The Energy-Position equation
The intention of this article is to let the interactive diagrams tell the story. You can choose to jump ahead to the interactive diagrams However, I strongly recommend you return to these paragraphs at some point; for overall understanding they are necessary.
The Work-Energy theorem and Hamilton's stationary action have the following in common: the physics taking place is described in terms of kinetic energy and potential energy. The Work-Energy theorem is derived from F=ma, which of course means that F=ma can be recovered from the Work-Energy theorem. As we know: Hamilton's stationary action also has the property that F=ma can be recovered from it. The purpose of this article is to demonstrate how it comes about that F=ma can be recovered both from the Work-Energy theorem and Hamilton's stationary action.
As an overarching name for both the Work-Energy theorem and Hamilton's stationary action I will use the expression 'energy mechanics'.
Requirement for well defined potential energy
In order to have a well defined expression for potential energy the force must have the property that the work done in moving from some point A to some point B is independent of the path taken between the two points. The validity of energy mechanics is limited to forces with that property.
(It may be that in terms of a deeper theory such as quantum physics all interactions actually have that potential-is-independent-of-the-path property, but we should not blindly assume that.)
As we know: the principle of conservation of energy is asserted without any restriction. That's why the concept is referred as a 'principle', it is a blanket statement.
The discussion in this article avoids the blanket statement. The discussion in this article is limited to the classes of cases where by way of experimental corroboration it is evident that the work done in moving from some point A to some point B is independent of the path taken between the two points.
The derivative of Hamilton's 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 following cannot be emphasized enough: what is evaluated is the derivative of Hamilton's action. The value of Hamilton's action is immaterial.
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 reaches a minimum. But there are also classes of cases such that when the trial trajectory coincides with the true trajectory Hamilton's action reaches a maximum. This tells us that for the evaluation being minimum or maximum is not relevant. The relevant property is the property that is always valid: when the trial trajectory coincides with the true trajectory the derivative of Hamilton's action is zero.
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. In the course of the derivation the following two relations will be used:
The integral for acceleration from a starting point s0 to a final point s
Intermediary step: change of the differential according to (1.1), with corresponding change of limits.
Change the order:
Change of the differential according to (1.2), with corresponding change of limits.
So we have:
Up to this point no physics has been stated. (1.7) follows from the definitions of velocity and acceleration, combined with the mathematical properties of differentiation/integration.
Multiplying both sides with m and combining with F = ma arrives at a physics statement: the Work-Energy theorem:
As we know: potential energy is defined as the negative of the integral of force over distance. From here on I will write in terms of potential energy.
We see that it follows from the Work-Energy theorem that the amount of change of kinetic energy will always match the amount of change of potential energy.
(1.9) implies: in cases where energy mechanics is valid there will be conservation of energy.
The order in which the steps of this derivation are executed emphasizes distinction between mathematics content and physics content. (1.7) represents the mathematics content. The physics content is F=ma.
The Work-Energy theorem is overall richer in content than F=ma, but not because it has more physics content. The additional content is mathematics content
Potential energy and kinetic energy are both entities that do not have an intrinsic zero point. Whenever a measurement of energy is made it is a measurement of energy difference. The only way to define potential energy at all is to define the potential difference between some start point and some end point. Kinetic energy satisfies Galilean relativity, so it is definable only as a difference between some starting velocity and some end velocity.
In any physics calculation what is (ultimately) used is not the potential energy itself, but the derivative of the potential energy.
In any physics calculation what is (ultimately) used is not the kinetic energy itself, but the derivative of the kinetic energy.
In energy mechanics the actual execution of taking the derivative of the energy is postponed, so that the manipulations of the expressions manipulate statements about energy. The expression for the kinetic energy can be treated as a scalar, that's great for more efficient manipulation. (Also great for using generalized coordinates)
In applying Energy mechanics: in the end, to obtain an actual trajectory the calculation moves to the derivative of the energy.
The validity of (1.9) extends down to infinitisimal change:
In the interactive diagram: in the energies panel the horizontal axis is the time axis; the slope of the trajectory is the time derivative.
From here on I will refer to the interactive diagrams as 'graphlets' (contraction of 'graphical applet'). The following three graphlets are three instances of the same graphlet, for three successive cases.
As you manipulate the sliders: the process to home in on the true trajectory is to manipulate the trial worldline such that over the entire trajectory (1.11) is satisfied. The graphlets show that when (1.11) is satisfied the derivative of Hamilton's action is zero.
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.
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.)
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.
In graphlet 1 the potential energy as a function of position is linear. Whenever the potential energy as a function of position is of lower order than the expression for the kinetic energy the action reaches a minimum when the derivative of the action is zero. In graphlet 2 the potential energy function and the kinetic energy function are both quadratic expressions, so the action is neither a minimum nor a maximum. Here in graphlet 3 the potential energy as a function of position is a higher order expression than the one for the kinetic energy (cubic versus quadratic), and consequently when the derivative of the action is zero the action reaches a *maximum*.
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.
How to operate the demonstration:
The graphlet is designed to be discoverable, but for the sake of completeness I provide a description.
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 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.
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.
The Work-Energy theorem:
To recover F=ma from the Work-Energy theorem we take derivative with respect to position.
Simplify to initial position coordinate zero and initial velocity zero.
the derivative with respect to position:
As stated earlier: potential energy is the negative of the integral of force over distance.
For the subsequent manipulations it is practical to move the minus sign inside the parentheses.
The validity of (2.5) extends down to infinitisimal change:
In the energies panel the horizontal axis is the time axis; the slope of the trajectory is the time derivative.
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:
From here on I will refer to (2.8) as the 'Energy-Position equation', as it evaluates the derivative with respect to position.
Kinetic energy and pythagoras' theorem
How do we begin to formulate theory of motion?
In the process of moving around and getting things done we have control over how fast things are moving through space. We don't have control over how fast things move through time. That is why theory of motion is fundamentally expressed in terms of time derivatives: velocity is the time derivative of position, acceleration is the time derivative of velocity.
The Work-Energy theorem shows we have the option to express theory of motion in terms of an auxillary quantity: Energy.
This quantity 'Energy' comes with a superpower: Pythagoras' theorem. When you have two linear independent velocity vectors the resultant kinetic energy is equal to the sum of the component kinetic energies. Hence it is enough to keep track of the resultant kinetic energy. That means you can treat the kinetic energy as a scalar.
Going back to the derivation of the Work-Energy theorem: (1.7) and (1.8): the quantity 'Energy' is constructed by evaluating position integral of acceleration.
(2.3) walks that that integration back: taking the derivative with respect to position recovers F=ma.
Variation of a trajectory is variation of position. To take the derivative with respect to variation and to take the derivative with respect to position is one and the same thing.
In the graphlet the global variation is implemented in the following way:
With pv as variational parameter:(global variation sweep) = (1 + pv) * (current trial trajectory)
We have the following property of integration: it is a linear operation:
The relation that is expressed by the Energy-Position equation, (2.8), 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:
Then, from start point to end point:
It follows from (2.10) that if the work-energy theorem holds good then Hamilton's stationary action will hold good.
In the graphlet: the move from (2.9) to (2.10) corresponds to the move from the energies sub-panel to the integral subpanel.
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.
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.
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Last time this page was modified: August 13 2021