Analytical mechanics
The principle of least action
Evaluate for trial trajectories and observe that the Newtonian path is a stationary-action path.
Equipment
- Interactive canvas showing trial and classical paths
- Sliders for path sag, particle mass, and gravitational acceleration
Procedure
Set the field
Choose a gravitational acceleration. The particle begins and ends at the same height during a fixed unit time.
Vary a trial path
Move the sag slider to select . The dashed curve is the trial path; the solid curve is the Newtonian solution.
Compare the actions
Read both action values and adjust the mass. The stationary path parameter is ; mass scales the action but does not move this path.
Simulation
Experiment history
In the 1740s, Pierre-Louis Moreau de Maupertuis argued that nature follows a principle of least action, although his formulation and physical interpretation were not the modern variational mechanics. The idea prompted debate over how a single principle might encompass optical and mechanical laws.
Around 1755, Leonhard Euler developed variational methods for mechanics, and Joseph-Louis Lagrange systematized analytical mechanics in his 1788 Mécanique analytique. The Euler–Lagrange equation made stationary action a precise route to equations of motion; in this lab a simple gravitational trajectory illustrates that later formulation.