Dynamic 3D Gaussian Splatting (3DGS) achieves photorealistic reconstruction of time-varying scenes, and recent physics-aware extensions improve extrapolation by explicitly predicting velocity fields. However, these extensions merely fit vector fields to visual deformations without satisfying Lagrangian mechanics, leading to three major issues: (i) physically inconsistent trajectories, (ii) lack of time-reversibility, and (iii) geometric collapse during long-term extrapolation. In this paper, we propose LagrangeGS, which formulates dynamic 3DGS as a non-conservative Lagrangian system. While this Lagrangian formulation fundamentally solves (i), a direct application of general LNNs to dynamic 3DGS requires a large velocity-Hessian inversion for millions of Gaussian particles. To overcome this computational bottleneck, we approximate the velocity-Hessian as an identity matrix, decoupling particle dynamics for computational tractability. For (ii), we restrict the non-conservative forces to be explicitly time independent, enabling consistent backward integration. Finally, to address (iii), we introduce local rigid alignment that regularizes particle trajectories. Extensive evaluations on dynamic scene benchmarks demonstrate that LagrangeGS enables stable long-term extrapolation, consistent time reversal, and counterfactual physics-based editing without retraining.
Deformable 3D Gaussian Splatting (D-3DGS) re-constructs dynamic scenes from monocular video by deforming a canonical set of 3D Gaussians through a positional-encoded MLP of frame time t. Although fitted to a continuous variable, the MLP couples no two values of t in its architecture and effectively predicts discrete per-frame offsets, leaving temporal smoothness to emerge only as a byproduct of optimisation. We redesign the deformation field as a stack of Closed-form Continuous-time (CfC) cells, a Liquid Neural Network (LNN), that is the closed-form solution of the Liquid Time-constant ODE while preserving every other part of the D-3DGS pipeline. Each cell exposes a sigmoidal time gate that interpolates between two candidate hidden states, baking a learned smooth response to t into the loss landscape without invoking any numerical solver. On the eight D-NeRF and seven NeRF-DS scenes the liquid field matches or exceeds the MLP baseline in aggregate, with its largest gains concentrated on the scenes with the most high-frequency articulated motion. The result is a near-zero-friction architectural design that turns the discrete MLP deformation field into an explicit continuous-time function of t.