Least action visualized

The standard notation when presenting the principle of least action is to define a quantity S, called the 'Action'. Also, I will follow the convention of denoting kinetic energy with 'T' and potential energy with 'V'.

$\displaystyle S = \int (T - V) dt$

(1)

We have that when the Action is evaluated for a range of trajectories the value of S is least for the case of the true trajectory. The natural queston is why the quantity (T-V) has that property. My goal is to show - for classical mechanics - that the principle of least action is an implication of the work-energy theorem. If the work-energy theorem holds good then so does the principle of least action.

I will proceed by evaluating the time integrals of T and V separately. To represent that in the notation I introduce action components S_T and S_V.

$\displaystyle S_T = \int T dt$

(2)

$\displaystyle S_V = \int V dt$

(3)

I will present the principle of least action in terms of differences between S_T and S_V.

The setup

Picture 1. Diagram

An object shoots upward, released to free motion. The object climbs, decelerated by gravity, and falls back again.

To work with round numbers I have selected the following conditions:
- Total duration: 2 seconds (from t=-1 to t=1)
- Gravitational acceleration: 2 m/s²
- Mass of the object: 1 unit of mass.

Those conditions imply that along the true worldline the starting velocity is 2 m/s, and the object climbs to a height of 1 meter.

Diagram 1 gives the height (vertical axis) of the object as a function of time (horizontal axis). The graph representing the motion is of course a parabola. Expression (4) gives that parabola.

(4)

Class of candidate worldlines

Picture 2. Animation

Animation 2 shows how the worldline will be varied, generating a class of trial worldlines. The startpoint and endpoint are fixed, in between the wordline is varied. The set of all possible variations of the worldline is much larger than this particular class of course, but for now I will use this one.

With this way of varying the worldline I need just a factor p_v which stands for 'variational parameter'. Now each worldline is a function of two variables: 't' and 'p_v'

(5)

As you can see, when p_v is zero expression (5) simplifies to expression (4).

Kinetic energy

To prepare the ground for comparing kinetic and potential energy animation 3 shows a trivial case: how the evaluation comes out if there is no force (and therefore no potential energy). The line on the left represents the motion of the object, the graph on the right represents the kinetic energy at each point in time.

I placed the zero point of the kinetic energy halfway up the diagram. Where the zero point is placed is not important to the calculation, as the calculation searches for a minimum among the entire class of possible worldlines.

The grey line sweeps along a range of trial worldlines, the red line represents the corresponding kinetic energy of each trial worldline. The shaded area underneath the red line is a measure of the integral of kinetic energy over time. Trivially, for the straight wordline that area is the smallest.

Picture 3. Animation

Kinetic energy and potential energy

Animation 4 depicts the case where a downwards force is present. With a force there is a potential. On the left once more the motion of the object. The red line represents the kinetic energy along each respective trial worldline. The green line deals with the potential energy.

The graph of potential energy has been flipped down. I'll explain in a minute why I flipped that down, first some remarks about the zero point. I opted to take the level of zero height as zero point of the potential energy. That is an arbitrary choice, it just happens to be convenient in this particular case.

I flipped the graph of the potential energy down to enable comparison. As the object is moving and changing velocity kinetic energy is converted to potential energy and back again. As we know, in this process the total energy is conserved. That means that at every point in time the rate of change of kinetic energy matches the rate of change of potential energy. In other words, in the case of the true worldline the graphs of the kinetic energy and minus potential energy are parallel to each other at every point in time.

The animation pauses for a moment at the frame that represents the true worldline.

Picture 4. Animation

Expression (5) gives the worldlines as a function of time t and the variational parameter p_v.

(5)

The derivative of function (5) with respect to time gives the velocity.

(6)

Expression (7), for the kinetic energy, is obtained by entering the velocity (6) in the formula ¹/₂mv². Expression (8) is obtained by multiplying the height (5) with the gravitational acceleration (which has the value 2 in this example.)

(7)

(8)

Both kinetic and potential energy are functions of p_v and t here. The difference is that the kinetic energy is proportional to the square of p_v whereas the potential energy is a linear function of p_v. Because of that there is only a single worldline with the property that the graphs of the kinetic and minus potential energy are parallel at every point in time.

Integrating over time

The Action of the principle of least action is the time integral of (T-V), but for now I integrate T and V separately.

The following integrals are from t=-1 to t=1.

$\int\limits_{-1}^{1} T(t,p_v) \, dt = \frac{4}{3}(1+p_v)^2$

(9)

$\int\limits_{-1}^{1} V(t,p_v) \, dt = \frac{8}{3}(1+p_v)$

(10)

As announced earlier I use the notation S_T for the 'Kinetic energy component of the action', and S_V for the 'Potential energy component of the action'.

The purpose of integrating is to simplify the expressions. (9) and (10) are simpler so it will be easier to find the value of p_v that corresponds with the true worldline.

Infinitisimals

I want to take some time now to compare with using differential calculus to find a trajectory. Differential calculus breaks motion down to infinitisimally small steps. To obtain an exact solution to the equation you take the limit of infinitisimally small steps.

Now consider expressions (9) and (10), both are integration from t=-1 to t=1, in notation ∫_-1¹f(t)dt. What if you would cut that up, and compute two adjoining integrals, ∫_-1⁰, and ∫₀¹?

The following property is so important that I'm putting it in a box section.

When ∫_-1¹f(t)dt is minimal then both ∫_-1⁰f(t)dt and ∫₀¹f(t)dt are minimal.

The above statement is specific, but clearly it generalizes to all cases. The integration to find an action is over a certain timeframe, we can call that: from t₁ to t₂. You can subdivide that timeframe in subsections. The overall integration will be minimal if and only if the integrals of each of the subsections are minimal. And there is no limit to how far you can subdivide, so it extends to infinitisimally short subsections.

What this shows is that in the end the principle of least Action has in common with using differential calculus that it evaluates a summation of infinitisimally small steps. In the appendix Feynman I give a transcript of Feynman's comments on this matter.

Graphs

In animation 5 the left panel still give the sequence of trial worldlines. But the panel on the right now gives the magnitude of the Action (vertical axis) as a function of p_v (horizontal axis.)

The red line is the graph of S_T. The red dot gives the value of S_T that corresponds to the trial worldline on the left. The green line graphs the negative of S_V (just as in animation 4 the green line is the negative of the potential energy). The blue line graphs the action as a function of p_v.

Animation 5 shows why the action has a minimum. The red parabola deals only with the kinetic energy. Note that when p_v is so negative that the worldline is a straight line the kinetic energy component of the action is minimal.

The action, the blue parabola, is the red parabola combined with the green line. In a graph, when you add a line to a parabola you get another parabola, but with the minimum shifted sideways.

Picture 5. Animation

The minimum of the action

Expressions (7) and (8) are functions of the variational parameter p_v and time t. Having performed the integration over time expressions (9) and (10) are functions of just p_v.

The graph of the Action as a function of p_v is a parabola. As is well known, an effective way to find the minimum of any graph is to differentiate the function. The minimum of the parabola is the point where its derivative is zero.

Expression (11) gives the condition that is satisfied when the blue line, the graph of the action, is at its minimum.

$\cfrac{dS_T}{dp_v} - \cfrac{dS_V}{dp_v} = 0$

(11)

$\cfrac{8}{3}(1+p_v) - \cfrac{8}{3} = 0$

(12)

Expression (12) shows that the Action is minimal when p_v is zero. This completes the demonstration that the work-energy theorem implies the principle of least action.

Surface area

Returning once more to the animation with the graphs of kinetic energy and potential energy. In the animation the area of the shaded region is a measure of the action.

Picture 6. Animation

When the red curve and the green curve are parallel everywhere then the area of the shaded region is minimized.

Java applet

For doublechecking the expressions, and to generate the animations, I used a Java Applet. Download the Least action Demo Applet. (800KB, requires JRE 1.5 or higher)
(If you are an EJS user, download the Least action Demo EJS source)

Picture 7. Image
Downsized screenshot of the supporting Java applet.

The applet controls include a slider to vary the worldline. The acceleration is adjustable (Main purpose: to cover the case of zero acceleration.) The applet computes the curves numerically and analytically. The numerical computation is on the fly. As you move the slider the applet computes the action for the current position of the slider; the dot moves in accordance, and the applet traces the dots, so gradually curves build up.
What I hope is that you will download EJS, that you will download and open the applet source file, and that you will examine the expressions for yourself. That is the applet's purpose.

Any acceleration

In the discussion above I used an acceleration of 2 units. Here is a version with the acceleration unspecified; it is represented with the symbol 'a'. As before the duration is set from t=-1 to t=1.

The position as a function of time is given by (h is the height that the object reaches):

$f(t) = -\frac{1}{2}at^2 + h$

(13)

Given the duration the height follows from the acceleration: h=¹/₂a

$f(t) = \frac{1}{2}a(-t^2 + 1)$

(14)

The worldline is varied, generating a class of trial worldlines.

$f(t,p_v) = (1 + p_v)(\frac{1}{2}a(-t^2 + 1))$

(15)

The time derivative to get the velocity:

(16)

The kinetic energy and the potential energy:

$T(t,p_v) = \frac{1}{2}(1 + p_v)^2a^2t^2$

(17)

$V(t,p_v) = a(1 + p_v)(\frac{1}{2}a(-t^2 + 1))$

(18)

The time integrals of the kinetic energy and the potential energy respectively.

$\int\limits_{-1}^{1} T(t,p_v) \, dt = \frac{1}{3}a^2(1 + p_v)^2$

(19)

$\int\limits_{-1}^{1} V(t,p_v) \, dt = \frac{2}{3}a^2(1 + p_v)$

(20)

Using the notation S_T for the 'Kinetic energy component of the action', and S_V for the 'Potential energy component of the action':

$\cfrac{dS_T}{dp_v} - \cfrac{dS_V}{dp_v} = 0$

(21)

$\frac{2}{3}a^2(1+p_v) - \frac{2}{3}a^2 = 0$

(22)

Appendix

Feynman lectures on physics, chapter 19, page 8

There is quite a difference in the characteristic of law which says that a certain integral from one place to another is a minimum - which tells you something about the whole path - and of a law which says that as you go along, there is a force that makes it accelerate. The second way tells how you inch your way along the path, and the other is a grand statement about the whole path. In the case of light, we talked about the connection of these two. Now, I would like to explain why it is true that there are differential laws when there is a least action principle of this kind. The reason is the following: Consider the actual path in space and time. As before, let's take only one dimension, so we can plot the graph of x as a function of t. Along the true path, S is a minimum. Let's suppose that we have the true path and that it goes through some point a in space and time, and also through another nearby point b.

Now if the entire integral from t₁ to t₂ is a minimum, it is also necessary that the integral along the little section from a to b is a minimum. It can't be that the part from a to b is a little bit more. Otherwise you could fiddle with just that piece of path and make the whole integral a little lower.
So every subsection of the path must also be a minimum. And this is true no matter how short the subsection. Therefore, the principle that the whole path gives a minimum can be stated also by saying that an infinitisimal section of path also has a curve such that it has minimum action. Now if we take a short enough section of path - between two points a and b very close together - how the potential varies from one place to another far away is not the important thing, because you are staying almost in the same place over the whole little piece of the path. The only thing that you have to discuss is the first-order change in the potential. The answer can only depend on the derivative of the potential and not on the potential everywhere. So the statement about the gross property of the whole path becomes a statement of what happens for a short section of the path - a differential statement. And this differential statement only involves the derivatives of the potential, that is the force at a point. That's the qualitative explanation of the relation between the gross law and the differential law.

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