Noether–Herglotz Results in Dissipative Mechanics

• Noether–Herglotz-type results are extensions of classical conservation laws that accommodate dissipation by incorporating a z-dependence in variational formulations.
• They unify variational, Hamiltonian, and optimal control approaches through contact geometry, yielding exponentially weighted invariants and modified energy-balance laws.
• Applications include systems such as damped pendulums and gas-piston models, where dissipated invariants replace traditional conservation concepts with dynamic balance conditions.

Noether–Herglotz-type results refer to the extension of Noether's theorem—central to variational and Hamiltonian mechanics—to frameworks characterized by non-conservative dynamics, specifically those governed by the Herglotz variational principle and contact geometry. In this setting, classical conservation laws are supplanted by balance laws for "dissipated" invariants reflecting the irreversible, dissipative character of the underlying evolution. These results unify and generalize variational, Hamiltonian, and optimal control formalisms in contexts such as contact Hamiltonian mechanics, Lie algebroids, dissipative mechanical systems, and non-conservative field theories (León et al., 2020, Machado et al., 2018, Simoes et al., 19 Dec 2025).

1. Contact Geometry and the Herglotz Principle

The geometric foundation for Noether–Herglotz-type results is contact geometry, which generalizes symplectic geometry to odd-dimensional manifolds admitting intrinsic dissipation. A contact manifold $(M,\eta)$ of dimension $2n+1$ is equipped with a 1-form $\eta$ such that $\eta\wedge(d\eta)^n\neq0$, supporting a unique Reeb vector field $R$ characterized by $i_R\eta=1$, $i_R d\eta=0$. Locally, Darboux coordinates $(q^i,p_i,z)$ realize $\eta=dz-p_i dq^i$, $R=\partial/\partial z$.

The dynamics generated by a contact Hamiltonian $H\in C^\infty(M)$ are governed by the bundle map $b(v)=i_v d\eta + \eta(v)\eta$ and the defining equation $b(X_H)=dH-(R(H)+H)\eta$, yielding evolution equations: $$\dot{q}^i = \frac{\partial H}{\partial p_i},\quad \dot{p}_i = -\left(\frac{\partial H}{\partial q^i} + p_i \frac{\partial H}{\partial z}\right),\quad \dot{z} = p_i\frac{\partial H}{\partial p_i}-H$$ (León et al., 2020, Simoes et al., 19 Dec 2025).

Herglotz's variational principle replaces the classical action integral with an evolution equation for a scalar "action-like" variable $z$: $\dot{q}^i = v^i,\qquad \dot{z} = L(q, v, z)$ where extremals satisfy the generalized Euler–Lagrange–Herglotz equations: $$\frac{d}{dt}\left(\frac{\partial L}{\partial v^i}\right)-\frac{\partial L}{\partial q^i} = \frac{\partial L}{\partial z}\,\frac{\partial L}{\partial v^i}$$ (León et al., 2020, Machado et al., 2018).

2. Pontryagin Maximum Principle in the Contact Setting

Optimal control problems with dissipation naturally fit into contact geometry. Consider a state manifold $Q$, control set $U\subset\mathbb{R}^k$, and a dissipation variable $z$. The system evolves by

$\dot{q}^i=f^i(q,u),\qquad \dot{z}=F(q,u,z)$

with costate variables $p_i$ conjugate to $q^i$ and with $z$ as an additional contact variable. The Pontryagin Hamiltonian is

$\mathcal{H}(q,p,z,u) = p_i f^i(q,u) + p_z F(q,u,z)$

The contact extension of Pontryagin's Maximum Principle (PMP) asserts that normal extremals $(q(t),p(t),z(t))$ satisfy

$$\dot{q}^i = \frac{\partial\mathcal{H}}{\partial p_i},\qquad \dot{p}_i = -\left(\frac{\partial\mathcal{H}}{\partial q^i} - p_i\frac{\partial\mathcal{H}}{\partial z}\right),\qquad \dot{z} = p_i\frac{\partial\mathcal{H}}{\partial p_i} - \mathcal{H}$$

with the pointwise maximization condition

$$\mathcal{H}(q,p,z,u)=\max_{v\in U}\mathcal{H}(q,p,z,v)$$

(León et al., 2020). This generalizes Herglotz's variational principle as a special case and demonstrates how dissipation (through the $z$-dependence) modifies the geometric structure of the control problem.

3. Noether–Herglotz Theorem and Dissipated Invariants

In dynamical systems derived from Herglotz-type principles, conserved quantities associated with symmetries become "dissipated" invariants. The Noether–Herglotz theorem states that if $\sigma$ is an infinitesimal symmetry of the Hamiltonian $H$ on a contact manifold (i.e., $\mathcal{L}_{\sigma^C}H=0$, $\mathcal{L}_{\sigma^C}\theta=0$ where $\sigma^C$ is the complete lift of $\sigma$), the associated momentum map

$$J_\sigma(x, y, p, z) = \langle p, \sigma(x) \rangle$$

evolves according to

$$\dot{J}_\sigma = -\frac{\partial H}{\partial z} J_\sigma$$

Integrating yields an exponentially weighted preserved quantity: $$\frac{d}{dt}\left(e^{\int_0^t (\partial H/\partial z)\,ds} J_\sigma\right) = 0$$ (Simoes et al., 19 Dec 2025). Thus, classical conservation laws are modified, reflecting the energetic exchange imposed by the $z$-dependence.

This phenomenon generalizes across geometric settings:

• Tangent bundles ($E=TQ$): Standard dissipative Hamiltonian systems.
• Lie algebras ($E=\mathfrak{g}$): Euler–Poincaré–Herglotz equations:

$$\dot{\mu}_\alpha = -C^\gamma_{\alpha\beta} \mu_\gamma \frac{\partial H}{\partial \mu_\beta} - \mu_\alpha \frac{\partial H}{\partial z}$$

with the dissipative momentum law as above.

• Lie algebroids: Noether functions $J_\sigma$ for sections $\sigma\in\Gamma(E)$ obey the same dissipative balance, with the contact structure determined by the canonical 1-form $\theta=dz-p_\alpha y^\alpha$ on $P=E\oplus E^*\oplus\mathbb{R}$ (Simoes et al., 19 Dec 2025).

4. Hamilton–Pontryagin–Herglotz Formulation on Manifolds

The unification of Lagrangian, Hamiltonian, and optimal control approaches with dissipation is achieved via the Hamilton–Pontryagin–Herglotz (HPH) variational principle. This principle introduces auxiliary variables and Lagrange multipliers to impose the evolution constraints, resulting in a system described by:

• Kinematic relations: $\dot{x} = v$, (higher order: $\nabla_t^M v = a$)
• "Action-like" variable evolution: $\dot{z} = H(x,v,a,z,p,q)$
• Stationarity conditions: $p=\partial_v L$, $q=\partial_a L$
• Adjoint equations encoding the dissipative correction via $\partial_z L$ in the form:

$$\nabla^M_t p = -\frac{\partial H}{\partial x} + (\partial_z H) p$$

(Machado et al., 2018). Elimination of the auxiliary (adjoint) variables produces covariant generalized Euler–Lagrange–Herglotz equations.

This methodology admits direct application to Riemannian manifolds (e.g., $S^n$) and higher-order systems, providing a fully intrinsic geometric encoding of dissipation and symmetry (Machado et al., 2018).

5. Applications and Concrete Examples

Noether–Herglotz-type results arise in diverse settings. Key examples include:

• Gas–Piston–Damper systems: A thermodynamic system of a gas in a piston with damping, modeled in the entropy representation. The phase space is $T^*Q\times\mathbb{R}_S$ with contact form $\eta=dS - p_V dV$. The Herglotz-type (contact) Hamiltonian incorporates external forces and dissipation, yielding nonnegative entropy production:

$\dot{S} \geq 0$

(León et al., 2020).

• Pendulum with linear drag: On $S^1$, the second-order Herglotz principle with Lagrangian $$L(x, v, a, z) = \tfrac{1}{2} m\|v\|^2 - mg\langle x, e_2 \rangle - \alpha z$$ produces the damped pendulum equation (in angular coordinate $\theta$):

$$\ddot{\theta} + \frac{\alpha}{m}\dot{\theta} + g\sin\theta = 0$$

(Machado et al., 2018).

• Lie algebroid systems: The Herglotz principle on a Lie algebroid $E\to M$ generalizes classical dissipative variational mechanics to geometrically rich settings involving reduction, symmetry, and gauge structure. Noether–Herglotz invariants here describe dissipated momentum maps compatible with the algebroid's anchor and bracket (Simoes et al., 19 Dec 2025).

Setting	Dissipated Invariant	Conservation Law Form
Standard tangent	$J_\sigma = p_i \sigma^i(q)$	$$\dot{J}_\sigma = -\frac{\partial H}{\partial z} J_\sigma$$
Lie algebra	$J_\sigma = \langle \mu, \sigma \rangle$	same as above
Lie algebroid	$J_\sigma = \langle p, \sigma(x)\rangle$	same as above

6. Energy-Balance Laws and Dissipative Dynamics

A characteristic feature of Noether–Herglotz-type frameworks is the presence of modified energy-balance laws. For the contact Hamiltonian $H(x,p,z)$, the evolution along solutions is

$$\frac{d}{dt} H(x,p,z) = -\frac{\partial H}{\partial z}(x,p,z)\ H(x,p,z)$$

or, equivalently,

$$\frac{d}{dt}\bigg(e^{\int_0^t (\partial H/\partial z)\,ds} H\bigg) = 0$$

(Simoes et al., 19 Dec 2025). This generalizes energy conservation to the dissipative setting: while $H$ is not conserved, the exponentially rescaled Hamiltonian remains constant, measured with respect to the dissipation coefficient $\partial H/\partial z$.

A similar balance law applies to Noether–Herglotz momentum invariants $J_\sigma$.

7. Geometric and Theoretical Unification

Noether–Herglotz-type results demonstrate that contact geometry, Herglotz-type variational principles, and optimal control with dissipation are facets of a unified geometric theory. The key insight is that dissipation, implemented by explicit $z$-dependence in the Lagrangian or Hamiltonian, modifies both the invariance and dynamical evolution so that classically conserved quantities acquire a dissipative multiplicative factor. In the presence of symmetries, this yields “dissipated invariants” instead of conserved quantities.

This unification extends to advanced structures such as Lie algebroids and principal bundle reductions, encompassing Euler–Lagrange–Herglotz, Euler–Poincaré–Herglotz, and Lagrange–Poincaré–Herglotz equations within the same conceptual and formal apparatus (Simoes et al., 19 Dec 2025, León et al., 2020). Applications span mechanical systems with damping, thermodynamic processes with entropy production, and geometric control systems on manifolds and homogeneous spaces.