PhD Averaging in systems with small Hamiltonian and much smaller non-Hamiltonian perturbations
PhD @Loughborough University posted 9 hours agoJob Description
Project details
For many applications, it is important to study the dynamics of systems that deviate from integrable systems due to small Hamiltonian and much smaller non-Hamiltonian perturbations. Typical examples include evolutionary problems in celestial mechanics. The unperturbed system is Hamiltonian, with its phase space foliated by invariant tori.
The goal of this project is to obtain asymptotic estimates to justify the following averaging principle: for an approximate description of evolution, one should average the perturbations over the unperturbed tori. Such averaging serves as a highly effective tool for studying evolution. It washes out the effects of Hamiltonian perturbations, resulting in a closed system of evolution governed by the averaged non-Hamiltonian perturbation. The rate of evolution is proportional to the magnitude of the non-Hamiltonian perturbation. Notably, the dimension of the phase space of the averaged system is reduced to half that of the original perturbed system.
The averaging principle is well established in cases where only Hamiltonian perturbations are present. Here, averaging predicts the absence of evolution. The Kolmogorov-Arnold-Moser (KAM) theory demonstrates that for most initial conditions, no evolution occurs. According to Nekhoroshev’s theory, on the remaining set of initial conditions, any evolution—when it exists—is extremely slow, with its speed decaying exponentially as the magnitude of the perturbation decreases linearly. It is generally believed that this evolution has a diffusive nature, known as Arnold diffusion.
The averaging principle is partially justified when Hamiltonian and non-Hamiltonian perturbations are of the same order of magnitude. In this scenario, the general result is that the solutions of the averaged system approximate the evolution of the exact system for most initial conditions over a time interval inversely proportional to the magnitude of the perturbation (as established by Anosov’s averaging theorem). Asymptotic estimates exist for one-frequency and two-frequency systems. However, systems with a larger number of frequencies pose significant challenges due to the complex structure of resonances between unperturbed frequencies in the phase space.
The case involving small Hamiltonian and much smaller non-Hamiltonian perturbations remains largely unexplored. Partial results are available for one-frequency and two-frequency systems.
This PhD project aims to prove (or disprove) the following hypothesis for systems with small Hamiltonian and much smaller non-Hamiltonian perturbations: outside a set of initial conditions whose measure tends to zero as the magnitude of the Hamiltonian perturbation approaches zero, the averaging principle accurately describes the evolution over time intervals inversely proportional to the magnitude of the non-Hamiltonian perturbation, with an error that tends to zero as the magnitude of the Hamiltonian perturbation tends to zero.
94% of Loughborough’s research impact is rated world-leading or internationally excellent. REF 2021
Supervisors
Primary supervisor:Â Anatoly Neishtadt
Entry requirements
Our entry requirements are listed using standard UK undergraduate degree classifications i.e. first-class honours, upper second-class honours and lower second-class honours. To learn the equivalent for your country, please choose it from the drop-down below.
Entry requirements for United Kingdom
Applicants should have, or expect to achieve, at least a 2:1 honours degree (or equivalent) or a 1st class honours degree.
English language requirements
Applicants must meet the minimum English language requirements. Further details are available on the International website.
Applicants must meet the minimum English language requirements. Further details are available on the International website.
Fees and funding
Tuition fees for 2025-26 entry
UK fee
£5,006 Full-time degree per annum
International fee
£22,360 Full-time degree per annum
Fees for the 2025-26 academic year apply to projects starting in October 2025, January 2026, April 2026 and July 2026.
Tuition fees cover the cost of your teaching, assessment and operating University facilities such as the library, IT equipment and other support services. University fees and charges can be paid in advance and there are several methods of payment, including online payments and payment by instalment. Fees are reviewed annually and are likely to increase to take into account inflationary pressures.
How to apply
All applications should be made online. Under programme name, select Mathematical Sciences. Please quote the advertised reference number: MA/AN – SF1/2025 in your application.
To avoid delays in processing your application, please ensure that you submit a CV and the minimum supporting documents.
The following selection criteria will be used by academic schools to help them make a decision on your application. Please note that this criteria is used for both funded and self-funded projects.
Please note, applications for this project are considered on an ongoing basis once submitted and the project may be withdrawn prior to the application deadline, if a suitable candidate is chosen for the project.
