This set of three graphlets represents the physics and mathematics of Hamilton's stationary action.

Picture 1. Graphlet
Potential energy increases linear with height; the trajectory is a parabola

Graphlet 1 is for the case of a uniform force, causing an acceleration of 2 m/s2.
As we know: with a uniform force the curve that represents the height as a function of time is a parabola.

In the 'energy' sub-panel the green curve represents the minus potential energy. In effect the potential energy curve has been flipped upside down. That way you can see directly that when the trial trajectory coincides with the true trajectory the red and green curve are parallel to each other everywhere.

(Any form of calculating the true trajectory uses the derivative of the potential energy. That is, in calculation the value of the potential energy itself is not used. Because of that the choice of zero point of potential energy is arbitrary.)

In the lower right subpanel:
The blue dot represents Hamilton's action.
The label of the horizontal axis is pv, which stands for 'variational parameter'. The positioning of the dots corresponds to the value of the main slider.

In the case of linear potential: when the trial trajectory coincides with the true trajectory the value of Hamilton's action is minimal/minimized. In the case implemented in this graphlet: any change of the trial trajectory results in raising the value of Hamilton's action.


Picture 2. Graphlet
Potential energy increases quadratic with height

Graphlet 2 is for the case where the force increases in linear proportion to the displacement. This case is commonly referred to as Hooke's law.
As we know: With Hooke's law the resulting motion is harmonic oscillation and the curve that represents the displacement as a function of time is the sine function.

With Hooke's law the potential energy increases with the square of the displacement. That is: with Hooke's law we have that the rate of change of potential energy is on par with that of the kinetic energy: with Hooke's law the expressions for the energies are both quadratic expressions.

As we know: with Hooke's law the amplitude and period of the resulting oscillation are independent from each other. Hamilton's stationary action corroborates that: when the trial trajectory is the harmonic oscillation function then if you change the amplitude the value of the action remains the same.

(In the diagram the actual curve is the cosine function; the point is that it is the harmonic oscillation function.)


Picture 3. Graphlet
Potential energy proportional to cube of height.

Graphlet 3 is for the case where the force increases proportional to the square of the displacement, hence the potential energy increases with the cube.



I recommend enlarging the graphlet to the full width of the browser window. Visibility of the navigation column of this page can be toggled. When the navigation column has been hidden: use the button 'Larger' and zoom in on the page as a whole to make the graphlet fill the entire width of the browser window.

Hamilton's stationary action

Hamilton's stationary action asserts that when the trial trajectory coincides with the true trajectory the derivative of Hamilton's action is zero. The value of Hamilton's action is immaterial. The following cannot be emphasized enough: what is evaluated is the derivative of Hamilton's action.

The name 'least action' arose because there are multiple classes of cases such that when the trial trajectory coincides with the true trajectory Hamilton's action is at a minimum. But there are also classes of cases such that when the trial trajectory coincides with the true trajectory Hamilton's action is at a maximum. It can be either way; it's not relevant. The true property is that the when the trial trajectory coincides with the true trajectory the derivative of Hamilton's action is zero.

Evaluation of slope

Let's say you have two ascending curves, and you know there is a single point where the two curves have the same slope. One method to find that equal-slope-point goes as follows:
- subtract one curve from the other, call this resultant curve the 'action'
- take the derivative of the 'action'
- the point where the derivative of the 'action' is zero is the point where the two curves have the same slope.

In the case of Hamilton's action: when the rate of change of kinetic energy matches the rate of change of potential energy the derivative of the action is zero.

The three graphlets demonstrate that Hamilton's stationary action will always hold good. The three graphlets as a set make it transparent why Hamilton's stationary action holds good.


How to operate the demonstration:

The main slider, located at the bottom of the graphlet, executes a global variation sweep. The value displayed in the "knob" of the main slider is a value to implement the variation, from here on I will refer to it as the 'variational parameter pv' In the 'integral' panel the variational parameter pv is along the horizontal axis.

Names for the 4 sub-panels:
- upper left: Control panel
- lower left: Height panel
- upper right: Energies panel
- lower right: Integrals panel

Note: The 3 sub-panels with a coordinate system are named after their vertical axis name: Height, Energy, Integral.

In the starting configuration the trajectory points (height panel) have been placed such that they coincide (to a very good approximation) with the true trajectory of the object.

When the object is moving along the true trajectory the rate of change of kinetic energy matches the rate of change of potential energy (work-energy theorem). This match of rate of change has been emphasized by turning the graph of the potential energy (green) upside down. When the object is moving along the true trajectory the red and green curve have the same slope along the entire trajectory.

The energies panel is where it happens. The energies panel represents how the equation that you are using solves for the true trajectory.

Hamilton's Action

S = \int (E_k - E_p) dt

Hamilton's Action is represented by the blue dot in the Integrals panel. Use the main slider to sweep out variation. The blue dot follows a curve; it is a curve in variation space. The slope of that curve represents the derivative of Hamilton's Action with respect to variation.

At the point where over the entire trajectory the derivative of the kinetic energy is equal to the derivative of the minus potential energy the derivative of the blue dot is zero. That is: the motion of the blue dot is horizontal when the variation coincides with the true trajectory.


In the control panel the two sliders on the far left are adjusters for the trajectory. The upper adjuster morphs the trajectory towards a curve that is more blunt than the true trajectory; the lower adjuster morphs the trajectory towards a triangle shape.

These adjusters allow morphing of the trial trajectory while maintaining that during the entire ascent the velocity never reverses from decreasing to increasing again, which would be unphysical.

The row of ten sliders is for adjustment of individual nodes. The three radio buttons toggle between three sets of node adjusting sliders.
The '× 1' button: for a ratio of 1-to-1 of moving the slider and movement of the corresponding node of the trajectory.
The '× 0.1' button: ratio of 10 to 1
The '× 0.01' button: ratio of 100 to 1, for fine adjustment.

In the energies panel: the grey point is draggable, and it drags the entire curve of the kinetic energy (red) with it. By shifting the kinetic energy curve over to the potential energy curve the user can verify that the red and green curve are parallel to each other along the entire trajectory. (The evaluation of the integral uses the unshifted position of the kinetic energy curve.)

In the integrals panel:
· Red/Green point: value of the integral of the red/green curve of the energies panel
· Blue point: the sum of the values of the red point and the green point

As you use the main slider to execute global variation sweep:
In graphlet 1 the action reaches a minimum when the trial trajectory coincides with the true trajectory. The change of potential energy is linear, and the change of kinetic energy is quadratic, hence the action reaches a minimum.
In graphlet 3 the action reaches a maximum when the trial trajectory coincides with the true trajectory. In graphlet 3 change of potential energy goes with the 3rd power (cubic), and the change of kinetic energy is quadratic, hence the action reaches a maximum.


The mathematics of stationary action

As you try out various configurations, using the sliders: when in the energies panel the two curves line up (parallel to each other the entire time), then in the integrals panel the motion of the blue dot is horizontal. Here we examine how that comes about.

Hamilton's stationary action uses kinetic/potential energy to express the physics taking place. Therefore this examination starts with the Work-Energy theorem:

\int_{s_0}^s F \ ds = \tfrac{1}{2}mv^2 - \tfrac{1}{2}mv_0^2
(1.1)

The work-energy theorem is only one step away from F=ma; the Work-Energy theorem is derived from F=ma as follows: for both the left hand side and the right hand side of F=ma the integral over distance is evaluated. To recover F=ma we take derivative with respect to position.

Simplify to initial position coordinate zero and initial velocity zero.

\int F \ ds = \tfrac{1}{2}mv^2
(1.2)

the derivative with respect to position:

\frac{\int F \ ds}{ds} = \frac{\tfrac{1}{2}mv^2}{ds} \qquad \Longleftrightarrow \qquad F = ma
(1.3)

From here on I will refer to the negative of the integral of force over distance as the 'potential energy'.

\Delta (E_k) = - \Delta (E_p)
(1.4)

It is essential to be aware that (1.3) comes with a fundamental restriction. The Work-Energy theorem is defined only when there is a well defined integral over distance, and that is not always the case.

A form of force with the property that the integral over distance is well defined is referred to as a conservative force. The Work-Energy theorem is applicable when the force that is involved is a conservative force.

By contrast, the principle of conservation of energy comes without any restriction, there is no limit to the scope of the principle of conservation of energy.

Here (1.4) is stated as a corollary of the Work-Energy theorem, and so it must be treated as subject to the same limitation of scope: conservative forces only.


For the subsequent manipulations it is practical to move the minus sign inside the parentheses.

\Delta (E_k) = \Delta (-E_p)
(1.5)

The validity of (1.5) extends down to infinitisimal change:

d(E_k) = d(-E_p)
(1.6)

In the energies panel the horizontal axis is the time axis; the slope of the trajectory is the time derivative.

\frac{d(E_k)}{dt} = \frac{d(-E_p)}{dt}
(1.7)

Here is the key to Hamilton's stationary action:
When the time derivative condition is satisfied the expression for the position derivative is satisfied too:

\frac{d(E_k)}{ds} = \frac{d(-E_p)}{ds}
(1.8)

From here on I will refer to (1.8) as the 'Energy-Position equation', as it evaluates the derivative with respect to position.


The variation

The global variation is implemented in the following way:

With pv as variational parameter:

    (global variation sweep) = (1 + pv) * (current trial trajectory)

The variation is linear, and with linear variation the following two are identical:
- derivative with respect to variation
- derivative with respect to position


We have the following property of integration: it is a linear operation:

\int a \ x^2 \ dx = a \int x^2 \ dx

The relation that is expressed by the Energy-Position equation, (1.8), has both a local effect and a global effect. Locally the relation solves for the trajectory at the infinitisimal level. Globally it affects the slope of the curve in the energies panel. The rate of change of that slope propagates to the rate of change of the integral.

Hence for variation between two fixed points:
If, from start point to end point:

\frac{d(E_k)}{ds} = \frac{d(-E_p)}{ds}
(1.9)

Then, from start point to end point:

\frac{d(\int E_k dt)}{ds} = \frac{d(\int -E_p dt)}{ds}
(1.10)

Again: with linear variation the following two are identical:
- derivative with respect to variation
- derivative with respect to position

It follows from (1.10) that if the work-energy theorem holds good then Hamilton's stationary action holds good.

Jacob's Lemma

When Johann Bernoulli had presented the Brachistochrone problem to the mathematicians of the time Jacob Bernoulli was among the few who was able to find the solution independently. The treatment by Jacob Bernoulli is in the Acta Eruditorum, May 1697, pp. 211-217

Jacob opens his treatment with an observation concerning the fact that the curve that is sought is a minimum curve.

Picture 2. Image
Jacob's Lemma

Lemma. Let ACEDB be the desired curve along which a heavy point falls from A to B in the shortest time, and let C and D be two points on it as close together as we like. Then the segment of arc CED is among all segments of arc with C and D as end points the segment that a heavy point falling from A traverses in the shortest time. Indeed, if another segment of arc CFD were traversed in a shorter time, then the point would move along ACFDB in a shorter time than along ACEDB, which is contrary to our supposition.


Jacob's lemma generalizes to all cases where the curve that you want to find is an extremum; either a maximum or a minimum. If the evaluation is an extremum for the entire curve, then it is also an extremum for any sub-section of the curve, down to infinitisimally short subsections.


Jacob's lemma is the basis of deriving the Euler-Lagrange equation.



Derivation of the Work-Energy theorem

The steps in the following derivation are all towards the end goal of obtaining an expression in terms of the velocity v.

Up to (3.7) the expressions follow from the following two definitions:

ds = v \ dt
(2.1)

dv = \frac{dv}{dt}dt
(2.2)

The integral for acceleration from a starting point s0 to a final point s

\int_{s_0}^s a \ ds
(2.3)

Use (2.1) to change the differential from ds to dt. Since the differential is changed the limits change accordingly

\int_{t_0}^t a \ v \ dt
(2.4)

Rearrange the order, and write the acceleration a as dv/dt:

\int_{t_0}^t v \ \frac{dv}{dt} \ dt
(2.5)

Use (2.2) for a second change of differential, the limits change accordingly.

\int_{v_0}^v v \ dv
(2.6)

Putting everything together:

\int_{s_0}^s a \ ds = \tfrac{1}{2}v^2 - \tfrac{1}{2}v_0^2
(2.7)

Up until this point no physics has been introduced. (2.7) follows from the definitions (2.1) and (2.2)

The final step is to obtain a dynamics statement by combining (2.7) and F = ma

\int_{s_0}^s F \ ds = \tfrac{1}{2}mv^2 - \tfrac{1}{2}mv_0^2
(2.8)




The interactive animation on this page is created with the Javascript Library JSXGraph. JSXGraph is developed at the Lehrstuhl für Mathematik und ihre Didaktik, University of Bayreuth, Germany.





Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 3.0 Unported License.

Last time this page was modified: June 20 2021