The Useful Layer
Classical mechanics is often introduced through forces and trajectories, but the deeper reusable layer is constraint and variation. Virtual work, generalized coordinates, Lagrangians, Hamiltonians, and action principles give a way to express a system through the quantities that remain meaningful under change of coordinates.
That is why the same mathematical machinery reappears outside the original mechanical examples.
Where It Reappears
Euler-Lagrange equations and Hamilton's principle show up in quantum mechanics, field theory, optimal control, reinforcement learning, and variational inference because those areas also need to optimize a functional under constraints.
The names change from displacement and velocity to state, action, policy, probability density, phase, or field. The structural question remains: what quantity is being varied, what constraints are respected, and what stationary condition defines the behavior?
Source Context
The original post linked a classical-mechanics lecture sequence. It remains useful as context for virtual work and variational mechanics.