Legendre Dynamics in Geometry & Mechanics

• Legendre dynamics is the study of dynamical systems using Legendre duality, transforming classical Lagrangian–Hamiltonian mechanics via geometric and analytic methods.
• It applies rigorous tools from symplectic and contact geometry to analyze deterministic and stochastic flows, preserving primal–dual structures in various applications.
• Recent methodologies utilize Legendre transforms in spectral methods and machine learning to solve kinetic equations and improve stability in dynamic models.

Legendre dynamics is the geometric and analytic study of dynamical systems whose structure and evolution are governed by Legendre duality, Legendre transforms, and Legendre submanifolds. The framework encompasses not only the classical correspondence between Lagrangian and Hamiltonian mechanics but also generalizations in contact and symplectic geometry, information geometry, stochastic processes, non-equilibrium thermodynamics, Lie group reductions, machine learning representations, and even novel expansions for kinetic equations.

1. Legendre Duality: Foundational Concepts

Legendre duality originates from the convex-analytic property that strictly convex, smooth functions $F : M \rightarrow \mathbb R$ (with $M$ a smooth $n$-manifold, e.g., $\mathbb R^n$) admit a well-defined Legendre transform. The Legendre transform $F^* : M^* \rightarrow \mathbb R$ is

$$F^*(p) = \sup_{x \in M} \{ \langle p, x \rangle - F(x) \}$$

For strictly convex $F$, the gradient map $\nabla F : M \to M^*$ is a global diffeomorphism, with the inverse $\nabla F^*$. These gradient relations establish a one-to-one correspondence between primal and dual variables, preserved under Legendre duality (Fong et al., 22 Dec 2025).

This structure endows certain submanifolds—namely, the images of $p = dF(x)$ in $T^*M$—with Lagrangian or Legendrian properties, forming the primary geometric objects of Legendre dynamics.

2. Legendre Dynamics in Symplectic and Contact Geometry

Symplectic Geometry

In the symplectic category, each $n$-dimensional Lagrangian submanifold of a $2n$-dimensional symplectic manifold is locally the graph of a closed 1-form. In cotangent bundle coordinates $(x,p)$, the graph of $p=dF(x)$ for a smooth $F$ is Lagrangian. The class of "Legendre dynamics" comprises development of dynamical systems (typically flows or stochastic processes) whose trajectories remain on such graphs at all times.

A symplectomorphism $F : T^*Q \to T^*Q$ mapping every Lagrangian graph $p=d\psi(x)$ to another is necessarily a cotangent lift followed by an exact translation: $F = \tau_{d\chi} \circ f^{\sharp}$ with $f: Q \rightarrow Q$ a diffeomorphism, $f^{\sharp}$ the cotangent lift, and $\tau_{d\chi}$ a fibre translation by $d\chi$ (Fong et al., 22 Dec 2025).

Contact Geometry

A contact manifold $(M, \eta)$ of dimension $(2n+1)$ is defined by a 1-form $\eta$ satisfying the non-integrability condition $\eta \wedge (d\eta)^n \neq 0$. In Darboux coordinates $(q^i, p_i, S)$, $\eta = dS - p_i dq^i$. Legendre submanifolds are $n$-dimensional integral submanifolds where $\eta$ vanishes, usually embedded as the graph of a generating function $\psi(q)$: $$\mathcal{A}_\psi = \{ (q, p_i = \partial_i \psi(q), S = \psi(q)) \}$$ where points on $\mathcal{A}_\psi$ correspond to equilibrium, and those off $\mathcal{A}_\psi$ represent nonequilibrium states (Goto, 2014, Esen et al., 2021, Garcia-Pelaez et al., 2014).

In this setting, Legendre transforms mediate between different potential functions generating the same Legendrian (the contact analogue of a Lagrangian submanifold), and dynamics are realized as flows tangent to these objects.

3. Legendre Dynamics: Flows, SDEs, and Relaxation

Deterministic and Stochastic Flows

Legendre dynamics encompasses ordinary and stochastic differential equations whose solutions preserve the Legendre structure of primal–dual variables. For instance, if $(X_t, P_t)$ evolves so that $P_t = \nabla F_t(X_t)$ for all $t$ (with possibly time-varying $F_t$), the process remains on a Legendre graph (Fong et al., 22 Dec 2025). The SDE constraint for preservation is: $$dP_t = D_x^2 F_t(X_t) \, dX_t + \partial_t \nabla F_t(X_t)\, dt$$

This framework recovers, for example:

• Linear-Gaussian inference dynamics (Kalman filtering, GP regression) as discrete Legendre dynamics on exponential families;
• Ornstein-Uhlenbeck flows as continuous-time Legendre dynamics with explicit ODEs for the mean and covariance.

Relaxation in Contact Geometry

In the contact setting, a class of contact Hamiltonian vector fields $X_H$ with $H(q, p, S) = h(\psi(q)-S)$ generates flows of the form: $$\begin{aligned} &\dot q^i = 0 \ &\dot p_i = (\partial_i\psi(q) - p_i) h'(\psi - S) \ &\dot S = h(\psi - S) \end{aligned}$$ For $h(\Delta) = \gamma \Delta$, these equations exponentially drive the system towards the Legendre submanifold $\mathcal{A}_\psi$ at rate $\gamma$. The function $\Delta = \psi(q) - S$ acts as a Lyapunov function; its strict monotonicity ensures that $\mathcal{A}_\psi$ is a global attractor (Goto, 2014).

4. Legendre Dynamics in Analytical Mechanics and Beyond

Classical Legendre Transform, Dual Potentials, and Generalizations

In standard Lagrangian–Hamiltonian mechanics, the Legendre transform transmutes the Lagrangian $L(q, \dot q)$ into the Hamiltonian $H(q,p) = p_i \dot q^i - L$, yielding canonical equations as first-order ODEs. This duality is reflected and generalized by viewing the Legendre map as a morphism of (generalized) tangent and cotangent bundles, including for Lie algebroids and in the presence of external forces or constraints (Arcuş, 2011, Hurtado, 2020).

Further, a full web of four mechanical potentials related by successive Legendre transforms—Lagrangian ($C'$), Hamiltonian ($H$), and two further potentials $Q$, $J$—mirrors the structure of the four standard thermodynamic potentials, with each level corresponding to a distinct set of equations of motion and conservation laws (Teruel, 2013).

Tulczyjew’s Triplet and Lie-Group/Contact Extensions

Tulczyjew’s triplet provides a fully geometric, symplectic-categorical construction of Legendre transformations, embedding Euler–Lagrange, Hamilton, Euler–Poincaré, and Lie–Poisson equations as Lagrangian submanifolds of specific symplectic or contact manifolds. In the Lie-group context, trivialized and reduced triplets yield explicit expressions for the generalized Legendre map, Morse families, and their critical-point conditions, unifying the dynamical equations via canonical geometric tools (Esen et al., 2015, Esen et al., 2021).

A contact version of Tulczyjew’s triple encapsulates contact Hamiltonian and Lagrangian theory, allowing Legendrian submanifolds to be generated via Morse families and displaying Legendre transforms as transitions between different generators of the same Legendrian (Esen et al., 2021).

5. Metrics, Information Geometry, and Thermodynamic Implications

A metric structure, such as the Mrugala metric on a contact manifold $(M, \eta)$ given by $G = dq^i \odot dp_i + \eta \otimes \eta$, induces on the Legendre submanifold $\mathcal{A}_\psi$ a Hessian metric $g_{ij} = \partial_i \partial_j \psi(q)$. This endows the equilibrium submanifold with a dually flat structure, placing Legendre dynamics at the intersection of thermodynamics, fluctuation theory, and information geometry (Goto, 2014).

In geometrothermodynamics, the requirement of infinitesimal Legendre invariance imposes strict constraints on admissible metrics: the induced metric is unique up to a single Legendre-invariant function $\Omega$, but this constraint is incompatible with the vanishing of scalar curvature for the ideal gas. As a result, the established interpretation of phase-space curvature as a measure of thermodynamic interaction does not hold in this maximally symmetric class (Garcia-Pelaez et al., 2014).

6. Applications in Modern Machine Learning: Symplectic Reservoirs

The structure of Legendre dynamics can be injected into machine learning architectures by requiring that recurrent updates are symplectomorphisms preserving Legendre graphs. The Symplectic Reservoir (SR) class of recurrent neural networks implements this: the update $x_{t+1} = W x_t + W_{in} u_t$ (with $W = \exp(A)$ for $A$ in the symplectic algebra) is a symplectomorphism that guarantees that any Legendre graph is mapped into another, preserving the primal–dual structure of information potentials at every step. This leads to more stable, interpretable, and geometrically principled recurrent learning systems, especially for online Bayesian filtering, phase-space tracking, and generative modeling (Fong et al., 22 Dec 2025).

7. Methodologies and Computational Techniques

Legendre dynamics also plays a crucial role in spectral methods for kinetic equations. For example, phase-space distributions $f(x,v,t)$ can be expanded in Hermite–Legendre polynomial bases to recast the Vlasov equation into a closed ODE system for the expansion coefficients

$$f(x,v,t) = \sum_{n,\ell} c_{n\ell}(t) G_{n\ell}(x,v) f_0(x,v)$$

where $G_{n\ell}$ are products of Hermite and Legendre polynomials orthogonal with respect to $f_0$. This representation is efficient for analyzing linearized dynamics in self-gravitating systems, enabling accurate tracking of mode evolution and validating spectral truncation properties via $N$-body simulations (Barnes et al., 2013).

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In summary, Legendre dynamics synthesizes the geometric, analytic, and computational structures stemming from Legendre duality, generating a multidimensional framework that underpins classical mechanics, thermodynamics, information geometry, stochastic flows, machine learning, and kinetic theory. Central features include the invariant characterization of dynamics via Legendre submanifolds, the structure-preserving nature of induced flows and transformations, and deep intersections between mechanics, geometry, and information theory (Goto, 2014, Fong et al., 22 Dec 2025, Esen et al., 2021, Esen et al., 2015, Garcia-Pelaez et al., 2014, Arcuş, 2011, Hurtado, 2020, Teruel, 2013, Barnes et al., 2013).