In the end of the lectures, the students
1) should have understood the fundamental laws of Mechanics and the rigorous mathematical framework that describes these laws and produces the new knowledge in the particular scientific field.
2) should be able to understand in details and build part of the theory based on the fundamental laws and by using mathematics
3) should have got advanced studies passing from the classical Newtonian approximation to the Lagrangian Mechanics and the modern Hamiltonian Mechanics.
4) should become familiar with new advanced methods for modeling and managing complicated mechanical systems, constructing the equations of motion and finding first integrals.
Course Content (Syllabus)
1. Newtonian mechanics: axioms, laws of dynamics and vector form of the differential equations of motion. Conservation laws.
2. Motion in intertial and non-inertial reference frames: non-inertial forces and equations of motion. Examples.
3. Coordinate systems: differential equation of motion in cartesian, spherical and cylindrical coordinates. Examples.
4. Dynamics: equilibria and their stability. Study of conservative 1 degree-of-freedom system, using the method of Potential. Phase diagrams.
5. Applications to 1 d.o.f systems: harmonic oscillator, pendulum, systems with friction, forced oscillations.
6. Central forces: conservation of angular momentum, effective potential and study of the equivalent 1 d.o.f system
7. Solutions of the equations of motion for basic central-force fields in Physics: gravity, Coulomb, Yukawa and the two-body problem.
8. Analytical mechanics: constraints and reaction forces – degrees of freedom. Classification of mechanical systems. Principle of virtual work.
9. The d'Alembert principle and Lagrange's equations: the Lagrangean function for conservative forces (scalar and vector potentials). Examples
10. Applications: finding equations of motion and conserved quantities (integrals of motion) with Lagrange's method.
11. The analytical method of Hamilton: The Hamiltonian function, canonical equations, phase space and integrals of motion. Applications.
12. The principle of least action: Hamilton's principle and axiomatic foundation of mechanics. Physical importance of the least-action principle and relation to other fields of Physics.
Additional bibliography for study
Goldstein H. Classical Mechanics, 2nd ed. Addison-Wesley, 1980
Sheck Fl. Mechanics, Springer, 1999
Γ.Καραχάλιος, Β. Λουκόπουλος. "Θεωρητική Μηχανική", 2014