The final exam will cover Chapters 1 through 9 of [HF]. Those Chapters contain some important concepts and ideas, as well as some not so important concepts, ideas and terminology. Here is a brief list of what might be the most important highlights from each chapter so that you know what to review particularly closely before the final.
1. LAGRANGIAN MECHANICS
Correct identification of the number of degrees of freedom in any mechanical
system. Configuration space.
Holonomic versus non-holonomic constraints. Definition of the
Lagrangian. Conservative forces. Euler-Lagrange equations. Definition of
the Hamiltonian and its relation to the energy. Conjugate momenta.
Ignorable coordinates.
2. VARIATIONAL CALCULUS AND ITS APPLICATION TO MECHANICS
Hamilton's principle. Derivation of the Euler-Lagrange equations from
Hamilton's principle. Difference between the variational problem and the
initial-value problem. Boundary conditions for Hamilton's principle.
Lagrange multipliers.
3. LINEAR OSCILLATORS
Static equilibrium. Stable versus unstable equilibrium from the Lagrangian
perspective. The harmonic oscillator: simple, and to a smaller degree of
importance also damped and forced. Green's
function method of solving linear problems.
4. ONE-DIMENSIONAL SYSTEMS: CENTRAL FORCES AND THE KEPLER PROBLEM
Solution of a generic system with one DoF. Phase portrait. Setting up
the Kepler problem. Reduction of the problem to one DoF by conservation
laws. Which of the three Kepler laws are true for any central force
potential, and which depend on the particular form of the Newton potential?
Why? Which of the three laws is exact and which is only approximate?
5. NOETHER'S THEOREM AND HAMILTONIAN MECHANICS
Relation between continuous symmetries and conservation laws in Lagrangian
mechanics (= Noether's theorem). Basic idea of Hamiltonian mechanics: physics
in phase space with coordinates and momenta treated
as independent variables.
Hamilton's equations, and their equivalence to the Euler-Lagrange equations
for regular Lagrangians. The condition of regularity. The Legendre
transform. Liouville theorem and the Poincare recurrence
theorem.
6. THEORETICAL MECHANICS: FROM CANONICAL TRANSFORMATIONS TO ACTION-ANGLE
VARIABLES
Canonical transformations, as symmetry transformations of the Hamiltonian
formulation of mechanics in phase space. Four types of generating functions.
Poisson brackets: their definition and their elementary properties. Notion
of a conjugate pair. The Hamilton-Jacobi equation, and its relation to
a particularly convenient canonical transformation. Various types of
solutions of the Hamilton-Jacobi equation. Separation of variables,
conservation laws. Action-angle variables. Adiabatic invariants.
Definition of an integrable system.
7. ROTATING COORDINATE SYSTEMS
Difference between inertial and non-inertial frames. Orthogonal
transformations. Angular velocity. Its relation to the orthogonal
matrix transforming between two frames. The concept of fictitious forces.
8. THE DYNAMICS OF RIGID BODIES
Definition of the rigid body. Its number of degrees of freedom. Split
between the translation and rotation parts of the kinetic energy. The
moment-of-inertia tensor. The angular momentum of the rigid body, its
relation to the angular velocity, Euler's equations. Conceptual understanding
of the definition of Euler angles.
9. THE THEORY OF SMALL VIBRATIONS
Static equilibrium for systems with N DoF. General Lagrangian near such a
point of static equilibrium. Normal modes. Identification of their
frequencies by solving the eigenvalue/eigenvector problem. Stability versus
instability of static equilibrium. Conceptual understanding of the
exceptional cases of two equal eigenvalues, or an eigenvale equal to zero.