ANALYTICAL DYNAMICS 1

This graduate level course on Analytical Dynamics is based on an integrated approach that combines the classical methodology of dynamical systems modelling (following Lagrange, Hamilton and Jacobi) with the analytical and geometrical developments of nonlinear mechanics (proposed by Lyapunov and Poincare'). This approach enables a comprehensive and rigorous treatment of both multiple rigid-body and continuous dynamical systems incorporating: i) formulation of constrained (non-holonomic) spatiotemporal models (via the variational Hamilton's principle), ii) solution derivation of nonlinear models via exact (integrals of motion, separatrices) and approximate singular perturbation (averaging, asymptotic multiple-scales) methods culminating with slowly varying evolution equations, iii) qualitative behavior of solutions (integrability, symmetry, periodicity, orbital stability, synchronization). Examples include applications to interdisciplinary problems governing the domain of dynamical systems that can only be described by several types of generalized coordinates/forces, such as robotics/mechatronics, thermo-visco-elastic initial-boundary-value problems, nano/micro-opto-electro-mechanical systems, and fluid-structure interaction.

SEMESTER START DATE: October 29, 2025

Contact Hours: 3

Day & Time: Wednesday, 14:30-17:30 (Israeli Time which is one hour ahead of Germany, France, Denmark, Switzerland, Czech Republic, and the Netherlands; same time zone as Estonia)

The objectives of this course are to: (i) introduce, (ii) develop and (iii) apply the fundamentals of analytical dynamics required for derivation, solution and analyses of nonlinear problems that govern engineering mechanics with applications in multiple rigid-body and continuous dynamical systems.

Topics:

• Introduction to calculus of variations and Hamilton's principle.
• Generalized forces, non-holonomic constraints and Lagrange multipliers.
• Lagrange's and Hamilton's equations (multi-rigid-body dynamical systems, ODEs).
• Lagrangian densities, Extended Hamilton's principle (initial-boundary-value problems, PDEs).
• Conservation laws, cyclic coordinates (Routh's equations).
• Stability of equilibria (local) and orbits (Floquet theory), Lyapunov functions (global stability).
• Integrability, separatrices and domains of attraction (radius of robustness).
• Generalized averaging (self-excited oscillations/periodic excitation).
• Multiple-scale asymptotics (singular perturbation near external/parametric resonant interactions).
• Evolution equations (primary/secondary/internal/combination resonances) and Poincare' maps.
• Canonical transformations, Poisson brackets, Hamilton-Jacobi equation.
• Introduction to bifurcation analysis (local/global) and chaos theory.

Literature [* - course text books, a - advanced reading] a- Arnold VI, Mathematical Methods of Classical Mechanics, 1989 [eBook].

• Goldstein H, Classical Mechanics, 1980/2002. a- Kevorkian JK and Cole JD, Multiple Scale and Singular Perturbation Methods, 1996 [eBook].
• Lanczos C, The Variational Principles of Mechanics, 1970/2020 [eBook].
• Meirovitch L, Methods of Analytical Dynamics, 1970/1988.
• Nayfeh AH and Mook DT, Nonlinear Oscillations, 1979/2004 [eBook]. a-Nayfeh AH and Balachandran B, Applied Nonlinear Dynamics 1995 [eBook]. a-Strogatz SH, Nonlinear Dynamics and Chaos 1994/2024.
• selected journal papers.

This hybrid course (in class & online) will be given (and recorded) in English. The course grade will consist of two individual projects [midterm 50%, final 50%].

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Lectures