexact/curl component: The residual $\Omega = \omega - df$ (sometimes termed the antiexact or curl component), which encodes non-conservative effects.

Darboux and Frobenius theorems classify these components:

• In 2D, any curl force can be expressed in terms of two generalized potentials; in 3D, up to three are required (Yavari et al., 22 May 2025).
• Any classical force may be decomposed locally as $F = -\nabla f +$ (non-conservative component), where the second term typically cannot be represented by any single scalar (Kycia, 13 Jul 2025).

This decomposition is instrumental both in constructing auxiliary conservative systems and in distinguishing contributions to physical observables that are path-dependent versus path-independent.

4. Physical Interpretation and Applications

Non-conservative potential terms arise in a multitude of practical and theoretical settings:

• Dissipation and Irreversibility: Linear or nonlinear damping (e.g., $$\mathcal{K}(\phi_1, \phi_2) = -(\gamma/2)(\phi_1-\phi_2)(\partial_t\phi_1 + \partial_t\phi_2)$$ in a damped scalar field) allows first-principles quantization of dissipative systems (Saha et al., 1 Sep 2025); tidal friction in astrophysical settings generates non-conservative contributions to orbital and rotational evolution (Migaszewski, 2012).
• Stochastic and Thermodynamic Systems: In Langevin and Fokker-Planck models, drift decompositions $F(x) = \Sigma\Sigma^T \nabla \ln p_S(x) + g(x)$ separate reversible from irreversible dynamics; the divergence-free constraint on $g(x)$ ensures preservation of stationary measures (Giorgini, 3 May 2025). In nonequilibrium settings, non-conservative potential terms encapsulate the minimal set of entropy-producing flux–force pairs subject to conservation laws (Rao et al., 2017).
• Non-conservative Machine Learning Models: Direct force prediction in atomistic ML may yield fast surrogate models, but failure to enforce the gradient structure leads to ill-posed geometry optimization and pathological molecular dynamics trajectories unless periodically corrected by conservative evaluations (Bigi et al., 16 Dec 2024).
• Fluid Dynamics and Non-conservative Products: Balance laws with non-conservative source terms (e.g., at geometric discontinuities in pipes) are handled by specialized mathematical frameworks assigning meaning to products involving discontinuous geometric variables and state vectors (Colombo et al., 2021).
• Non-equilibrium Quantum Field Theory: In Schwinger–Keldysh/in-in formalisms and path-integral approaches, non-conservative potential terms shift the poles of propagators, reflect loss of time-reversal symmetry, and underlie finite-lifetime spectral properties (Saha et al., 1 Sep 2025).

5. Impact on Symmetry, Conservation, and Noether’s Theorem

The introduction of a non-conservative potential term generically breaks the conservation laws derived from Noether’s theorem. In the doubled-variable framework, the usual conserved Noether currents acquire explicit extra terms proportional to $K$ and its derivatives.

For example, the energy function becomes

$$\mathcal{E} = E + \dot{q}^i \kappa_i \quad \text{with}\quad \kappa_i = \left[\frac{\partial K}{\partial \dot{q}_{-,i}}\right]_{PL}$$

and its rate of change includes non-conservative sources. In non-conservative systems, this results in a modified balance law rather than exact conservation. When extended to quantum systems, this structure underpins the selection of retarded Green’s functions and loss of time-reversal symmetry (Galley et al., 2014, Saha et al., 1 Sep 2025).

Notably, in extended systems (e.g., Navier–Stokes fluids or viscoelastic materials), this methodology supports the direct variational inclusion of dissipative processes at the action level.

6. Numerical and Data-Driven Methods Incorporating Non-conservative Terms

The explicit treatment of non-conservative potential terms is essential not just in analytic theory but also in practical numerics and data-driven model reduction:

• In network analysis, non-conservative centrality (e.g., Alpha-Centrality) encodes broadcast-like growth on graphs, aligning more closely with empirical influence in social networks than conservative metrics (Ghosh et al., 2011).
• PINNs designed to handle shocks in non-conservative PDEs use adaptive viscosity and loss re-weighting to stabilize the learning and faithfully represent discontinuities regardless of conservative or non-conservative form (Neelan et al., 27 Jun 2025).
• Data-driven model reduction leverages drift-matrix decomposition and score estimation to faithfully reconstruct both equilibrium (conservative) and non-equilibrium (irreversible) dynamics from time series, ensuring preservation of invariant measures and correctly attributing circulation to the non-conservative sector (Giorgini, 3 May 2025).

Hybrid approaches are also emerging—for instance, periodical correction of fast non-conservative models with conservative evaluations in atomistic ML simulations to balance computational speed with physical fidelity (Bigi et al., 16 Dec 2024).

7. Summary and Outlook

Non-conservative potential terms provide a rigorous and flexible formalism to represent a wide spectrum of physical phenomena inaccessible to strictly conservative theory. Their inclusion gives rise to time-irreversible and/or path-dependent effects, enables the consistent quantization and simulation of dissipative mechanics and field theories, clarifies the geometric and algebraic structure of non-gradient forces, and catalyzes progress in model reduction, fluid dynamics, machine learning, and stochastic thermodynamics. Ongoing research is focused on the refinement of variational principles, advanced geometric decompositions via differential forms in arbitrary dimensions, and the development of hybrid computational strategies that reconcile the efficiency and fidelity trade-offs encountered in large-scale simulations (Galley et al., 2014, Giorgini, 3 May 2025, Yavari et al., 22 May 2025, Kycia, 13 Jul 2025, Bigi et al., 16 Dec 2024, Saha et al., 1 Sep 2025).