Part I: Lagrangian Mechanics
The reformulation of Newtonian mechanics that replaces forces with energies, unlocking the power of symmetry, generalized coordinates, and variational principles.
Why Lagrangian Mechanics?
Newton's laws work beautifully for simple systems, but they become unwieldy when constraints are present. A bead sliding on a rotating wire, a double pendulum, or a charged particle in an electromagnetic field — all of these are far more naturally described by the Lagrangian approach. Instead of tracking vector forces and constraint reactions, we write down a single scalar function \(L = T - V\) and let the mathematics do the rest.
Lagrangian mechanics is not merely a computational shortcut. It reveals the deep connection between symmetries and conservation laws (Noether's theorem), provides the foundation for quantum field theory via the path integral, and generalizes seamlessly to fields, relativity, and gauge theories.
Topics in Part I
Principle of Least Action
The single most powerful principle in physics: all of classical mechanics follows from extremizing the action \(S = \int L\,dt\). Derive the Euler-Lagrange equations from\(\delta S = 0\), understand Hamilton's principle, and see how Fermat's principle for light is a special case.
Euler-Lagrange Equations
Master the workhorse equations of analytical mechanics. Learn to choose generalized coordinates, handle holonomic and nonholonomic constraints, and apply Lagrange multipliers. Solve classic problems: the simple pendulum, bead on a wire, double pendulum, and the Atwood machine.
Symmetries & Conservation Laws
Noether's theorem is one of the most beautiful results in mathematical physics: every continuous symmetry of the action yields a conserved quantity. Derive the theorem in full, then see how time translation gives energy conservation, spatial translation gives momentum, and rotational symmetry gives angular momentum.
Key Equations of Lagrangian Mechanics
The Lagrangian
\[L(q, \dot{q}, t) = T - V\]
The Action
\[S[q] = \int_{t_1}^{t_2} L(q, \dot{q}, t)\,dt\]
Euler-Lagrange Equation
\[\frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} = 0\]
Noether's Conserved Charge
\[Q = \sum_i \frac{\partial L}{\partial \dot{q}_i}\delta q_i - L\,\delta t\]