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Dirac-Feynman-Berezin sums 1 - Action

By Chris Austin. 26 November 2020.

An earlier version of this post was published on another website on 26 July 2012.

Hello, and welcome to this blog. I am Chris Austin, I work independently on high energy physics theory, in Maryport on the north-west coast of England, UK.

In this series of posts, I would like to tell you something about the foundation of our understanding of the way the physical world works, which I'll call Dirac-Feynman-Berezin sums.

I'll show you some formulae and things like that along the way, but I'll try to explain what all the parts mean as we go along, so you don't need to know about that sort of thing in advance.

First I have to tell you about action.

About 60 years after Sir Isaac Newton published his laws of motion in 1687, Pierre-Louis Moreau de Maupertuis discovered that Newton's laws follow from the requirement that a quantity he called "action", that depends on the positions and motions of a collection of objects over a period of time, should be relatively unaltered by small changes in the positions and motions of those objects.

For an example, let's consider a collection of objects, such that each object behaves approximately as though its mass is concentrated at a single point, the objects are moving slowly compared to the speed of light, and the forces between the objects arise from a contribution to their total energy, called their potential energy $V$, that depends on their positions but not on their motions. This is a useful first approximation for many things in the everyday world, for the Sun and the planets in the Solar System, for the motions of most of the stars in our galaxy, and for the atoms in a solid, liquid, gas, or living thing.

If $T$ denotes the sum of the kinetic energies of the objects, or in other words, the energy due to their motion, where the kinetic energy of a pointlike object of mass $m$ moving at a speed $v$ is $\frac{1}{2} mv^2$ if $v$ is small compared to the speed of light, then the formula for the action of the objects over a period of time that begins at $t_1$ and ends at $t_2$ is:

\begin{displaymath}S = \int_{t_1}^{t_2} \left( T - V \right) \mathrm{d} t. \end{displaymath}

Here $S$ is the action. The symbol $\int$ was once an alternative form of the letter $s$, and is nowadays called the integral sign. It means a sum of what follows it. The upright letter $\mathrm{d}$ in $\mathrm{d} t$ means "a tiny amount of," and was introduced in this context by Gottfried Wilhelm Leibniz. The idea is that the period from $t_1$ to $t_2$ is divided up into a great number of tiny time intervals, each so small that $T$ and $V$ only change by a tiny amount during any one time interval. The notation $\int^{t_2}_{t_1} \left( T - V \right) \mathrm{d} t$ means that $S$ is the sum of a contribution from each of these tiny time intervals, and the contribution from a tiny time interval of length $\mathrm{d} t$ is the product of $\left( T - V \right)$ and the tiny amount of time $\mathrm{d} t$, where $T$ and $V$ are evaluated at an arbitrary moment during that interval. Then for motions such that $T$ and $V$ vary smoothly with time, the sum of all the contributions changes by an ever smaller amount as the period from $t_1$ to $t_2$ is divided up ever more finely, and $S$ is the limit of the sum of all the contributions, as the period is divided up so finely that the length of the longest tiny interval tends to 0.

We can specify the position of an object at a particular moment in time by 3 numbers, called coordinates. For example, we can specify the position of an aeroplane by its latitude and longitude, in degrees, and its altitude, in meters. We can write these 3 numbers as a list, for example (latitude, longitude, altitude).

If the number of objects in our example is $N$, then their positions, at a particular moment in time, can be specified by a table of numbers with 3 columns and $N$ rows. Each row gives the coordinates of a different one of the objects. The motions of the $N$ objects over a period of time can be represented as accurately as desired by a series of such tables, one for each of a closely spaced sequence of moments in time.

It is convenient to let a single symbol, say $x$, represent this entire collection of data. Then if the letter $a$, for example, represents one of the numbers 1, 2, 3, and the letter $I$, for example, represents one of the numbers $1, 2, 3, \ldots , N$, the value of position coordinate number $a$, of object number $I$, at time $t$, can be represented as $x_{a I t}$, where the subscripts $a$, $I$, and $t$ are called indexes. Alternative notations for the same thing, which can be used as convenient, are $x \left( a, I, t \right)$, and $x_{a I} \left( t \right)$, for example.

The speed $v$ of an object is proportional to the rate at which its coordinates change with time. If a collection of data that gives the value of a quantity at each moment in time is represented by a symbol or expression $X$, then the collection of data that gives the rate of change of that quantity with time, at each moment in time, is often represented as $\frac{\mathrm{d} X}{\mathrm{d} t}$, or alternatively $\frac{\mathrm{d}}{\mathrm{d} t} X$. The Leibniz $\mathrm{d}$ in the numerator of $\frac{\mathrm{d}}{\mathrm{d} t}$ means, "the change in the following expression, when the time changes by the tiny amount $\mathrm{d} t$," and indicates that the formula is to be taken in the limit where the size of the time interval $\mathrm{d} t$ tends to 0. The value of $\frac{\mathrm{d} X}{\mathrm{d} t}$ at time $t$ is $\left( \frac{\mathrm{d} X}{\mathrm{d} t} \right)_t = \frac{X \left( t + \mathrm{d} t \right) - X \left( t \right)}{\mathrm{d} t}$, which is well-defined if $X$ changes smoothly with $t$. The coordinates $x_{a I t}$ of the objects in our example will change smoothly with $t$ for a sensible choice of position coordinates, since we assumed that $v$ is small compared to the speed of light, and thus finite.

To calculate the speed $v_I$ of object number $I$, we need to know to the distance travelled for a small change in its coordinates $x_{a I}$. For example the distance travelled by an aeroplane, for a $1^{\circ}$ change in its longitude at fixed latitude and altitude, is smaller, the closer the aeroplane is to the north or south pole.

For the flat 2-dimensional world of Euclidean geometry, we can choose as coordinates the distances from 2 fixed straight lines, at right-angles to one another. These coordinates were introduced by René Descartes, and are called Cartesian coordinates. The distance $r$ between two points whose $x_1$ coordinates differ by $p$ and whose $x_2$ coordinates differ by $q$ is then given by Pythagoras as $r = \sqrt{p^2 + q^2}$, since directly from the diagram, $r^2 = \left( p + q \right)^2 - 4 \left( \frac{1}{2} pq \right) = p^2 + q^2$.

For the flat 3-dimensional generalization of Euclidean geometry, called 3-dimensional Euclidean space, we can choose as coordinates the distances from 3 fixed flat planes, each at right-angles to the other two. These are also called Cartesian coordinates. The distance $r$ between two points whose $x_a$ coordinates differ by $p_a$, for $a = 1, 2,$ and $3$, is then $r = \sqrt{p^2_1 + p^2_2 + p^2_3}$, which follows from applying Pythagoras first to the $x_1$ and $x_2$ coordinates, and then to $\sqrt{p^2_1 + p^2_2}$ and $p_3$.

The assumptions of our example imply that any gravitational fields present are sufficiently weak that if we use Cartesian coordinates, then distances are given by Pythagoras to a good approximation, for otherwise the objects would not continue to move slowly compared to the speed of light. Then in Cartesian coordinates, the speed $v_I$ of object number $I$ is $v_I = \sqrt{\left( \frac{\mathrm{d} x_{1 I}}{\mathrm{d} t} \right)^2 + \left( \... ...hrm{d} t} \right)^2 + \left( \frac{\mathrm{d} x_{3 I}}{\mathrm{d} t} \right)^2}$, and the sum of the kinetic energies of the objects is given by:

\begin{displaymath}T = \frac{1}{2} \sum_{I = 1}^N \left( m_I \sum_{a = 1}^3 \left( \frac{\mathrm{d} x_{a I}}{\mathrm{d} t} \right)^2 \right) . \end{displaymath}

The symbol $\sum$ is the upper-case Greek letter Sigma, and indicates a sum of what follows it. The idea is that each contribution to the sum is obtained from the expression that follows the $\sum$, by substituting a specific value for one of the indexes in the expression, and the notations below and above the $\sum$ show which index is to be substituted, and the range of values of that index, for which terms are to be included in the sum. Thus the meaning of $\sum$ is quite similar to the meaning of $\int$ as above. The difference is that $\sum$ is used for a sum over a discrete index such as $a$ or $I$, while $\int$, together with a tiny factor such as $\mathrm{d} t$, is used for a sum over a continuous index such as $t$.

Let's now consider a small change to the positions and motions of the $N$ objects during the period of time from $t_1$ to $t_2$. I'll represent the change, or "perturbation", of the positions and motions by the Greek letter $\varepsilon$, pronounced epsilon, which is often used to represent a small quantity, so the modified positions and motions are represented by $x + \varepsilon$. Here $\varepsilon$, like $x$, represents an entire collection of data, for example it could introduce different types of wobbles to the motions of each of the objects. I shall assume that $\varepsilon$ is 0 at $t_1$ and $t_2$, or in other words, that $\varepsilon_{a I} \left( t_1 \right) = \varepsilon_{a I} \left( t_2 \right) = 0$ for all values $1, 2, 3$ of $a$ and all values $1, 2, \ldots, N$ of $I$, while for times $t$ between $t_1$ and $t_2$, I shall assume only that all the $\varepsilon_{a I t}$ are small, and change smoothly with time.

Near the start of the post, above, I said that de Maupertuis's requirement, which implies Newton's laws, is that the action should be relatively unaltered by small changes to the positions and motions of the objects. What I meant by that is that as $\varepsilon$ tends to 0, or in other words, as $\varepsilon_{a I t}$ approaches 0 for all relevant values of $a$, $I$, and $t$, the change to the action should tend to 0 more rapidly than in proportion to $\varepsilon$.

The change to the contribution $\int_{t_1}^{t_2} T \mathrm{d} t$ to the action, that results from the replacement of $x$ by $x + \varepsilon$, is:

\begin{displaymath}\int_{t_1}^{t_2} \left( \frac{1}{2} \sum_{I = 1}^N \left( m_I... ...x_{a I}}{\mathrm{d} t} \right)^2 \right) \right) \mathrm{d} t \end{displaymath}

\begin{displaymath}\simeq \int_{t_1}^{t_2} \sum_{I = 1}^N \left( m_I \sum_{a = 1... ...varepsilon_{a I}}{\mathrm{d} t} \right) \right) \mathrm{d} t, \end{displaymath}

since $\left( \frac{\mathrm{d} \left( x_{a I} + \varepsilon_{a I} \right)}{\mathrm{d} ... ...thrm{d} t} + \left( \frac{\mathrm{d} \varepsilon_{a I}}{\mathrm{d} t} \right)^2$, and the contribution from the first expression, usually called a term, in the right-hand side of this, cancels against the last expression in the left-hand side of the above equation for each value of the indexes $a$ and $I$, while the third term, proportional to $\varepsilon^2$, is much smaller than the second term for very small $\varepsilon$, so can be neglected. The symbol $\simeq$ means "approximately equal to".

Let's now consider the rate of change with time of a product $X_t Y_t$, where $X$ and $Y$ represent collections of data that give the values, at each moment in time, of quantities that change smoothly with time. The expression $XY$ represents the collection of data that gives the value of the product $X_t Y_t$ at each moment in time, so from above, the collection of data that gives the rate of change with time of the product $X_t Y_t$, at each moment in time, can be represented as $\frac{\mathrm{d}}{\mathrm{d} t} \left( XY \right)$. And:

\begin{displaymath}\left( \frac{\mathrm{d}}{\mathrm{d} t} \left( XY \right) \rig... ...m{d} t \right) - X \left( t \right) Y \left( t \right) \right) \end{displaymath}

\begin{displaymath}= \frac{1}{\mathrm{d} t} \left( \left( X \left( t \right) + \... ...m{d} t \right) - X \left( t \right) Y \left( t \right) \right) \end{displaymath}

\begin{displaymath}= \left( \frac{\mathrm{d} X}{\mathrm{d} t} \right)_t Y \left(... ...t \right) \left( \frac{\mathrm{d} Y}{\mathrm{d} t} \right)_t . \end{displaymath}

The second line here follows from noting that if $X$ is any expression that varies smoothly with $t$, then since $\left( \frac{\mathrm{d}}{\mathrm{d} t} X \right)_t = \frac{X \left( t + \mathrm{d} t \right) - X \left( t \right)}{\mathrm{d} t}$, we have $X \left( t + \mathrm{d} t \right) = X \left( t \right) + \left( \frac{\mathrm{d} X}{\mathrm{d} t} \right)_t \mathrm{d} t$. The third line follows because the last contribution in the second line cancels part of the contribution before it, and the ratio $\frac{\left( \mathrm{d} t \right)^2}{\mathrm{d} t} = \mathrm{d} t$ is 0 in the limit where $\mathrm{d} t$ tends to 0.

The above formula is true at every value of the time $t$, so it can be summarized as:

\begin{displaymath}\frac{\mathrm{d}}{\mathrm{d} t} \left( XY \right) = \frac{\ma... ...d} X}{\mathrm{d} t} Y + X \frac{\mathrm{d} Y}{\mathrm{d} t} . \end{displaymath}

This is called Leibniz's rule for the rate of change of a product.

Applying this to the product $\frac{\mathrm{d} x_{a I}}{\mathrm{d} t} \varepsilon_{a I}$, we have:

\begin{displaymath}\frac{\mathrm{d}}{\mathrm{d} t} \left( \frac{\mathrm{d} x_{a ... ...rm{d} t} \frac{\mathrm{d} \varepsilon_{a I}}{\mathrm{d} t} . \end{displaymath}

From this result and the previous one, the change to $\int_{t_1}^{t_2} T \mathrm{d} t$ that results from the replacement of $x$ by $x + \varepsilon$, is:

\begin{displaymath}\simeq \int_{t_1}^{t_2} \sum_{I = 1}^N \left( m_I \sum_{a = 1... ...ight) \right) \varepsilon_{a I} \right) \right) \mathrm{d} t. \end{displaymath}

From now on, if an expression $X$ represents a collection of data that gives the value of a quantity at each moment in time, I shall for brevity just say that $X$ is a time-dependent quantity.

Let's now consider the expression $\int^{t_2}_{t_1} \frac{\mathrm{d} X}{\mathrm{d} t} \mathrm{d} t$, where $X$ is any time-dependent quantity whose value $X_t$ changes smoothly with time. From the description I gave near the start of the post above, this expression is given by dividing the period from $t_1$ to $t_2$ up into a great number of tiny time intervals, each so small that $\frac{\mathrm{d} X}{\mathrm{d} t}$ only changes by a tiny amount during any one time interval, and adding together a contribution from each of these tiny time intervals. The contribution from a tiny time interval of length $\mathrm{d} t$ is $\frac{\mathrm{d} X}{\mathrm{d} t} \mathrm{d} t$, where $\frac{\mathrm{d} X}{\mathrm{d} t}$ is evaluated at an arbitrary moment during that interval, and the expression is the limit of the sum of all the contributions, as the period is divided up so finely that the length of the longest tiny interval tends to 0.

For a tiny time interval that starts at time $t_s$ and finishes at time $t_f$, where $t_f - t_s = \mathrm{d} t$, we have $\frac{\mathrm{d} X}{\mathrm{d} t} = \frac{X \left( t_s + \mathrm{d} t \right) -... ...}{\mathrm{d} t} = \frac{X \left( t_f \right) - X \left( t_s \right)}{t_f - t_s}$ in the limit where $\mathrm{d} t$ tends to 0, so the contribution of that interval is $\frac{\mathrm{d} X}{\mathrm{d} t} \mathrm{d} t = X \left( t_f \right) - X \left( t_s \right)$. When we add together the contributions of all the tiny intervals, the $X \left( t_f \right)$ term in the contribution of each interval except the last one cancels the $- X \left( t_s \right)$ term in the contribution of next interval, so that:

\begin{displaymath}\int^{t_2}_{t_1} \frac{\mathrm{d} X}{\mathrm{d} t} \mathrm{d} t = X \left( t_2 \right) - X \left( t_1 \right) . \end{displaymath}

In words, this means that the integral of the rate of change of a quantity is equal to the net change of that quantity.

This is true, in particular, if $X$ is $\frac{\mathrm{d} x_{a I}}{\mathrm{d} t} \varepsilon_{a I}$, so from the previous result, the change to $\int_{t_1}^{t_2} T \mathrm{d} t$ that results from the replacement of $x$ by $x + \varepsilon$, is:

\begin{displaymath}\simeq \sum_{I = 1}^N \left( m_I \sum_{a = 1}^3 \left( \int_{... ...\right) \right) \varepsilon_{a I} \mathrm{d} t \right) \right) \end{displaymath}

\begin{displaymath}= \sum_{I = 1}^N \left( m_I \sum_{a = 1}^3 \left( \left( \fra... ...ght) \right) \varepsilon_{a I} \mathrm{d} t \right) \right) . \end{displaymath}

However we assumed above that $\varepsilon$, which is a small modification to the positions and motions of the objects, is 0 at $t_1$ and $t_2$. So the first two terms inside the outer two pairs of parentheses in the above expression are 0, so the change to $\int_{t_1}^{t_2} T \mathrm{d} t$ that results from the replacement of $x$ by $x + \varepsilon$, is:

\begin{displaymath}\simeq - \sum_{I = 1}^N \sum_{a = 1}^3 \int_{t_1}^{t_2} m_I \... ...\mathrm{d} t} \right) \right) \varepsilon_{a I} \mathrm{d} t. \end{displaymath}

The remaining contribution to the change of the action in our example, that results from modifying the positions and motions of the objects by the small perturbation $\varepsilon$, is the contribution from the change of $- \int^{t_2}_{t_1} V \mathrm{d} t$, where $V$ is the potential energy, which depends on the positions of the objects, but not on their motions. The symbol $V$ represents the collection of data that includes the value of the potential energy at each moment in time, and the value of the potential energy at time $t$, which we can write as $V_t$ or $V \left( t \right)$ as convenient, depends on the values $x_{a I t}$ of the coordinates at time $t$, but not on the values of the coordinates at any other time. For example if the objects are stars or planets then the significant potential energy is their gravitational potential energy, given by:

\begin{displaymath}V = - \sum_{I = 1}^N \left( \sum_{J = 1}^{I - 1} \frac{Gm_I m_J}{r_{I J}} \right), \end{displaymath}

where $r_{I J} = \sqrt{\sum^3_{a = 1} \left( x_{a I} - x_{a J} \right)^2}$ is the distance between object $I$ and object $J$, and $G = 6.674 \times 10^{- 11} \hspace{0.8em} \mathrm{{{metres}}}^3 / \left( \mathrm{{{kg}}} \hspace{0.8em} \mathrm{\sec}^2 \right)$ is Newton's constant of gravitation.

If a quantity, such as the potential energy, depends on a number of quantities $y_p$ that can vary continuously, where $y$ represents the collection of those quantities, and the index $p$ distinguishes the quantities in the collection, and if the collection of data that gives the value of the dependent quantity at each $y$, or in other words, at each set of values of the quantities $y_p$, is represented by a symbol $X$, then the collection of data that gives the rate of change of the dependent quantity as the quantity $y_q$ changes, while all the other quantities in $y$ have fixed values, is usually represented as $\frac{\partial X}{\partial y_q}$, or alternatively as $\frac{\partial}{\partial y_q} X$. The symbol $\partial$ is an alternative notation for Leibniz's $\mathrm{d}$, and $\left( \frac{\partial X}{\partial y_q} \right)_{y_q} = \frac{X \left( y_q + \mathrm{d} y_q \right) - X \left( y_q \right)}{\mathrm{d} y_q}$, where the quantities in the collection $y$ other than $y_q$ all have the same values in both terms in the numerator in the right-hand side as they have in the left-hand side, so their values don't need to be displayed.

If the value $X \left( y \right)$ changes smoothly with $y$, or in other words, smoothly with the quantities $y_p$ in the collection $y$, then for $y$ near a reference collection $y_{\left( 0 \right)}$, in the sense that all the quantities $y_p - y_{\left( 0 \right) p}$ are small in magnitude, the value $X \left( y \right)$ can be represented approximately as:

\begin{displaymath}X \left( y \right) \simeq X \left( y_{\left( 0 \right)} \righ... ...eft( 0 \right)}} \left( y_p - y_{\left( 0 \right) p} \right), \end{displaymath}

where as the magnitudes of all the $y_p - y_{\left( 0 \right) p}$ tend to 0, the error of this approximate representation tends to 0 more rapidly than in proportion to those magnitudes. For the two sides of the above formula are equal when $y = y_{\left( 0 \right)}$. And by using the above definition of $\frac{\partial X}{\partial y_q}$ for the case when $X$ is $y_p$, we find that $\frac{\partial y_p}{\partial y_q} = \delta_{p q}$, where $\delta$ is the Greek letter delta, and $\delta_{p q}$, which is called the Kronecker delta after Leopold Kronecker, is 1 when $p = q$, and 0 otherwise. Thus applying $\frac{\partial}{\partial y_q}$ to the right-hand side of the above formula gives $\left( \frac{\partial X}{\partial y_q} \right)_{y_{\left( 0 \right)}}$ for all $y$, which is in agreement with the application of $\frac{\partial}{\partial y_q}$ to the left-hand side when $y = y_{\left( 0 \right)}$. Thus the two sides of the above formula would be in agreement for all $y$ if the quantities $\frac{\partial X}{\partial y_q}$ were independent of $y$. This is not so in general, but the assumption that $X \left( y \right)$ changes smoothly with $y$ implies that the differences $\left( \frac{\partial X}{\partial y_q} \right)_y - \left( \frac{\partial X}{\partial y_q} \right)_{y_{\left( 0 \right)}}$ tend to 0 at least as fast as the differences $y_p - y_{\left( 0 \right) p}$ as $y$ approaches $y_{\left( 0 \right)}$, so the error of the above formula tends to 0 at least as fast as products of two of those differences, and thus more rapidly than in proportion to those differences.

Applying the above formula to the potential energy, with $y_{\left( 0 \right)}$ taken as $x$, and $y - y_{\left( 0 \right)}$ taken as $\varepsilon$, we have:

\begin{displaymath}V \left( x + \varepsilon \right) - V \left( x \right) \simeq ... ...ac{\partial V}{\partial x_{a I}} \right)_x \varepsilon_{a I}, \end{displaymath}

where the error of this formula tends to 0 more rapidly than in proportion to $\varepsilon$, as $\varepsilon_{a I t}$ tends to 0 for all relevant values of $a$ and $I$.

Combining this formula for the change of the potential energy with the formula for the change of the kinetic energy we obtained before it, we find that the change to the action $\int_{t_1}^{t_2} \left( T - V \right) \mathrm{d} t$ that results from the replacement of $x$ by $x + \varepsilon$, is:

\begin{displaymath}\simeq - \sum_{I = 1}^N \sum_{a = 1}^3 \int_{t_1}^{t_2} \left... ...al x_{a I}} \right)_x \right) \varepsilon_{a I} \mathrm{d} t, \end{displaymath}

where the error of this formula tends to 0 more rapidly than in proportion to $\varepsilon$, as $\varepsilon$ tends to 0.

De Maupertuis's principle requires that the change to the action should tend to 0 more rapidly than in proportion to $\varepsilon$, as $\varepsilon$ tends to 0. But from the above formula, this is only possible for all perturbations $\varepsilon$ such that $\varepsilon$ is 0 at $t_1$ and $t_2$, and all the $\varepsilon_{a I t}$ change smoothly with time, if:

\begin{displaymath}m_I \frac{\mathrm{d}}{\mathrm{d} t} \left( \frac{\mathrm{d} x... ...) + \left( \frac{\partial V}{\partial x_{a I}} \right)_x = 0, \end{displaymath}

for all relevant values of $a$, $I$, and $t$.

We are using Cartesian coordinates, so $\frac{\mathrm{d} x_{a I}}{\mathrm{d} t}$ is the $a$'th component of the velocity of the $I$'th object, and $\frac{\mathrm{d}}{\mathrm{d} t} \left( \frac{\mathrm{d} x_{a I}}{\mathrm{d} t} \right)$, which is usually written as $\frac{\mathrm{d}^2 x_{a I}}{\mathrm{d} t^2}$, is the $a$'th component of the acceleration of the $I$'th object. And by the definition of potential energy, the $a$'th component of the force on the $I$'th object is $- \frac{\partial V}{\partial x_{a I}}$. Thus the above equation is Newton's second law of motion.

Let's now consider the rate of change with time of the total energy $T + V$, when the objects move in accordance with Newton's second law of motion, which we have just derived from de Maupertuis's principle. From Leibniz's rule for the rate of change of a product, which we proved above, the rate of change of $\left( \frac{\mathrm{d} x_{a I}}{\mathrm{d} t} \right)^2$ is $\left( \frac{\mathrm{d}}{\mathrm{d} t} \frac{\mathrm{d} x_{a I}}{\mathrm{d} t} ... ...{\mathrm{d} x_{a I}}{\mathrm{d} t} \frac{\mathrm{d}^2 x_{a I}}{\mathrm{d} t^2}$, so the rate of change with time of the kinetic energy $T$ is:

\begin{displaymath}\frac{\mathrm{d} T}{\mathrm{d} t} = \sum_{I = 1}^N \sum_{a = ... ...}{\mathrm{d} t} \frac{\mathrm{d}^2 x_{a I}}{\mathrm{d} t^2} . \end{displaymath}

And by choosing $\varepsilon_{a I}$ to be $\frac{\mathrm{d} x_{a I}}{\mathrm{d} t} \mathrm{d} t$, in the formula we derived above, for the change of the potential energy $V$ when $x$ is replaced by $x + \varepsilon$, we obtain the rate of change with time of the potential energy $V$ as:

\begin{displaymath}\frac{\mathrm{d} V}{\mathrm{d} t} = \frac{V_{t + \mathrm{d} t... ...l x_{a I}} \right)_x \frac{\mathrm{d} x_{a I}}{\mathrm{d} t}, \end{displaymath}

since $V$ only depends on time through the dependence on time of the coordinates $x_{a I}$ of the objects.

From the sum of the above two formulae, we obtain the rate of change with time of the total energy $T + V$ of the objects, when they move in accordance with Newton's second law of motion, as:

\begin{displaymath}\frac{\mathrm{d} \left( T + V \right)}{\mathrm{d} t} = \sum_{... ...right)_x \right) \frac{\mathrm{d} x_{a I}}{\mathrm{d} t} = 0. \end{displaymath}

Thus the total energy $T + V$ of the objects never changes. This is usually referred to as the conservation of total energy. (The everyday phrase, "conservation of energy," refers to trying to reduce the rate at which some particular forms of energy, such as the potential energy associated with the arrangement of the atoms within the lattice structure or molecules of a chemical fuel, are converted to other forms of energy.)

Stationary action for a planet

To illustrate the practical application of de Maupertuis's discovery, which is sometimes called the principle of stationary action, let's consider a planet in orbit around the Sun, neglecting the gravitational effects of the other planets, which are relatively small. The mass $m_S$ of the Sun is much greater than the mass $m_p$ of the planet, so to a good approximation, we can treat the Sun as fixed in position, and just consider the motion of the planet around the Sun. The gravitational force on the planet is always in the direction of the straight line from the planet to the Sun, so the planet stays in the 2-dimensional plane defined by the straight line from the planet's initial position to the Sun, and the direction of the planet's initial velocity, which I shall assume is not exactly along that line.

It is convenient to specify the planet's position in this plane by the distance $r$ from the planet to the Sun, and the angle between the straight line from the planet to the Sun, and the initial direction of that line. I shall represent that angle by $\theta$, which is the Greek letter theta. To keep the formulae as simple as possible, the angle $\theta$ will be measured not in degrees but in "radians", where 1 radian is the angle turned through when something moving along a circular path has travelled a distance along the circle equal to the radius of the circle. Thus a full $360^{\circ}$ rotation is $2 \pi \simeq 6.283$ radians, and 1 radian is approximately $57.30^{\circ}$.

Due to measuring the angle $\theta$ in radians, the distance travelled by the planet when $\theta$ increases by a small amount $\mathrm{d} \theta$ at fixed $r$ is $r \mathrm{d} \theta$, and in the limit when $\mathrm{d} \theta$ tends to 0, this is in the direction perpendicular to the straight line from the planet to the Sun. Thus the square of the planet's speed can be calculated from $r \frac{\mathrm{d} \theta}{\mathrm{d} t}$ and $\frac{\mathrm{d} r}{\mathrm{d} t}$ using Pythagoras, so the kinetic energy of the planet is $T = \frac{1}{2} m_p \left( r^2 \left( \frac{\mathrm{d} \theta}{\mathrm{d} t} \right)^2 + \left( \frac{\mathrm{d} r}{\mathrm{d} t} \right)^2 \right)$. The gravitational potential energy is $V = - \frac{Gm_S m_p}{r}$, so the action is:

\begin{displaymath}S = \int_{t_1}^{t_2} \left( \frac{1}{2} m_p \left( r^2 \left(... ... \right)^2 \right) + \frac{Gm_S m_p}{r} \right) \mathrm{d} t. \end{displaymath}

From above, the total energy:

\begin{displaymath}E = T + V = \frac{1}{2} m_p \left( r^2 \left( \frac{\mathrm{d... ...rm{d} r}{\mathrm{d} t} \right)^2 \right) - \frac{Gm_S m_p}{r} \end{displaymath}

is independent of time, and thus partly characterizes the planet's orbit. De Maupertuis's principle leads to one independent equation of motion for each coordinate of each moving object. We have already obtained the time-independence of the total energy $E$ from one combination of the equations of motion, so we only need to obtain one of the two equations of motion directly by requiring that the action $S$ is relatively unaltered by a small modification to the time dependence of the coordinates. Using the same method as above, we find that if a small time-dependent perturbation $\varepsilon_{\theta}$ is added to $\theta$, such that $\varepsilon_{\theta} \left( t \right)$ depends smoothly on $t$, and $\varepsilon_{\theta} \left( t_1 \right) = \varepsilon_{\theta} \left( t_2 \right) = 0$, then the modification to the action is:

\begin{displaymath}S \left( \theta + \varepsilon_{\theta} \right) - S \left( \th... ...thrm{d} t} \right) \right) \varepsilon_{\theta} \mathrm{d} t, \end{displaymath}

where the error of this formula is proportional to $\varepsilon^2_{\theta}$, and thus tends to 0 more rapidly than in proportion to $\varepsilon_{\theta}$, as $\varepsilon_{\theta}$ tends to 0. To obtain this formula, we used the equality of the integral of the rate of change and the net change, as above, applied to the expression $r^2 \frac{\mathrm{d} \theta}{\mathrm{d} t} \varepsilon_{\theta}$, together with Leibniz's rule for the rate of change of a product, as above, applied to the product of $r^2 \frac{\mathrm{d} \theta}{\mathrm{d} t}$ and $\varepsilon_{\theta}$.

De Maupertuis's principle requires that the change to the action should tend to 0 more rapidly than in proportion to $\varepsilon_{\theta}$, as $\varepsilon_{\theta}$ tends to 0. But from the above formula, this is only possible for all perturbations $\varepsilon_{\theta}$ such that $\varepsilon_{\theta}$ is 0 at $t_1$ and $t_2$, and $\varepsilon_{\theta}$ changes smoothly with time, if $\frac{\mathrm{d}}{\mathrm{d} t} \left( r^2 \frac{\mathrm{d} \theta}{\mathrm{d} t} \right) = 0$, for all relevant values of $t$. This means that $r^2 \frac{\mathrm{d} \theta}{\mathrm{d} t}$ is independent of time.

For a tiny amount of time $\mathrm{d} t$, the area swept out by the straight line from the Sun to the planet during the time interval $\mathrm{d} t$ is approximately $\frac{1}{2} r^2 \frac{\mathrm{d} \theta}{\mathrm{d} t} \mathrm{d} t$, which is the area of the right-angled triangle made by the straight lines from the Sun to the planet at the times $t$ and $t + \mathrm{d} t$, together with the straight line tangential to the circle of radius $r$ centred at the Sun, that meets that circle at the position of the planet at time $t$. The difference between $\frac{1}{2} r^2 \frac{\mathrm{d} \theta}{\mathrm{d} t} \mathrm{d} t$, and the area swept out by the straight line from the Sun to the planet during the time interval $\mathrm{d} t$, tends to 0 in proportion to $\left( \mathrm{d} t \right)^2$ as $\mathrm{d} t$ tends to 0, and thus more rapidly than in proportion to $\mathrm{d} t$, so the rate at which the straight line from the Sun to the planet sweeps out area is $\frac{1}{2} r^2 \frac{\mathrm{d} \theta}{\mathrm{d} t}$. We found above from de Maupertuis's principle that this is independent of time, so the straight line from the Sun to the planet sweeps out equal areas in equal times. This is the second of the three laws of planetary motion, which Johannes Kepler discovered by studying the astronomical measurements made by Tycho Brahe.

The product $m_p r^2 \frac{\mathrm{d} \theta}{\mathrm{d} t}$, which is also independent of time since the planet's mass $m_p$ is constant, is called the orbital angular momentum of the planet. I shall represent it by $J$. The value of $J$, like the value $E$ of the planet's total energy, partly characterizes the orbit of the planet.

To find the possible shapes of the planet's orbit, we can convert the rate of change of $r$ with time to the rate of change of $r$ with $\theta$, using the relation $J = m_p r^2 \frac{\mathrm{d} \theta}{\mathrm{d} t}$. The time interval during which $\theta$ changes by a tiny amount $\mathrm{d} \theta$ is $\mathrm{d} t = \frac{m_p r^2}{J} \mathrm{d} \theta$, so $\frac{\mathrm{d} r}{\mathrm{d} t} = \frac{J}{m_p r^2} \frac{\mathrm{d} r}{\mathrm{d} \theta}$. Using this result and also the relation $r^2 \left( \frac{\mathrm{d} \theta}{\mathrm{d} t} \right)^2 = \frac{J^2}{m^2_p r^2}$ in the above formula for the planet's total energy $E$, we find:

\begin{displaymath}E = \frac{1}{2} m_p \left( \frac{J^2}{m^2_p r^2} + \frac{J^2}... ...}{\mathrm{d} \theta} \right)^2 \right) - \frac{Gm_S m_p}{r} . \end{displaymath}

Rearranging this formula, we find:


\begin{displaymath}\left( \frac{\mathrm{d} r}{\mathrm{d} \theta} \right)^2 = \frac{2 Em_p r^4}{J^2} + \frac{2 Gm_S m^2_p r^3}{J^2} - r^2 . \end{displaymath}

To use this formula to find the possible orbits of the planet, it is helpful to know about the Cartesian coordinates of something moving around a circle, and their rate of change with angle. If something is moving along a circular path, and $\theta$ is the angle in radians, as above, between the straight line from the centre of the moving object to the centre of the circle, and a fixed straight line in the plane of the circle though the centre of the circle, then the traditional names for the Cartesian coordinates of the centre of the moving object, relative to the centre of the circle, in units of the radius of the circle, are $\mathrm{\cos} \left( \theta \right)$ for the coordinate parallel to the fixed straight line, and $\mathrm{\sin} \left( \theta \right)$ for the coordinate perpendicular to the fixed straight line in the plane of the circle. The directions of the coordinates are chosen so that $\mathrm{\cos} \left( 0 \right) = 1$ and $\mathrm{\sin} \left( \frac{\pi}{2} \right) = 1$. From Pythagoras, we have $\left( \mathrm{\cos} \left( \theta \right) \right)^2 + \left( \mathrm{\sin} \left( \theta \right) \right)^2 = 1$, for all $\theta$.

If the object starts at angle $\theta$ and goes round the circle $n$ times, so that $\theta$ increases by $2 \pi n$, where $n$ is any whole number, then the Cartesian coordinates of the centre of the object come back to their initial values, so that $\mathrm{\cos} \left( \theta + 2 \pi n \right) = \mathrm{\cos} \left( \theta \right)$, and $\mathrm{\sin} \left( \theta + 2 \pi n \right) = \mathrm{\sin} \left( \theta \right)$, for all whole numbers $n$. This diagram shows $\mathrm{\cos} \left( \theta \right)$, for $\theta$ in the range $- 4 \pi$ to $4 \pi$.

When the angle $\theta$ increases by a tiny amount $\mathrm{d} \theta$, the changes to the coordinates of the centre of the object are approximately the same as they would be if the object moved a distance $r \mathrm{d} \theta$ along the straight line tangential to the circle at $\theta$ instead of exactly along the circle, where $r$ is the radius of the circle, and the relative error of this approximation tends to 0 as $\mathrm{d} \theta$ tends to 0. And from this diagram, the change to the Cartesian coordinate parallel to the fixed straight line, when the centre of the object moves a distance $l$ along the tangential straight line in the direction of increasing $\theta$, is $- l \mathrm{\sin} \left( \theta \right)$, and the change to the Cartesian coordinate perpendicular to the fixed straight line is $l \mathrm{\cos} \left( \theta \right)$. Thus:

\begin{displaymath}\frac{\mathrm{d}}{\mathrm{d} \theta} \mathrm{\cos} \left( \th... ...r \mathrm{d} \theta} = - \mathrm{\sin} \left( \theta \right), \end{displaymath}

and

\begin{displaymath}\frac{\mathrm{d}}{\mathrm{d} \theta} \mathrm{\sin} \left( \th... ...{r \mathrm{d} \theta} = \mathrm{\cos} \left( \theta \right) . \end{displaymath}

Returning to the planet in orbit around the Sun, we can now confirm that the above formula for $\left( \frac{\mathrm{d} r}{\mathrm{d} \theta} \right)^2$ implies that the orbit of the planet is an ellipse with the Sun at one focus, in agreement with Kepler's first law of planetary motion. An ellipse is the curve formed by all the points in a plane such that the sum of the distances from a point on the ellipse to two fixed points, called the focuses of the ellipse, has a fixed value. With the Sun at one focus, the distance from the planet to that focus is $r$. I shall represent the distance between the two focuses by $a$, and choose the fixed direction corresponding to $\theta = 0$ to be along the straight line from the Sun to the other focus. Then using Cartesian coordinates for a moment, the Cartesian coordinates of the other focus are $\left( a, 0 \right)$ and the Cartesian coordinates of the planet are $\left( r \, \, \mathrm{\cos} \left( \theta \right), r \: \mathrm{\sin} \left( \theta \right) \right)$, so by Pythagoras, the distance from the planet to the second focus is:

\begin{displaymath}\sqrt{\left( a - r \: \mathrm{\cos} \left( \theta \right) \ri... ...rt{a^2 + r^2 - 2 ar \, \mathrm{\cos} \left( \theta \right)} . \end{displaymath}

I shall represent the fixed sum of the distances from the planet to the two focuses by $b$, so if the planet's orbit is an ellipse characterized by the distances $a$ and $b$, then $r$ and $\theta$ are related by:

\begin{displaymath}r + \sqrt{a^2 + r^2 - 2 ar \, \mathrm{\cos} \left( \theta \right)} = b. \end{displaymath}

Rearranging this formula, we find:

\begin{displaymath}r = \frac{b^2 - a^2}{2 \left( b - a \, \, \mathrm{\cos} \left( \theta \right) \right)} . \end{displaymath}

We can calculate the rate of change of $r$ with $\theta$ from this formula by using Leibniz's rule for the rate of change of a product, which we obtained above, since the product $\frac{b^2 - a^2}{2 \left( b - a \, \, \mathrm{\cos} \left( \theta \right) \right)} \left( b - a \, \, \mathrm{\cos} \left( \theta \right) \right)$ is constant, so its rate of change is 0. Thus:

\begin{displaymath}0 = \frac{\mathrm{d}}{\mathrm{d} \theta} \left( \frac{b^2 - a... ...- a \, \, \mathrm{\cos} \left( \theta \right) \right) \right) \end{displaymath}

\begin{displaymath}= \left( \frac{\mathrm{d}}{\mathrm{d} \theta} \left( \frac{b^... ...left( b - a \, \, \mathrm{\cos} \left( \theta \right) \right) \end{displaymath}

\begin{displaymath}= \left( \frac{\mathrm{d}}{\mathrm{d} \theta} \left( \frac{b^... ...a \right) \right)} a \: \mathrm{\sin} \left( \theta \right) . \end{displaymath}

So we find:

\begin{displaymath}\frac{\mathrm{d} r}{\mathrm{d} \theta} = \frac{\mathrm{d}}{\m... ... b - a \, \, \mathrm{\cos} \left( \theta \right) \right)^2} . \end{displaymath}

To compare this formula for $\frac{\mathrm{d} r}{\mathrm{d} \theta}$ for an ellipse with the formula for $\left( \frac{\mathrm{d} r}{\mathrm{d} \theta} \right)^2$ for the planet's orbit that we obtained above from de Maupertuis's principle, we square both sides of the ellipse formula, and then use the above relation between $r$ and $\theta$ for the ellipse to express the right-hand side in terms of $r$ instead of $\theta$:

\begin{displaymath}\left( \frac{\mathrm{d} r}{\mathrm{d} \theta} \right)^2 = \fr... ...\theta \right) \right)^2 \right)}{\left( b^2 - a^2 \right)^2} \end{displaymath}

\begin{displaymath}= \frac{r^2 \left( 4 a^2 r^2 - \left( 2 br - \left( b^2 - a^2... ... - \frac{4 r^4}{b^2 - a^2} + \frac{4 br^3}{b^2 - a^2} - r^2 . \end{displaymath}

This exactly matches the formula for $\left( \frac{\mathrm{d} r}{\mathrm{d} \theta} \right)^2$ for the planet's orbit that we obtained above from de Maupertuis's principle, if $\frac{2 Em_p}{J^2} = - \frac{4}{b^2 - a^2}$ and $\frac{2 Gm_S m^2_p}{J^2} = \frac{4 b}{b^2 - a^2}$, so that $b = \frac{Gm_S m_p}{\left\vert E \left\vert \right. \right.}$, and $a = \sqrt{\frac{G^2 m^2_S m^2_p}{E^2} - \frac{2 J^2}{\left\vert E \left\vert m_p \right. \right.}}$, where $\left\vert E \right\vert$ denotes the absolute value of $E$. The value of $E = T + V$ is negative because the planet is gravitationally bound to the Sun.

The time taken for the planet to complete one orbit is called the orbital period, and I shall represent it by $P$. We found above that the rate at which the straight line from the Sun to the planet sweeps out area is $\frac{1}{2} r^2 \frac{\mathrm{d} \theta}{\mathrm{d} t} = \frac{J}{2 m_p}$. Thus

\begin{displaymath}P = \frac{2 m_p}{J} A, \end{displaymath}

where $A$ represents the area enclosed by the orbit. To calculate $A$, it is helpful to use Cartesian coordinates centred at the centre of the ellipse, halfway between the two focuses. Then with $x_1$ representing the coordinate parallel to the line between the two focuses and $x_2$ representing the coordinate perpendicular to this line in the plane of the ellipse, the distances from the point $x = \left( x_1, x_2 \right)$ to the focuses of the ellipse are $\sqrt{\left( x_1 + \frac{a}{2} \right)^2 + x^2_2}$ and $\sqrt{\left( x_1 - \frac{a}{2} \right)^2 + x^2_2}$ by Pythagoras, so the equation of the ellipse is:

\begin{displaymath}\sqrt{\left( x_1 + \frac{a}{2} \right)^2 + x^2_2} + \sqrt{\left( x_1 - \frac{a}{2} \right)^2 + x^2_2} = b. \end{displaymath}

Squaring both sides and rearranging, this becomes:

\begin{displaymath}2 \sqrt{\left( x^2_1 + x^2_2 + \frac{a^2}{4} + x_1 a \right) ... ...ht)} = b^2 - 2 \left( x^2_1 + x^2_2 + \frac{a^2}{4} \right) . \end{displaymath}

Squaring both sides of this and rearranging, it becomes:

\begin{displaymath}\frac{x^2_1}{b^2} + \frac{x^2_2}{\left( b^2 - a^2 \right)} = \frac{1}{4} . \end{displaymath}

If we rewrite this in terms of a rescaled $x_2$ coordinate $x_2'$ such that $x_2 = \sqrt{\frac{b^2 - a^2}{b^2}} x'_2$, it becomes the equation of a circle of radius $\frac{b}{2}$, with area $\frac{\pi}{4} b^2$. Thus the actual area of the ellipse is:

\begin{displaymath}A = \sqrt{\frac{b^2 - a^2}{b^2}} \frac{\pi}{4} b^2 = \frac{\pi}{4} b \sqrt{b^2 - a^2} . \end{displaymath}

Thus $P = \frac{\pi m_p}{2 J} b \sqrt{b^2 - a^2}$. We found above that $\frac{2 Gm_S m^2_p}{J^2} = \frac{4 b}{b^2 - a^2}$, so:

\begin{displaymath}P = \pi \sqrt{\frac{b^3}{2 Gm_S}} . \end{displaymath}

$b$ is the length of the major axis of the ellipse, so this shows that the square of the orbital period $P$ is equal to the cube of the length of the major axis of the orbit, multiplied by a quantity that is the same for all the planets. This is Kepler's third law of planetary motion.

One of the clues that led to the discovery of Dirac-Feynman-Berezin sums came from the attempted application to electromagnetic radiation of discoveries about heat and temperature. In the next post in this series, Multiple Molecules, we'll look at some of those discoveries.

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