# Classical Mechanics

### PHYSICS761

Newtonian, Lagrangian, and Hamiltonian methods for classical systems; symmetry and conservation laws; rigid body motion; normal modes; nonlinear oscillations; canonical transformations; Lagrangian and Hamiltonian methods for continuous systems and fields. 3 units

#### Possible principal texts:

- H. Goldstein,
*Classical Mechanics*, 3^{rd}ed. (Addison-Wesley, 2001). - J. V. Jose and E. J. Saletan,
*Classical Dynamics – a Modern Approach*(Cambridge Univ. Press, 1998).

#### Other texts to consider:

- L. D. Landau and E. Lifschitz,
*Mechanics*(vol. 1). - V. I. Arnold,
*Mathematical Methods of Classical Mechanics*, (Springer, 1978). - A. Fetter and J. Walecka,
*Theoretical Mechanics of Particles and Continua*, (Dover, 2003); A. Fetter and J. Walecka,*Nonlinear Mechanics: A Supplement to Theoretical Mechanics of Particles and Continua,*(Dover, 2006).

#### Prerequisites

The prerequisite is at least one semester of an intermediate undergraduate classical mechanics course at the level of J.B Marion and S.T. Thornton's textbook.

In Duke Physics, there is an undergraduate "Intermediate Mechanics" course (PHYSICS 181), with the synopsis:

Newtonian mechanics at the intermediate level, Lagrangian mechanics, linear oscillations, chaos, dynamics of continuous media, motion in noninertial reference frames.

#### Syllabus

- Variational calculus.
- Generalized coordinates and constraints.
- Hamilton’s principle and the Lagrange equations of motion.
- Conservation theorems and symmetries.
- Central forces; scattering in a central force.
- Hamiltonian formulation and the Principle of Least Action.
- Canonical transformations and Poisson brackets.
- Hamilton-Jacobi method; action-angle variables.
- Adiabatic invariance; Liouville’s theorem.
- Kinematics of rigid body motion – tensor notation.
- Rotation matrices, rotating frames, Coriolis force.
- Rigid body dynamics
- Linear oscillations - normal modes, symmetry groups.
- Nonlinear oscillations.
- Secular perturbation theory.
- Dissipative nonlinear systems and chaos.
- Transition from discrete to continuous systems; the Lagrangian density.
- Lagrangian formulation for continuum systems; waves on a one-dimensional string, sound waves in three dimensions.
- Hamiltonian formulation for continuum systems.
- Stress-energy tensor and conservation theorems.

*Sample* lecture schedule

(based on 25 lectures each of duration 75 minutes).

*Lecture 1:*__Review__of basic Newtonian mechanics.*Lecture 2:*Variational calculus.*Lecture 3:*Generalized coordinates and constraints.*Lecture 4:*Hamilton’s principle and the Lagrange equations of motion.*Lecture 5:*Conservation theorems and symmetries.*Lecture 6:*Central forces; scattering in a central force.*Lecture 7:*Hamiltonian formulation and the Principle of Least Action.*Lecture 8:*Canonical transformations and Poisson brackets.*Lecture 9:*Hamilton-Jacobi method; action-angle variables.*Lecture 10:*Adiabatic invariance; Liouville’s theorem.*Lecture 11:*Kinematics of rigid body motion – tensor notation.*Lecture 12:*Rotation matrices, rotating frames, Coriolis force.*Lecture 13:*Rigid body dynamics 1.*Lecture 14:*Rigid body dynamics 2.*Lecture 15:*Linear oscillations 1 – normal modes.*Lecture 16:*Linear oscillations 2 – the role of symmetries and symmetry groups.*Lecture 17:*Nonlinear oscillations.*Lecture 18:*Secular perturbation theory.*Lecture 19:*Dissipative nonlinear systems and chaos.*Lecture 20:*Transition from discrete to continuous systems; the Lagrangian density.*Lecture 21:*Lagrangian formulation for continuum systems; waves on a one-dimensional string, sound waves in three dimensions.*Lecture 22:*Hamiltonian formulation for continuum systems.*Lecture 23:*Stress-energy tensor and conservation theorems.*Lecture 24:*Special topics.*Lecture 25:*Special topics