More on the Principle of Least Action

   It appears that the way the variation is done affects the minimum of the action integral. If one arranges the variation so that it is normal to tangent of the curve the number of terms contributing to the action integral is reduced to two.

Here's an exaggerated variation to make it more visible.

The variation in the segments contributing to the variation of the action integral show a fairly uniform variation with positive signs.

The resulting action integral now has a minimum when the magnitude of the variation is zero and we can rightfully refer to it as Least Action.

Formulas for v·dr

The formulas for v·dr modified to contain the variations are,

with t being the tangent unit vector to the trajectory gives,

The Principle of Least Action

 Maupertuis loosely introduced the Principle of Least Action in 1741. A few years later in 1744 in Methodus inveniendi Euler associated Action with an integral that was a minimum for the path a mass actually followed, he wrote ∫ds√v = min with √v corresponding to the speed of the mass. He wrote,

I tried to check if the integral was actually a minimum for the mass moving in a uniform gravitational field as in fig 26 (see translation of Methodus inveniendi in Wikisource),

but the minimum didn't correspond to the motion of the particle even when the integral was replaced by δ∫v·dr.

The numerical integration was fairly straightforward. We start by defining some constants.

The variation of δy, was assumed to be a simple some function with k cycles on the interval of x equal to (0,b) with b=200. 

p=dy/dx was determined by fitting the values of x, x², and y, which enabled the components of the velocity to be calculated.

We were then able to calculate δv•dr for each point of the trajectory of the mass. To numerically integrate this over each Δx a three point fit of δv•dr was evaluated and used.

An exaggerated variation of δy is shown in the following figure.

When the δ∫v·dr is computed for small values of the amplitude of the variation the minimum is slightly offset from the zero.