Fundamental concepts and characteristics of modern stability definitions. Sensitivity and variational equations; linear variational equations; phase space analysis; Liapunov’s direct method. Autonomous and non-autonomous systems; stability in first approximation; the effect of force type on stability; frequency method.

**Lectures Outline:**

- Review of fundamentals of Analytical Mechanics: Hamilton=s principle, Lagrange=s equations of motion, Hamilton=s canonical equations, motion in the phase space.
- Fundamentals and common characteristics of stability definitions: equilibrium and asymptotic stability; stability in the large; conditional stability; equilibrium and equations of perturbed motion.
- The direct Liapunov method for autonomous systems: Sylvester=s criterion; Liapunov functions; Liapunov=s theorem of stability; asymptotic stability; theorems of instability; methods to obtain Liapunov functions; applications.
- Equilibrium states and stationary motions of conservative systems: Lagrange=s Theorem and its invertibility; cyclic coordinates; the Routh transform; stability of stationary motion; applications.
- Stability in first approximation: general formulation of the problem; theorems of stability in first approximation; Hurwitz=s criterion; applications
*.* - Linear autonomous systems: matrices and matrix operations; elementary divisors; stability of autonomous linear systems; stability of resonance.
- Direct Liapunov method and stability of control systems: governing differential equations of perturbed motion of automatic control systems; canonical equations of perturbed motion; Liapunov functions; absolute stability.
- The frequency method of stability analysis: transfer functions and frequency characteristics; Nyquist stability criterion; nonlinear systems; applications.

**Suggested References:**** **(on reserve at Carleton University Library)

- Methods of Analytical Dynamics, by: Leonard Meirovitch, McGraw-Hill 1970
- Stability Theory, by: H.H.E. Leipholz, Academic Press, 1970
- Introduction to Dynamics and Control, by: Leonard Meirovitch, John Wiley and Sons,1985
- Matrix Methods in Stability Theory, by: S. Barnett, C. Storey, Thomas Nelson & Sons Ltd., 1970.
- Introduction to Perturbation Techniques, by: A.H. Nayfeh, John Wiley & Sons, 1981