The diagrams on the page have a slider for active exploration. Moving the slider sweeps out a range of trial trajectories. As you change the trial trajectory the diagram shows how the graphs of the energies come out accordingly.
In this resource Hamilton's stationary action is introduced in a two-stage process.
First stage:
We have the Work-Energy theorem, which we can apply with equal validity in infinitisimal form. The true trajectory has the following obvious property: at every instant in time the rate of change of potential energy matches the rate of change of kinetic energy. Demanding this match as a condition we identify the true trajectory among the range of trial trajectories. That is, this initial stage is already variational approach, but it doesn't yet use the concept of action.
Second stage:
Demonstration of moving in a single step from the first stage to Hamilton's stationary action.
The demonstration is for the simplest case: a uniform force, hence a linear potential. The reasoning generalizes to all cases.
I have an educational resource for introduction to Hamilton's stationary action. The title is "Least action visualized".
http://www.cleonis.nl/physics/phys256/least_action.php
The diagrams on the page have a slider for active exploration. Moving the slider sweeps out a range of trial trajectories. As you change the trial trajectory the diagram shows how the graphs of the energies come out accordingly.
In this resource Hamilton's stationary action is introduced in a two-stage process.
First stage: We have the Work-Energy theorem, which we can apply with equal validity in infinitisimal form. The true trajectory has the following obvious property: at every instant in time the rate of change of potential energy matches the rate of change of kinetic energy. Demanding this match as a condition we identify the true trajectory among the range of trial trajectories. That is, this initial stage is already variational approach, but it doesn't yet use the concept of action.
Second stage: Demonstration of moving in a single step from the first stage to Hamilton's stationary action.
The demonstration is for the simplest case: a uniform force, hence a linear potential. The reasoning generalizes to all cases.