Classical Mechanics | Physics - Wyatt's Notes

sources:

• text: Halliday, Resnick, Walker - Fundamentals of Physics

graph TD
    A[Newton's Laws] --> B[Force Analysis]
    B --> C[Motion Equations]
    C --> D[Energy Methods]

1. Newtonian Mechanics Review
2. Generalised Coordinates and Constraints
3. Lagrangian Mechanics
4. Hamiltonian Mechanics
5. Noether”s Theorem and Conservation Laws
6. Central Force Problems
7. Small Oscillations and Normal Modes
8. Rigid Body Dynamics
9. Problem Set
10. Rigid Body Dynamics: Advanced Topics
11. Hamiltonian Mechanics: Advanced Topics
12. Nonlinear Dynamics and Chaos
13. Classical Field Theory

University-level classical mechanics notes covering Lagrangian, Hamiltonian, and advanced topics.

• Lagrangian Mechanics: Generalised coordinates, Euler-Lagrange equations, constraints. Lagrangian mechanics reformulates Newton’s laws using energy rather than forces, making constrained systems tractable.
• Hamiltonian Mechanics: Phase space, canonical transformations, Hamilton-Jacobi theory. Hamiltonian mechanics provides the foundation for quantum mechanics and statistical mechanics.
• Central Force Problems: Kepler orbits, scattering, potential theory. Central forces (gravity, electrostatics) lead to conservation of angular momentum and closed orbits.
• Rigid Body Dynamics: Rotations, inertia tensor, Euler equations. Rotational motion requires tensor mathematics — the inertia tensor relates angular velocity to angular momentum.

• Newtonian mechanics (forces, energy, momentum)
• Multivariable calculus (partial derivatives, integrals)
• Linear algebra (vectors, matrices)
• Basic differential equations

Start with Lagrangian mechanics to build foundational knowledge, then progress to Hamiltonian mechanics and advanced topics. Each section includes worked examples and practice problems.

Use the sidebar to browse topics, or start with the introductory pages linked from the sidebar.

Each section includes:

• Detailed explanations of key concepts
• Worked examples with step-by-step solutions
• Practice problems with answers
• Common pitfalls and how to avoid them
• Connections to other areas of physics

1. Master Lagrangian mechanics: Understand the principle of least action and generalised coordinates. The Lagrangian formulation is more powerful than Newtonian mechanics for constrained systems.
2. Practise problems: Work through many problems to build intuition. Classical mechanics requires careful application of conservation laws and coordinate transformations.
3. Draw diagrams: Visualise coordinate systems and phase space. Phase portraits reveal the qualitative behaviour of dynamical systems.
4. Learn symmetries: Understand Noether’s theorem and conservation laws. Every continuous symmetry corresponds to a conserved quantity — this insight extends to quantum mechanics and field theory.
5. Connect to advanced physics: Relate classical mechanics to quantum mechanics and field theory. The Hamiltonian formulation is the starting point for quantum mechanics; Lagrangian field theory underpins the Standard Model.

• Thermal Physics: Statistical mechanics and thermodynamics; kinetic theory connects classical mechanics to temperature.

• Electromagnetism: Electromagnetic forces and fields; charged particle motion in electromagnetic fields.

• Quantum Mechanics: Classical limit of quantum theory; the correspondence principle connects classical and quantum mechanics.

• Mathematics: Multivariable calculus, linear algebra, and differential equations are the mathematical tools of classical mechanics.

• Calculus

• Linear Algebra

• Vector Calculus

• Quantum Computing

• Confusing the Lagrangian with the Hamiltonian: The Lagrangian is $L = T - V$ (kinetic minus potential); the Hamiltonian is $H = T + V$ (kinetic plus potential, for conservative systems). Using the wrong sign gives incorrect equations of motion.
• Forgetting that Noether’s theorem requires continuous symmetries: Discrete symmetries (like parity) do not produce conserved quantities via Noether’s theorem. Only continuous symmetries (translations, rotations, time shifts) yield conserved momenta and energy.
• Applying $F = ma$ in non-inertial frames without fictitious forces: Newton’s second law holds in inertial frames. In rotating or accelerating frames, you must add Coriolis and centrifugal forces, or use the Lagrangian formulation which handles generalised coordinates automatically.
• Assuming energy conservation applies to non-conservative systems: Energy is conserved only when all forces are conservative or when you account for work done by non-conservative forces (friction, drag). Friction converts mechanical energy to heat, so mechanical energy decreases.

Classical mechanics is the physics of how things move — from cannonballs to planets to spacecraft — and its power lies in its ability to predict the future given the present. Newton’s formulation (F = ma) tells you that forces cause acceleration, and that’s the whole story for most practical purposes. But Lagrangian and Hamiltonian mechanics reveal something deeper: the laws of physics can be derived from a single principle — that nature follows the path of least action. The Lagrangian (kinetic minus potential energy) encodes everything about a system’s dynamics, and the Euler-Lagrange equations extract the equations of motion from it. This reformulation isn’t just mathematically elegant; it’s practically indispensable for systems with constraints (pendulums on moving supports, molecules with fixed bond lengths, robotic arms with joints).

The Hamiltonian reformulation takes this further by recasting mechanics in terms of positions and momenta rather than positions and velocities. Phase space — the space of all possible positions and momenta — becomes the natural arena for dynamics. Every point in phase space represents a complete state of the system, and the Hamiltonian equations describe how the system flows through phase space over time. This viewpoint is the direct ancestor of quantum mechanics, where the Hamiltonian operator determines time evolution, and statistical mechanics, where the structure of phase space determines thermodynamic properties.

Noether’s theorem is perhaps the deepest insight in classical mechanics: every continuous symmetry of a physical system corresponds to a conserved quantity. Translational symmetry gives conservation of momentum. Rotational symmetry gives conservation of angular momentum. Time-translation symmetry gives conservation of energy. This isn’t just a mathematical curiosity — it’s the organising principle of all physics. When you understand why angular momentum is conserved, you understand why gyroscopes work, why planets orbit in ellipses, and why quantum spin is quantised. Classical mechanics isn’t just the starting point for physics; it’s the framework that all subsequent theories must reduce to in the appropriate limit.