Code
Φ-501
Level
Graduate
Category
B
Teacher
G. C. Psaltakis
ECTS
6
Hours
5
Semester
Spring
Display
Yes
Offered
No
Teacher Webpage
Goal of the course
The present course, for graduate and advanced undergraduate students, provides the extension of an introductory course on Newtonian classical mechanics. Its main subject matter includes Lagrange's equations, Hamilton's principle, Hamilton's equations, canonical transformations, and the Hamilton-Jacobi theory. The course emphasizes analytic techniques and includes also useful applications to the kinematics and dynamics of rigid bodies.
Program
Tuesday, 11:00-13:00, 1st floor semminar room
Thursday, 11:00-13:00, 1st floor semminar room
Thursday, 11:00-13:00, 1st floor semminar room
Syllabus
Survey of the elementary principles:
Mechanics of a particle. Mechanics of a system of particles. Constraints.Actual displacements and virtual displacements. D' Alembert's principleand Lagrange's equations. Velocity-dependent potentials and the dissipationfunction. Simple applications of the Lagrangian formulation.
Variational Principles and Lagrange's equations:
Calculus of variations, Euler equations. Hamilton's principle (least action principle). Derivation of Lagrange's equations from Hamilton's principle.Extension of Hamilton's principle to systems with additional holonomic constraints. Conservation theorems and symmetry properties, Noether's theorem.Energy function and the conservation of energy.
Kinematics of rigid body:
The independent coordinates of a rigid body. Translations and rotations.Space axes and body axes. The rotation matrix R and the angular velocity vectorω. Rate of change of a vector. Euler's theorem. The Euler angles ($(phi, theta,psi)$).Expressions of R in terms of Euler angles.
Dynamics of rigid body:
Kinetic energy and angular momentum of motion about a point. The inertia tensor. Principal axes and eigenvalues of the inertia tensor.The Euler equations of motion, solution of rigid body problems. Stabilityof rotation about a principal axis. Motion of the heavy symmetric top: spinning,precession, nutation.
The Hamilton equations:
Legendre transformations and the Hamilton equations of motion. Conservation theorems and ignorable coordinates. Derivation of Hamilton's equations from a variational principle: the modified Hamilton's principle (least action principle). Poisson brackets. Equations of motion and conservation theorems in the Poisson bracket formulation.
Canonical transformations:
The equations of canonical transformation. Examples of canonical transformations.The harmonic oscillator. Canonical transformations and Poisson brackets.Infinitesimal canonical transformations.
Hamilton-Jacobi theory and action-angle variables:
The Hamilton-Jacobi equation for Hamilton's principal function. Applicationto the harmonic oscillator. The Hamilton-Jacobi equation for Hamilton's characteristic function. Separation of variables in the Hamilton-Jacobi equation. Action-angle variables in systems of one degree of freedom.Action-angle variables for completely separable systems.
Mechanics of a particle. Mechanics of a system of particles. Constraints.Actual displacements and virtual displacements. D' Alembert's principleand Lagrange's equations. Velocity-dependent potentials and the dissipationfunction. Simple applications of the Lagrangian formulation.
Variational Principles and Lagrange's equations:
Calculus of variations, Euler equations. Hamilton's principle (least action principle). Derivation of Lagrange's equations from Hamilton's principle.Extension of Hamilton's principle to systems with additional holonomic constraints. Conservation theorems and symmetry properties, Noether's theorem.Energy function and the conservation of energy.
Kinematics of rigid body:
The independent coordinates of a rigid body. Translations and rotations.Space axes and body axes. The rotation matrix R and the angular velocity vectorω. Rate of change of a vector. Euler's theorem. The Euler angles ($(phi, theta,psi)$).Expressions of R in terms of Euler angles.
Dynamics of rigid body:
Kinetic energy and angular momentum of motion about a point. The inertia tensor. Principal axes and eigenvalues of the inertia tensor.The Euler equations of motion, solution of rigid body problems. Stabilityof rotation about a principal axis. Motion of the heavy symmetric top: spinning,precession, nutation.
The Hamilton equations:
Legendre transformations and the Hamilton equations of motion. Conservation theorems and ignorable coordinates. Derivation of Hamilton's equations from a variational principle: the modified Hamilton's principle (least action principle). Poisson brackets. Equations of motion and conservation theorems in the Poisson bracket formulation.
Canonical transformations:
The equations of canonical transformation. Examples of canonical transformations.The harmonic oscillator. Canonical transformations and Poisson brackets.Infinitesimal canonical transformations.
Hamilton-Jacobi theory and action-angle variables:
The Hamilton-Jacobi equation for Hamilton's principal function. Applicationto the harmonic oscillator. The Hamilton-Jacobi equation for Hamilton's characteristic function. Separation of variables in the Hamilton-Jacobi equation. Action-angle variables in systems of one degree of freedom.Action-angle variables for completely separable systems.
Bibliography
1. "Classical Mechanics" -- H. Goldstein, C. Poole, and J. Safko (3rd edition, Addison Wesley, San Francisco, 2002).
2. "Classical Dynamics: A Contemporary Approach" -- J. V. Jose and E. J. Saletan (Cambridge University Press, Cambridge, 1998).
3. "Analytical Mechanics" -- L. N. Hand and J. D. Finch (Cambridge University Press, Cambridge, 1998).
4. "Theoretical Mechanics, Vol. B" -- J. D. Chatzidimitriou, (3rd edition, Giahoudis Press, Thessaloniki, 2000).
2. "Classical Dynamics: A Contemporary Approach" -- J. V. Jose and E. J. Saletan (Cambridge University Press, Cambridge, 1998).
3. "Analytical Mechanics" -- L. N. Hand and J. D. Finch (Cambridge University Press, Cambridge, 1998).
4. "Theoretical Mechanics, Vol. B" -- J. D. Chatzidimitriou, (3rd edition, Giahoudis Press, Thessaloniki, 2000).
