Applications of Analytical Mechanics

Analytical solutions for double pendulums, central force problems, and coupled systems.

General idea of the topic

The Lagrange and Hamilton formalisms are justified by what they allow us to solve. Applied to a harmonic oscillator, a pendulum and systems with constraints, they reproduce Newton's results in fewer steps and reveal more structure, such as natural frequencies or conserved quantities.

The topic develops the three cases: the harmonic oscillator, where the Lagrangian is quadratic; the pendulum, with a single angular coordinate that removes the tension; and problems with constraints, in which generalized coordinates sidestep the constraint forces.

What you will truly understand

By the end you will be able to set up and solve with the analytical method an oscillator, a pendulum and a system with a simple constraint, and compare the result with the one obtained by forces.

You will also be able to identify which kinds of problems gain most from this method.

Recommended learning route

1. Harmonic Oscillator: Lagrangian and Hamiltonian of the oscillator, frequency and phase-space trajectories.
2. Pendulum: equation of motion of the pendulum with one coordinate and the small-oscillation limit.
3. Small Oscillations and Normal Modes: linearization about equilibrium, natural frequencies and normal modes of coupled systems.
4. Problems with Constraints: choice of coordinates that remove the constraint and obtaining the equations without knowing the reaction forces.

Physical criterion you must acquire

The essential criterion is always to check with a known case: the oscillator must give ω = √(k/m) and the pendulum ω = √(g/l) for small angles. Choosing the coordinate well is half the work, because a good choice removes the constraint from the start.

Connection with real problems

• clocks and mechanical oscillators.
• cranes and articulated arms.
• vibrations in structures.
• robot dynamics.

Final goal of the topic

The final goal is to apply analytical mechanics to concrete systems, verify the results against the vector method and judge when it is worth using.