Hamilton–Pontryagin–Herglotz Theory

• Hamilton–Pontryagin–Herglotz theory is a unified geometric framework that generalizes classical variational principles to include dissipative and nonconservative systems.
• It introduces an additional scalar action variable z and leverages contact Hamiltonian dynamics to derive generalized Euler–Lagrange and Herglotz equations.
• The theory integrates optimal control and extends Pontryagin's Maximum Principle, demonstrating applications in reduced mechanical systems and dissipative control.

The Hamilton–Pontryagin–Herglotz (HPH) theory is a unified geometric and variational framework accommodating dissipative and nonconservative systems, optimal control with action-dependent costs, and generalizes contact Hamiltonian mechanics, Pontryagin's Maximum Principle (PMP), and the Herglotz variational principle within both classical and algebroid settings. By incorporating an additional scalar action variable $z$—itself governed by a contact-type ODE—the theory accounts intrinsically for dissipation and irreversibility, formulating equations and conservation-like laws on tangent bundles, Lie algebras, and general Lie algebroids via geometric constructions, including E-connections and contact geometry (Simoes et al., 19 Dec 2025, León et al., 2020).

1. Herglotz Variational Principle and Contact Formalism

The Herglotz variational principle extends the classical variational calculus by introducing a dynamical variable $z(t)$, subject to

$$\dot z(t) = L\left(t, q(t), \dot{q}(t), z(t)\right), \quad z(a) = z_0,$$

where the Lagrangian $L$ may depend explicitly on $z$. Extremals are curves $q(t)$ yielding stationary terminal $z(b)$ under variations with fixed $q(a), q(b), z(a)$. The corresponding Euler–Lagrange–Herglotz equations are

$$\frac{\partial L}{\partial q^i} - \frac{d}{dt}\frac{\partial L}{\partial \dot{q}^i} + \frac{\partial L}{\partial z}\frac{\partial L}{\partial \dot{q}^i} = 0, \qquad \dot z = L.$$

This reduces to the standard variational framework when $L$ is independent of $z$.

Herglotz-type equations are Legendre-equivalent to contact Hamiltonian dynamics. On a contact manifold $(M, \eta)$ with $M = T^*Q \times \mathbb R$ and $\eta = dz - p_i dq^i$, the contact Hamilton equations are

$$\dot q^i = \frac{\partial H}{\partial p_i}, \qquad \dot p_i = -\frac{\partial H}{\partial q^i} - p_i \frac{\partial H}{\partial z}, \qquad \dot z = p_i \frac{\partial H}{\partial p_i} - H.$$

This formulation enables the direct treatment of dissipative systems in a Hamiltonian language (León et al., 2020).

2. Hamilton–Pontryagin–Herglotz Principle on Lie Algebroids

Given a Lie algebroid $\tau: E \to Q$ with anchor $\rho$ and bracket $[\cdot,\cdot]$, the HPH variational principle generalizes contact variational mechanics to a geometric setting encompassing reduction and symmetry. Let $L:E \times \mathbb R \to \mathbb R$ be a Herglotz-type Lagrangian. The action variable $z(t)$ evolves via a contact-type ODE,

$$\dot z(t) = L\bigl(x(t), v(t), z(t)\bigr) + \langle p(t), a(t) - v(t) \rangle,$$

where $a(t)$ is an admissible $E$-path, $v(t) \in E_{x(t)}$ an unconstrained velocity, and $p(t) \in E^*_{x(t)}$ a Lagrange multiplier enforcing $a = v$. The variational principle considers the endpoint value $z(T)$,

$$z(T) = z(0) + \int_0^T \left[ L\bigl(x(t), v(t), z(t)\bigr) + \langle p(t), a(t) - v(t) \rangle \right] dt,$$

with variations vanishing at endpoints in $x$ and $z$ (Simoes et al., 19 Dec 2025).

Stationarity yields the implicit HPH equations,

$$\begin{cases} \dot x = \rho(a) & \text{(admissibility)} \ a = v & \text{(velocity matching)} \ p = dL_{\mathrm{ver}}(a, z) & \text{(momentum relation)} \ \overline{\nabla}^*_a p - \rho^*(dL_{\mathrm{hor}}(a, z)) = \frac{\partial L}{\partial z}(a, z)\, p & \text{(horizontal dynamics)} \ \dot z = L(a, z) & \text{(Herglotz ODE)} \end{cases}$$

where the covariant derivatives utilize connections induced on $E$ and $E^*$, and $$dL = dL_{\mathrm{hor}} + dL_{\mathrm{ver}} + (\partial L/\partial z) dz$$ is the intrinsic decomposition of fiber derivatives. In coordinates, this reproduces the generalized Herglotz–Euler–Lagrange equations for each fiber type (Simoes et al., 19 Dec 2025).

3. Covariant Connection-Based and Intrinsic Formulations

Fixing a linear $TQ$-connection $\nabla$ on $E$, the theory employs induced E-connections and their duals: $$\nabla_a u = \nabla_{\rho(a)}u, \quad \overline{\nabla}_a u = \nabla_{\rho(u)}a + [a, u],$$ with the dual E-connection on $E^*$ given by

$$\langle \nabla^*_a p, u \rangle = \rho(a)\langle p, u \rangle - \langle p, \nabla_{a}u \rangle.$$

This enables a coordinate-free, intrinsic statement of the HPH equations: $$\overline{\nabla}^*_a p - \rho^*(dL_{\mathrm{hor}}(a, z)) = \frac{\partial L}{\partial z}(a, z) p, \qquad p = dL_{\mathrm{ver}}(a, z),$$ supplemented by $\dot x = \rho(a)$ and $\dot z = L(a, z)$ (Simoes et al., 19 Dec 2025).

The local expression in the presence of structure functions $C^\gamma_{\alpha\beta}$ and anchors $\rho^i_\alpha$ is

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

4. Relation to Pontryagin Maximum Principle and Optimal Control

The framework naturally incorporates optimal control with dissipation via an extension of the Pontryagin Maximum Principle. For systems with state $q$, control $u$, and dissipation, the costate equations and the maximization condition arise from a presymplectic formalism: $$\dot q^i = \frac{\partial H}{\partial p_i} = f^i, \qquad \dot z = \frac{\partial H}{\partial p_z} = F, \qquad \dot p_i = -\frac{\partial H}{\partial q^i} = -p_j \frac{\partial f^j}{\partial q^i} - p_z \frac{\partial F}{\partial q^i}, \qquad \dot p_z = -\frac{\partial H}{\partial z} = -p_z \frac{\partial F}{\partial z},$$ with $$H(t, q, z, u, p, p_z) = p \cdot f(t, q, u) + p_z F(t, q, u, z)$$. The maximization is imposed on $u$: $$H\left(t, q, z, u^*, p, p_z\right) = \max_v H\left(t, q, z, v, p, p_z\right).$$

For normal extremals with $p_z = -1$, this Hamiltonian system is contact, and the resulting extremals are precisely the Reeb-normalized contact Hamiltonian trajectories (León et al., 2020).

A unified Hamilton–Pontryagin–Herglotz functional is

$$\mathbb S[q, p, z, u] = \int_a^b \left( p_i \dot q^i + p_z \dot z - H(t, q, p, z, u) \right) dt,$$

with $p_z=-1$, directly yielding both Herglotz Euler–Lagrange equations and the contact Pontryagin system upon variation.

5. Energy Balance Laws and Noether–Herglotz Theorem

The HPH setting generalizes conservation laws to dissipative variants. The Herglotz energy function

$$E(x, y, z) = \left\langle \frac{\partial L}{\partial y}, y \right\rangle - L(x, y, z)$$

evolves along solutions as

$\dot E = \frac{\partial L}{\partial z} E,$

so the rescaled quantity $e^{-\int^t (\partial_z L) d\tau}\, E(t)$ is exactly conserved (Simoes et al., 19 Dec 2025).

Noether's theorem is adapted: if a section $\sigma \in \Gamma(E)$ has its complete lift preserving $L$, $\mathcal L_{\sigma^C} L = 0$, then the momentum

$$J_\sigma(t) = \left\langle \frac{\partial L}{\partial y}(x, y, z), \sigma(x) \right\rangle$$

obeys

$$\dot J_\sigma = \frac{\partial L}{\partial z} J_\sigma \implies \frac{d}{dt}\left( e^{-\int^t (\partial_z L) d\tau} J_\sigma \right) = 0,$$

promoting dissipated invariants in place of strict conservation laws.

6. Specializations: Tangent Bundles, Lie Algebras, and Reduction

The HPH theory encompasses several classical and modern instances:

• Tangent Bundle ($E = TQ$): The anchor is identity, yielding the classical Euler–Lagrange–Herglotz equations,

$$\frac{d}{dt} \frac{\partial L}{\partial \dot q^i} - \frac{\partial L}{\partial q^i} - (\partial_z L)\frac{\partial L}{\partial \dot q^i} = 0, \quad \dot z = L.$$

• Lie Algebra ($E = \mathfrak g$): The base is a point, and the equations are the Euler–Poincaré–Herglotz equations,

$$\frac{d}{dt} \left( \partial_\xi \ell \right) + \mathrm{ad}_\xi^* (\partial_\xi \ell) = \frac{\partial \ell}{\partial z} \partial_\xi \ell, \quad \dot z = \ell(\xi, z).$$

• Atiyah Algebroid ($E = TQ/G$): For a principal $G$-bundle with connection $\mathcal A$, the Lagrange–Poincaré–Herglotz equations, expressed in a $G$-invariant frame, reduce to contact-reduced equations, capturing reduced dissipative dynamics.

These specializations demonstrate the generality of the HPH principle in unifying geometric variational formulations across a spectrum of dissipative and nonconservative mechanical systems.

7. Applications and Implications in Dissipative Mechanics and Control

The HPH framework finds direct application in optimal control problems featuring dissipation or action-dependent costs, and naturally extends to thermodynamically irreversible and contact Hamiltonian dynamics. The triangle of equivalences—Herglotz variational principle, contact Hamiltonian dynamics, and the Herglotz–Pontryagin Maximum Principle—permits seamless transition between Lagrangian variational, Hamiltonian contact, and optimal control perspectives, all within a single geometric formalism (León et al., 2020).

Concrete applications include the thermomechanical evolution of gas–piston–damper systems, where the contact formalism enables the treatment of control and dissipation on equal footing with conservative dynamics. The trajectory arising from extremizing the contact Hamiltonian agrees with the optimal trajectory for entropy-minimal evolution. This suggests the HPH framework is a canonical geometric setting for analysis, reduction, and optimality in nonconservative mechanics and control systems.