VGP Hamiltonians: Variational Galerkin Insights

• VGP Hamiltonians are discrete Hamiltonians derived by projecting the continuous variational principle onto a finite-dimensional ansatz using quadrature rules.
• They yield structure-preserving integrators that are symplectic and conserve momentum through a discrete Noether theorem and group invariance.
• The framework generalizes symplectic Runge–Kutta methods, enabling high-order, stable numerical integration for Hamiltonian systems.

VGP Hamiltonians are discrete Hamiltonians arising from a variational Galerkin projection (VGP) of Hamilton’s phase-space variational principle, specifically by discretizing the continuous variational problem using a finite-dimensional ansatz and quadrature approximations. As a result, VGP Hamiltonians define structure-preserving, symplectic integrators for Hamiltonian dynamical systems, with deep connections to group invariance and the discrete Noether theorem. They are of particular significance in numerical analysis, geometric integration, and theoretical mechanics for their ability to guarantee preservation of symplectic structure and momentum maps.

1. Continuous and Discrete Variational Principles

The foundational principle for VGP Hamiltonians is Hamilton’s variational principle written with Type II boundary data in phase space: $$\mathcal{S}(q(\cdot),p(\cdot)) = p(T)\cdot q(T) - \int_0^T \left[ p(t)\cdot \dot{q}(t) - H(q(t),p(t)) \right] dt$$ Variations with $q(0)$ and $p(T)$ fixed yield Hamilton’s canonical equations. The extremal value of this action with these boundary conditions defines a Type II generating function: $$\mathcal{S}(q_0,p_T) = \mathrm{ext}_{(q(\cdot),p(\cdot))\,:\,q(0)=q_0,\,p(T)=p_T} \mathcal{S}(q(\cdot),p(\cdot))$$ For a time step $h$, the corresponding exact discrete right Hamiltonian is: $$H_d^+(q_0,p_1) = \mathrm{ext}_{(q(\cdot),p(\cdot))\,:\,q(0)=q_0,\,p(h)=p_1} \left[ p_1\cdot q(h) - \int_0^h \left[ p(t)\cdot \dot{q}(t) - H(q(t),p(t)) \right] dt \right]$$ Differentiating the generating function $\mathcal{S}$ with respect to boundary data recovers initial and final positions and momenta, e.g., $q_1 = D_2 \mathcal{S}(q_0,p_1)$, $p_0 = D_1 \mathcal{S}(q_0,p_1)$.

2. Variational Galerkin Projection and VGP Hamiltonians

The exact discrete Hamiltonian is generally not computable: it requires extremizing over all smooth phase-space trajectories. VGP Hamiltonians are constructed by projecting onto a finite-dimensional ansatz space using basis functions $\{\psi_i(\tau)\}$ and approximating the action with quadrature. With $\tau\in[0,1]$, the ansatz takes the form: $$\dot{q}_d(\tau h) = \sum_{i=1}^s V^i\psi_i(\tau),\qquad q_d(\tau h) = q_0 + h \sum_{i=1}^s V^i \int_0^\tau \psi_i(\rho)d\rho$$ A quadrature rule with nodes $c_i$, weights $b_i$ is applied: $$H_d^+(q_0,p_1) \approx \mathrm{ext}_{V^i,P^i} \left\{ p_1 \cdot (q_0 + h \sum_i B_i V^i) - h \sum_{i=1}^s [ P^i(\sum_j V^j \psi_j(c_i)) - H(q_0 + h \sum_j A_{ij} V^j, P^i) ] \right\}$$ with $B_i = \int_0^1 \psi_i(\tau)d\tau$ and $A_{ij} = \int_0^{c_i} \psi_j(\tau) d\tau$.

This generalized Galerkin integrator produces a discrete Hamiltonian map; with a hyperregular $H$, the resulting map agrees whether one starts from a discrete Lagrangian or discrete Hamiltonian.

3. Group Invariance and Discrete Noether Theorem

If the Hamiltonian (or Lagrangian) is invariant under a Lie group $G$ acting on $Q$, its exact flow (and suitable discrete approximation) conserves the momentum map $J$. The paper shows that the discrete Hamiltonian $H_d^+$ inherits invariance properties if defined via $G$-equivariant interpolatory functions. This leads to a discrete Noether theorem: if $R_d(q_0,q_1,p_1) = p_1\cdot q_1 - H_d^+(q_0,p_1)$ is $G$-invariant, then

$p_1 \cdot \xi_Q(q_1) - p_0 \cdot \xi_Q(q_0) = 0$

for any $\xi$ in the Lie algebra $\mathfrak{g}$; this is the discrete analogue of continuous momentum conservation. $G$-equivariant interpolants (constructed, e.g., using natural charts from the exponential map) guarantee this invariance and hence momentum preservation by the discrete map.

4. SPRK Methods from the VGP Framework

The VGP approach yields integrators that are algebraically equivalent to symplectic partitioned Runge–Kutta (SPRK) methods. Reformulating the stationarity conditions results in SPRK update formulas: $$\begin{align*} q_1 &= q_0 + h \sum_{i=1}^s b_i \left.\frac{\partial H}{\partial p}\right|_{(Q^i, P^i)} \ p_1 &= p_0 - h \sum_{i=1}^s b_i \left.\frac{\partial H}{\partial q}\right|_{(Q^i, P^i)} \end{align*}$$ with the internal stages $Q^i, P^i$ determined from the quadrature and basis coefficients. The resulting SPRK method is symplectic by construction and, when $G$-invariance has been enforced via the ansatz, also preserves discrete momentum maps.

The key advantage of the SPRK realization is that it unites powerful numerical properties of partitioned Runge–Kutta methods (such as high-order accuracy, stability, and efficiency) with structure preservation derived from the underlying variational principle.

5. Structure Preservation and Numerical Implications

VGP Hamiltonian integrators inherit symplecticity automatically due to discretization of the action functional. When group invariance is present, momentum conservation holds at the discrete level via the discrete Noether theorem. The integrators thus exhibit bounded energy error and stable qualitative long-time behavior, which are critical for Hamiltonian system simulation, especially over many timesteps.

The generalized Galerkin construction accommodates arbitrary polynomial order and can systematically produce high-order integrators by increasing basis dimension and quadrature accuracy. For hyperregular systems, the equivalence between discrete Lagrangian and Hamiltonian formalisms ensures compatibility across geometric integration algorithms.

6. Summary and Significance

VGP Hamiltonians are a class of discrete Hamiltonians derived by variational Galerkin projection of the continuous phase-space variational principle. They:

• Provide structure-preserving numerical integrators (symplectic, momentum-conserving),
• Generalize symplectic Runge–Kutta methods via discrete generating functions,
• Systematically incorporate group invariance to guarantee momentum conservation,
• Offer a framework for high-order and efficient computational schemes for Hamiltonian systems,
• Underpin modern geometric integration practice with rigorous variational foundations.

The synthesis of variational discretization, symmetry, and algebraic implementation embodied by VGP Hamiltonians is central to both the theory and application of geometric numerical integration (Leok et al., 2010).