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Integrable systems
Most non-linear differential equations cannot be solved exactly. The exception are integrable systems. They possess some structure that allows exact solutions to be found. This structure usually leads to ordered dynamics, so, in a sense, integrability is the opposite of chaotic dynamics
Integrable systems come in many forms, varying from ordinary or partial differential equations in the continuous setting to lattice equations (difference equations) in the discrete setting. They all share the feature that some hidden mathematical structure leads to orderly dynamics and makes it possible to obtain exact solutions. However, there is no universal definition of what this mathematical structure consists of.
One of the main notions of integrability is expressed in terms of Hamiltonian dynamics and involves the existence of a sufficiently large hierarchy of compatible Hamiltonians. In classical mechanics, Hamiltonian systems are equivalent to Lagrangian systems (under mild conditions), but in integrable systems there has long been a notable absence of the Lagrangian perspective.
Lagrangian multiforms
Context
Lagrangian multiform theory is a recent development in integrable systems, the central idea of which comes from physics. Almost every physical theory can be described by the fact that something is minimised. Such a description is called a variational principle. In optics, for example, a ray of light will always take the fastest possible path. In other cases, the quantity that is minimised may be less intuitive, but a variational principle always provides powerful mathematical tools.
In 2009, a concept of integrability based on Lagrangian dynamics was proposed [Lobb & Nijhoff 2009] and became known under the names “Lagrangian multiforms” and “pluri-Lagrangian systems”. Initially it was studied in the context of discrete integrable systems [Boll et al 2014], but it can be applied in the continuous case as well [Suris 2013], where it provides a Lagrangian counterpart to the Hamiltonian theory of integrable ordinary differential equations (ODEs) and partial differential equations (PDEs), as summarised in [Vermeeren 2021].
The theory of Lagrangian multiforms has had some successes, but much of its potential remains unexplored. There are strong indications that it will be useful in connecting and classifying different types of integrable systems. It is also becoming clear that Lagrangian multiforms are not just an attribute of integrability, but can be used more broadly to capture Lagrangian systems with symmetries. Furthermore, some Lagrangian multiforms show a deep, but thus far poorly understood, connection to differential geometry.
Mathematical introduction
The main object in the (continuous) theory of Lagrangian multiforms is a differential \(d\)-form \(\mathcal L[u] \in \Omega^d(\mathbb R^N)\), where the square brackets denote dependence on a function \(u: \mathbb R^N \rightarrow Q\) and its derivatives. Here, \(\mathbb R^N\) is the space of independent variables, often referred to as “multi-time”, \(Q\) is the configuration space, and \(d < N\) is the dimension of the individual equations in the hierarchy. If \(d = 1\) this setup describes \(N\) commuting ODEs.
The variational principle of Lagrangian multiform theory requires that the integral of the \(d\)-form \(\mathcal L\) over an arbitrary \(d\)-dimensional submanifold \(\Gamma\) of \(\mathbb R^N\) is critical with respect to variations of \(u\), i.e.
\begin{equation}
\frac{\partial}{\partial \varepsilon} \bigg|_{\varepsilon = 0} \int_\Gamma \mathcal L[u + \varepsilon v] = 0
\tag{1}
\end{equation} for all smooth functions \(v: \mathbb R^N \rightarrow \mathbb C\) such that \(v\) and all its derivatives vanish on the boundary of \(\Gamma\).
This variational principle is sometimes called the “pluri-Lagrangian” principle. In addition, Lagrangian multiform theory requires that the action integral is invariant under variations of the surface \(\Gamma\). By stokes theorem this is equivalent to the fact that the \(d\)-form is closed, \(\mathrm d \mathcal L = 0\), on solutions. This property can be sued to establish equivalence to other notions of integrability, such as Hamiltonian structures and variational symmetries. Often, the exterior derivative also provides useful information within the Lagrangian framework itself.
As an example, consider the potential Korteweg-de Vries (KdV) hierarchy. It is described by a Lagrangian 2-form
\begin{equation}
\mathcal L = L_{12} \,\mathrm d t_1 \wedge \mathrm d t_2 + L_{13} \,\mathrm d t_1 \wedge \mathrm d t_3 + L_{23} \,\mathrm d t_2 \wedge \mathrm d t_3 + \ldots
\end{equation} with coefficients
\begin{align*}
L_{12} &= -u_{1}^3 – \tfrac{1}{2} u_{1} u_{111} + \tfrac{1}{2} u_{1} u_{2}, \\
L_{13} &= -\tfrac{5}{2} u_{1}^4 + 5 u_{1} u_{11}^2 – \tfrac{1}{2} u_{111}^2 + \tfrac{1}{2} u_{1} u_{3}, \qquad \ldots
\end{align*} where subscripts denote derivatives, \(u_1 = \frac{\partial u}{\partial t_1}\) etc.
The coefficients \(L_{ij}, \ i,j>1\), have more complicated expressions, but methods for their explicit construction exist. The Euler-Lagrange (EL) equation of \(L_{12}\) is a differentiated version of the potential KdV equation:
\[
u_{12} = 6 u_{1} u_{11} + u_{1111}.
\] This corresponds to the variational principle (1) where \(\Gamma\) is a plane spanned by \(t_1\) and \(t_2\). Similarly, if \(\Gamma\) is a plane spanned by \(t_1\) and \(t_3\), we recover
\[
u_{13} = 30 u_{1}^2 u_{11} + 20 u_{11} u_{111} + 10 u_{1} u_{1111} + u_{111111}.
\] The equations that characterize critical functions of Lagrangian multiforms are called multi-time Euler-Lagrange equations and include additional equations corresponding to other surfaces $\Gamma$. The full set of multi-time Euler-Lagrange equations for the 2-form above is equivalent to
\[\tag{2}\begin{split}
u_{2} &= 3 u_{1}^2 + u_{111} , \\
u_{3} &= 10 u_{1}^3 + 5 u_{11}^2 + 10 u_{1} u_{111} + u_{11111}, \\
& \phantom{/}\vdots
\end{split}\] Remarkably, the Lagrangian multiform produces the potential KdV hierarchy in its evolutionary (i.e. first order) form.
The coefficients of the exterior derivative of \(\mathcal L\) can be written as a sum of factorised terms, for example:
\begin{align*}
\mathrm d \mathcal L(\partial_{t_1},\partial_{t_2},\partial_{t_3}) &= \tfrac{1}{2} {\left(30 u_{1}^2 u_{11} + 20 u_{11} u_{111} + 10 u_{1} u_{1111} + u_{111111} – u_{13}\right)} {\left(3 u_{1}^2 + u_{111} – u_{2}\right)} \\
&\qquad – \tfrac{1}{2} {\left(10 u_{1}^3 + 5 u_{11}^2 + 10 u_{1} u_{111} + u_{11111} – u_{3}\right)} {\left(6 u_{1} u_{11} + u_{1111} – u_{12}\right)} .
\end{align*} This expression attains a double zero on solutions of the multi-time Euler-Lagrange equations (2), so infinitesimal variations of \(\mathrm d \mathcal L\) vanish if the multi-time Euler-Lagrange equations are satisfied. In general, the equations obtained by taking variations of the exterior derivative \(\mathrm d \mathcal L\) are equivalent to the multi-time Euler-Lagrange equations. This gives us a second way of deriving equations from the variational principle. In some examples, the equivalence between these two sets of equations yields a nontrivial result.
For further reading on continuous Lagrangian multiforms, see e.g. [Suris 2013], [Suris & Vermeeren 2016], [Sleigh et al 2021], [Vermeeren 2021].
The discrete version of Lagrangian multiform theory is analogous to its continuous counterpart. It arose in the context of multi-dimensionally consistent quad equations. The Lagrangian in this case is a discrete differential form, which we sum over an arbitrary discrete submanifold of a higher-dimensional lattice. We require each of these action sums (i.e. discrete integrals) to be critical.
More on discrete Lagrangian multiforms can be found e.g. in [Lobb & Nijhoff 2009], [Boll et al 2014], [Richardson & Vermeeren 2024].
Connections and applications
Relations between integrable systems of different types
The two ways of deriving multi-time Euler-Lagrange equations can lead to nontrivial equivalences. One such example is found in the context of semi-discrete systems (which involve both discrete and continuous variables), providing a connection to integrable partial differential equations [Sleigh & Vermeeren, 2022]. This is only one of several contexts in which Lagrangian multiforms provide relations between integrable systems of different types. I am working on a broad investigation of this phenomenon, with the aim of transferring insights between equations of different types and classifying equations of interest.
Geometry — intrinsic and extrinsic
In the theory of Lagrangian multiforms, parameters describing symmetries of the system are treated in the same way as the time-variable. Together they form “multi-time”. In the simplest version of Lagrangian multiform theory, multi-time is a euclidean space, but there are nontrivial geometric structures within multi-time, related to special solutions of the integrable system.
When Lagrangian multiform theory is applied to non-commuting flows, the geometry of multi-time itself becomes more interesting. Taking it to be a Lie group allows us to capture a larger class of differential equations and transfer the insights of Lagrangian multiforms beyond the realm of integrable systems [Caudrelier et al 2023].
Yet another geometric aspect of Lagrangian multiform theory is that many of the equations they describe have a differential-geometric interpretation. This is a well-known aspect of integrable systems, but the relations between integrable systems that are established through Lagrangian multiform theory offer exciting glimpses into new connections between various types of differential geometry.
Applications
Lagrangian multiforms have potential applications in numerical mathematics and in fundamental physics. Variational principles have many applications in both these areas, but not all are fully understood from a rigorous mathematical perspective.
In numerical integration, variational integrators are an important class of structure-preserving integrators based on variational prinicples. Lagrangian multiform theory, especially its semi-discrete version, is exactly the right framework to study these integrators as discrete maps with continuos symmetries.
In fundamental physics, Lagrangian multiforms could provide a novel way to approach quantisation (path integrals) and gauge theories (emphasising the fact that Lagrangian multiforms capture symmetries in general, rather than only the integrable cases). Initial steps in this area have been made [King & Nijhoff 2019], but much work remains to be done.