Euler–Lagrange–Herglotz Equations

• Euler–Lagrange–Herglotz equations are a set of variational equations that extend classical Euler–Lagrange principles by incorporating dissipative and state-dependent effects through an evolving action variable.
• They utilize a non-additive action variable defined by a differential equation, enabling extensions to higher-order, fractional, and geometric variational problems.
• These equations underpin practical modeling in contact systems, damped oscillators, and optimal control scenarios involving time delays and memory effects.

The Euler–Lagrange–Herglotz equations generalize classical Euler–Lagrange variational equations to include non-conservative (dissipative) and state-dependent effects via an auxiliary “action” variable $z(t)$. Under this variational principle, instead of extremizing an integral, one considers systems in which the action evolves according to a non-additive ordinary differential equation, typically of the form $\dot z(t)=L(t,x(t),\dot x(t),z(t))$. The resulting necessary conditions—Euler–Lagrange–Herglotz equations—govern optimality for fundamental, higher-order, time-delayed, and even fractional and geometric extensions of variational problems. These equations underpin the mechanics of contact (action-dependent Lagrangian) systems, dissipative dynamical models, and their geometric generalizations.

1. Fundamental Herglotz Principle and Classical Equations

For a $C^1$ trajectory $x:[a,b]\to\mathbb R^n$ and initial action $z(a)=z_a$, the Herglotz variational principle seeks extremals of the terminal action $z(b)$, where $z(t)$ evolves as $\dot z(t)=L(t,x(t),\dot x(t),z(t))$ and $L\in C^1$ is the (possibly $z$-dependent) Lagrangian. Variations $x\to x+\varepsilon\eta$ with $\eta(a)=\eta(b)=0$ (and corresponding $z_\varepsilon(t)$) induce a first variation that leads to the Euler–Lagrange–Herglotz equation: $$\frac{d}{dt} \left( \frac{\partial L}{\partial \dot x_j} \right) - \frac{\partial L}{\partial x_j} + \frac{\partial L}{\partial z} \frac{\partial L}{\partial \dot x_j} = 0, \qquad j=1,\ldots,n$$ This is the archetypal contact Euler–Lagrange equation, reducing to the classical conservative case for $\partial L/\partial z=0$ (Santos et al., 2014, Santos et al., 2015, López-Gordón et al., 2022). The extra term (proportional to $\partial L/\partial z$) represents the instantaneous rate of nonconservative feedback between the action variable and the dynamics.

The associated transversality (natural boundary) condition, for free final state, is $\frac{\partial L}{\partial \dot x}(b) = 0$.

2. Higher-Order Herglotz Variational Problems

Higher-order Herglotz problems involve Lagrangians that depend on higher derivatives $(x, \dot x, \ldots, x^{(n)})$ and $z$, with $\dot z = L(t,x,\dot x,\ldots,x^{(n)},z)$. The Euler–Lagrange–Herglotz equations for such problems are: $$\sum_{j=0}^n (-1)^j \frac{d^j}{dt^j}\left( \mu(t) \frac{\partial L}{\partial x^{(j)}} \right)=0,\qquad \mu(t)=\exp\left(-\int_a^t \frac{\partial L}{\partial z}(\tau)\,d\tau\right)$$ For each order, the weighting factor $\mu(t)$ encodes the $z$-dependence. Explicit transversality conditions describe free boundary values at endpoints. For example, in the second-order case, the equations yield fourth-order necessary conditions intertwining acceleration, action, and nonconservative contributions (Santos et al., 2013, Santos et al., 2015, Machado et al., 2018).

3. Fractional and Memory-Dependent Extensions

The fractional Herglotz variational principle incorporates generalized Caputo-type derivatives or arbitrary memory-kernels into the Lagrangian. For $B_P^\alpha[x]$ a generalized Caputo operator and $L=L(t,x,B_P^\alpha[x],z)$, the Euler–Lagrange–Herglotz equations become: $$\lambda(t) \frac{\partial L}{\partial x_j}(t,\ldots) + A_{P^*}^\alpha\left( \lambda(t) \frac{\partial L}{\partial(B_P^\alpha x_j)}(t,\ldots) \right) = 0$$ where $$\lambda(t)=\exp\left( -\int_a^t \partial L/\partial z \right)$$ and $A_{P^*}^\alpha$ is the dual Caputo operator. The generalized integration-by-parts formulas required for the derivation naturally yield fractional transversality conditions (Garra et al., 2017, Almeida et al., 2014). For classical kernels and $\alpha\to 1$, the equations reduce to standard (integer-order) Herglotz forms.

The application to dissipative oscillators with memory and time-dependent parameters illustrates the efficacy of the framework. When the fractional order parameter interpolates between no memory and full memory, the resulting equation models exponential attenuation and history dependence simultaneously.

4. Geometric and Hamiltonian Formulations

The contact Lagrangian/Herglotz principle has geometric generalizations on Lie algebroids, manifolds, and in the presence of nonholonomic constraints. For a Lie algebroid $E$ with anchor $\rho$ and structure functions $C^\gamma{}_{\alpha\beta}$, the Euler–Lagrange–Herglotz system reads

$$\frac{d}{dt}\Bigl(\frac{\partial L}{\partial y^\alpha}\Bigr) + C^\gamma_{\alpha\beta}\,y^\beta\,\frac{\partial L}{\partial y^\gamma} - \rho^i_\alpha\,\frac{\partial L}{\partial x^i} - \frac{\partial L}{\partial z} \frac{\partial L}{\partial y^\alpha} = 0,$$

with evolution equations for $x^i$ and $z$ (Simoes et al., 19 Dec 2025).

In the Hamiltonian setting, one employs the contact Legendre transform and the resulting dynamics on the extended contact phase space. For regular Lagrangians, the contact bracket gives rise to the contact Hamiltonian equations, with energy dissipation governed by the action-dependence of the Lagrangian. Impact problems in non-smooth contexts admit a generalization of the principle, enforcing matching of tangential momentum and dissipated energy at collision surfaces (López-Gordón et al., 2022).

5. Delay, Multi-Objective, and Vakonomic Extensions

For time-delay systems, piecewise smooth, and multi-objective Herglotz variational problems, the structure of the equations adapts with additional variables and coupling:

• Time delay: The higher-order Euler–Lagrange–Herglotz equations incorporate both present and delayed arguments, yielding coupled difference-differential necessary conditions for each segment (Santos et al., 2016).
• Multi-objective: The Herglotz system extends to vector-valued actions $u(s)\in\mathbb R^n$ with linear coupling via a matrix $A$, producing an $n$-coupled system of optimality conditions of Herglotz-type, which underpin cooperative weakly coupled Hamilton–Jacobi systems (Cheng et al., 2021).
• Vakonomic (constrained) Herglotz: Nonholonomic constraints are incorporated via Lagrange multipliers, giving rise to constrained Euler–Lagrange–Herglotz equations in the contact setting (León et al., 2021).

6. Noether-Type Theorems, Energy Dissipation, and Invariants

The extension of Noether’s theorem to the Herglotz variational context replaces conservation laws by dissipated invariants, weighted by the integrating factor corresponding to the $z$-dependence: $$\frac{d}{dt} E_L = \frac{\partial L}{\partial z} E_L$$ for the energy $E_L$ associated to the Lagrangian. Along extremal trajectories, the rescaled quantity $e^{-\int \partial L/\partial z\,dt} E_L$ is strictly conserved (Simoes et al., 19 Dec 2025, López-Gordón et al., 2022, Santos et al., 2015, Garra et al., 2017, Almeida et al., 2014). Similar weighted conservation laws hold for momenta associated to infinitesimal symmetries, generalizing Noether’s results to dissipative and memory-affected systems.

In the contact and fractional frameworks, Noether-type results involve bilinear operators and multiple derivative types (classical and fractional), yielding conservation or dissipation laws appropriately adapted to the variational structure.

7. Applications and Representative Examples

The Euler–Lagrange–Herglotz equations are implemented in a range of contexts:

• Damped harmonic oscillators: Fractional and integer-order Herglotz principles yield evolution equations modeling time-dependent dissipation and memory, interpolating between classical damped and undamped cases (Garra et al., 2017, Almeida et al., 2014).
• Riemannian cubics and higher-order geometry: Second-order contact variational principles on $S^n$ produce equations governing cubic polynomials and dissipative extensions on spheres, with direct relevance to geometric mechanics (Machado et al., 2018).
• Optimal control and weakly coupled systems: The analogy with Pontryagin’s Maximum Principle enables direct identification of adjoints, transversality conditions, and DuBois-Reymond identities. For multi-objective cases, this yields the connection to cooperative Hamilton–Jacobi PDEs (Santos et al., 2015, Santos et al., 2014, Cheng et al., 2021).

The flexibility of the Herglotz variational framework supports both dissipative and non-conservative physical models and their geometric and analytical generalizations, ensuring its prominence in modern variational research.