research

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Arash Yavari         Amit Acharya

Curl forces are position-dependent, non-conservative, and non-dissipative forces that, in general, cannot be derived from an ordinary potential energy. Consequently, their equations of motion do not, in general, follow from a standard variational principle. In this paper, we present a dual variational formulation for particle dynamics under curl forces. By introducing variables dual to position and velocity and an auxiliary function, we construct a pre-dual action in which the equations of motion act as constraints. Stationarity with respect to the primal variables defines a dual-to-primal mapping, whose substitution into the pre-dual action gives an action expressed entirely in terms of the dual variables. The Euler–Lagrange equations of the dual action recover both the original equations of motion and their prescribed initial conditions. We also introduce an auxiliary dual Hamiltonian that is conserved along stationary dual trajectories, although it does not represent the physical energy. The formulation is illustrated using two nonlinear curl force fields in two and three dimensions and the classical Ziegler column. These examples demonstrate that non-conservative curl-force dynamics can admit variational descriptions even in the absence of an ordinary potential energy or a conventional Lagrangian.

Our latest paper is now freely accessible for the next 50 days from this link: https://authors.elsevier.com/a/1nUCG3In-v8hqF 

Someone asked how "Thermodynamic state index in the Unified Mechanics Theory relates, conceptually or mathematically, to conditional phase-space entropy in finite Hamiltonian systems." I posed the question to Gemini AI. Here is its answer in the attached file. 

The lemma of Hill (or of Hill-Mandel) is crucial to the consistent treatment of effective properties in the theory of composites. Building on various classical results (notably by Maxwell and Airy), I've recently applied the lemma sort of "out of context" to characterize (count really) the (infinitesimal) isometric deformations of periodic surfaces.

Dear colleagues,

I am pleased to invite you to the next webinar in the Extreme Mechanics Letters (EML) Webinar Series.

Our upcoming seminar will be delivered by Prof. Miguel Bessa (Brown University):

"Disentangling Uncertainty in AI for Engineering"

Date: Wednesday, 15 July 2026

Time: 10:00 am Boston / 3:00 pm London / 4:00 pm Paris / 10:00 pm Beijing

A variational minimum principle for linear elastodynamics of a possibly heterogeneous material without a stored energy function is developed. It involves a change of variables to dual fields, and results in a degenerate elliptic Euler-Lagrange system, even when the primal elastodynamics is hyperbolic. Uniqueness assertions for the dual dynamic and static problems and implications of the degenerate ellipticity are sketched. Some implications pertaining to heterogeneous materials and ones with indefinite elastic moduli are discussed.

Elastomers are among the most familiar and most deceptive solids. They stretch enormously, recover their shape, and appear forgiving in a way that glass or ceramic never would. Yet they are held together by covalent bonds whose intrinsic strength is measured in GPa. The macroscopic strength of rubber-like networks, however, is usually only in the MPa range. How does a material made of very strong bonds become so weak?

Dear colleagues,

I am honoured to be helping restart the EML Webinar Series as Special Editor for Extreme Mechanics Letters (EML).

Our first seminar of the new season will be by Prof. Alain Goriely, University of Oxford, on:

“Tilings and Mosaics: Soft Cells and the Tainted Love of the Nautilus”

Date: Friday, 19 June 2026

Time: 10 am Boston / 10 pm Beijing / 4 pm Paris / 3 pm London

Discussion leader: Prof. Ellen Kuhl, Stanford University