Drag the red control point to deform the trajectory x(t) while keeping the endpoints fixed.
The action S=\int L\,dt is recomputed in real time for a particle moving in one dimension
in a uniform gravitational field, with
L=\tfrac12 mv^2-mgx.
Click any point on the blue trajectory to display the local value of the Lagrangian.
Current trajectory
Stationary trajectory
Draggable control point
Fixed endpoints
Selected point
This demo uses a one-parameter family of smooth trial curves:
each trajectory is a quadratic curve passing through the two fixed endpoints and the draggable midpoint.
That is enough to illustrate the stationary-action idea very clearly in this simple case.
System
Lagrangian:L(x,v)=\tfrac12 mv^2-mgx
Potential:V(x)=mgx
Equation of motion:\ddot x=-g
Parameters
m = 1
g = 0.8
t_i = 0, x_i = 3.5
t_f = 4, x_f = 1.0
Action
S = ...
ΔS = ...
Control point: ...
Selected point on the blue curve
Click on the current trajectory to inspect a point.
Controls
The green dashed curve is the exact stationary trajectory for the fixed endpoints.
In this example it also gives the minimum action among the displayed trial curves.