American Journal of Physics, Vol. 72, No. 4, pp. 428–435, April 2004
©2004 American Association of Physics Teachers. All rights reserved.
Symmetries and conservation laws:^{ }Consequences of Noether's theorem
Technical University, Vysokoskolska 4, 042 00 Kosice, Slovakia
Slavomir Tuleja^{b)}
Gymnazium arm.^{ }gen. L. Svobodu, Komenskeho 4, 066 51 Humenne, Slovakia
Martina Hancova^{c)}
P. J. Safarik University,^{ }Jesenna 5, 040 11 Kosice, Slovakia
Received: 30 December 2002; accepted: 23 May 2003We derive conservation^{ }laws from symmetry operations using the principle of least action.^{ }These derivations, which are examples of Noether's theorem, require only^{ }elementary calculus and are suitable for introductory physics. We extend^{ }these arguments to the transformation of coordinates due to uniform^{ }motion to show that a symmetry argument applies more elegantly^{ }to the Lorentz transformation than to the Galilean transformation. ©^{ }2004 American Association of Physics Teachers. ^{ }
Contents
I. INTRODUCTION
"It is increasingly clear^{ }that the symmetry group of nature is the deepest thing^{ }that we understand about nature today" (Steven Weinberg).^{1} Many of^{ }us have heard statements such as for each symmetry operation^{ }there is a corresponding conservation law. The conservation of momentum^{ }is related to the homogeneity of space. Invariance under translation^{ }in time means that the law of conservation of energy^{ }is valid. Such statements come from Noether's theorem, one of^{ }the most amazing and useful theorems in physics. ^{ }
When the^{ }German mathematician Emmy Noether proved her theorem,^{2}^{,}^{3} she uncovered the^{ }fundamental justification for conservation laws. This theorem tells us that^{ }conservation laws follow from the symmetry properties of nature. Symmetries^{ }(called "principles of simplicity" in Ref. 1) can be regarded^{ }as a way of stating the most fundamental properties of^{ }nature. Symmetries limit the possible forms of new physical laws.^{ }The deep connection between symmetry and conservation laws requires the^{ }existence of a minimum principle in nature: the principle of^{ }least action. In classical mechanics, symmetry arguments are developed using^{ }high level mathematics. On the other hand, the corresponding physical^{ }ideas often are much easier to understand than the mathematical^{ }derivations. ^{ }
In this paper we give an elementary introduction to^{ }the relation between symmetry arguments and conservation laws, as mediated^{ }by the principle of least action. We shall use only^{ }elementary calculus so that our approach can be used in^{ }introductory university physics classes. ^{ }
Because the paper deals mainly with^{ }symmetry, it is important how we define or characterize this^{ }concept in the framework of introductory physics. We adopt Feynman's^{ }simple description of symmetry from his lectures on physics,^{4} which^{ }says that anything is symmetrical if one can subject it^{ }to a certain operation and it appears exactly the same^{ }after the operation. ^{ }
Like Feynman, we will concentrate on symmetry^{ }in physical laws. The question is what can be done^{ }to a physical law so that this law remains the^{ }same. Noether's theorem derives conservation laws from symmetries under the^{ }assumption that the principle of least action governs the motion^{ }of a particle in classical mechanics. This principle can be^{ }phrased as "The action is a minimum for the path^{ }(worldline) taken by the particle,"^{5} which leads to the reformulation^{ }of our basic question about symmetry: What changes can we^{ }make in the worldline that do not lead to changes^{ }in either the magnitude or the form of the action?^{ }^{ }
We will explore and apply symmetry operations to the action^{ }along an infinitesimally small path segment. Because the action is^{ }additive, conclusions reached about a path segment apply to the^{ }entire path. The simplest examples of symmetry show the independence^{ }of the action on the difference in some quantity such^{ }as position, time, or angle.^{6} When such a symmetry exists,^{ }Noether's theorem tells us that a physical quantity corresponding to^{ }this symmetry is a constant of the motion that does^{ }not change along the entire path of the particle.^{7} The^{ }existence of such a constant implies a conservation law, which^{ }we then identify. ^{ }
Section II briefly describes our software that^{ }helps students study the action and its connection to conservation^{ }laws. Section III analyzes four examples of symmetry operations: translation^{ }in space and time, rotation through a fixed angle, and^{ }symmetry under uniform linear motion, namely the Galilean transformation. The^{ }first three symmetries lead to three conservation laws: momentum, energy,^{ }and angular momentum. Section IV extends the analysis to symmetry^{ }in relativity, showing that these conservation laws exist in that^{ }realm. Moreover, for uniform linear motion the symmetry argument applies^{ }more elegantly to the Lorentz transformation than to the Galilean^{ }transformation. ^{ }
In the following we often talk about variations or^{ }changes in the action. Consistent with standard practice, we will^{ }only be interested in variations representing infinitesimal firstorder changes in^{ }the action. To keep the arguments simple, we also assume^{ }that the particle's invariant mass m (rest mass) does not^{ }change during the motion to be studied. ^{ }
II. SOFTWARE
We start with the wellknown definition of action for^{ }a particle of mass m that moves from some initial^{ }position at time t_{1} to some final position at time^{ }t_{2}:
or equivalently
Here KE_{av} denotes the time averaged kinetic^{ }energy and PE_{av} the time averaged potential energy between t_{1}^{ }and t_{2}. We use the notation KE and PE as^{ }symbols for kinetic and potential energies, respectively, because they are^{ }more mnemonic than the traditional symbols T and V. ^{ }
Action^{ }is not a familiar quantity^{8} for many students, so we^{ }employ an interactive computer program^{9} to help them develop an^{ }intuition about the nature of the action and the principle^{ }of least action. By using an interactive computer display, the^{ }student cannot only explore the operation of the principle of^{ }least action, but also study the relation between this principle^{ }and conservation laws in specific cases (Fig. 1). In carrying^{ }out this manipulation, the student naturally works with the central^{ }concepts of a worldline (a graph of the time dependence^{ }of a particle's position) and an event (a point on^{ }a worldline). Unlike the trajectory in space, the worldline specifies^{ }completely the motion of a particle. For background on the^{ }symmetry properties of nature, we suggest that our students read^{ }a selection from Ref. 10. ^{ }
Figure 1. III. SYMMETRY AND CONSERVATION LAWS IN NEWTONIAN MECHANICS
A.^{ }Translation in space
We first examine the symmetry related to translation^{ }in space. When we perform an experiment at some location^{ }and then repeat the same experiment with identical equipment at^{ }another location, then we expect the results of the two^{ }experiments to be the same. So the physical laws should^{ }be symmetrical with respect to space translation. ^{ }
As a simple^{ }example, consider the action of a free particle (in zero^{ }potential or uniform potential) moving along the x axis between^{ }two events 1 [t_{1},x_{1}] and 2 [t_{2},x_{2}] infinitesimally close to one another^{ }along its worldline. Because the worldline section is considered to^{ }be straight, the particle moves at constant velocity v = (x_{2}–x_{1})/(t_{2}–t_{1}) and^{ }therefore with a constant kinetic energy (1/2)mv^{2}. According to Eq.^{ }(1b), the action along this straight segment in zero potential^{ }is (the consideration for uniform potential is analogous)
^{ }
If we^{ }change the positions of both observed events by a fixed^{ }displacement a, the action remains unchanged (invariant), because the value^{ }of the action depends only on the difference between the^{ }positions: x_{2} + a–(x_{1} + a) = x_{2}–x_{1}. The principle of least action is symmetrical with^{ }respect to a fixed displacement of the position. Noether's theorem^{ }implies that this symmetry is connected with a conservation law.^{ }In the following, we demonstrate that the conservation law related^{ }to symmetry under space translation is conservation of momentum. ^{ }
1.^{ }Principle of least action and momentum
Think of the motion of^{ }a free particle along the x axis. To explore the^{ }connection between the principle of least action and the conservation^{ }of momentum, we take advantage of the additive property of^{ }the action to require that the action along an arbitrary^{ }infinitesimal section of the true worldline have a minimal value.^{11}^{ }Thus we consider three successive infinitesimally close events, 1, 2,^{ }and 3 on the particle's worldline and approximate a real^{ }worldline by two connected straight segments, A and B (see^{ }Fig. 2). ^{ }
Figure 2. Because we are considering translation in space, we^{ }fix the first and last events, 1 and 3, and^{ }change the space coordinate x_{2} of the middle event 2^{ }so as to minimize the value of the total action^{ }S. This minimum condition corresponds to a zero value of^{ }the derivative of S with respect to x_{2}:
^{ }
Because the^{ }action is an additive quantity, the total action equals the^{ }sum of the actions for segments A and B, so^{ }S = S(A) + S(B). If we use Eq. (2), we can write
^{ }
If^{ }we perform in Eq. (4) the derivative indicated in Eq.^{ }(3), we obtain the condition:
The expression on the lefthand^{ }side of Eq. (5) is the momentum p_{A} for segment^{ }A while the expression on the righthand side is the^{ }momentum p_{B} for segment B, so p_{A} = p_{B}. We could continue^{ }and add other segments C, D, E, to cover the^{ }entire worldline that describes the particle motion. For all these^{ }segments the momentum will have the same value, which yields^{ }the conservation law of momentum. The action for this free^{ }particle depends only on the change of the coordinate x^{ }and the result of this dependence is the conservation of^{ }the particle's momentum. ^{ }
However, this derivation uses only the displacement^{ }of one event on the worldline. Therefore, we have not^{ }yet demonstrated the relation between the conservation of momentum and^{ }the symmetry of translation in space in which all three^{ }events are displaced. ^{ }
2. Symmetry and the conservation of momentum
Now^{ }we show the straightforward relation between the symmetry of translation^{ }in space and conservation of momentum. Again consider three infinitesimally^{ }close events on the worldline x(t) of the free particle^{ }shown in Fig. 3 (the extension to the entire worldline^{ }will be discussed later). ^{ }
Figure 3. We shift the worldline x(t) so^{ }that every event changes its position by a fixed infinitesimal^{ }displacement a. The new events create a shifted worldline which^{ }we indicate by an asterisk: x^{*}(t). As pointed out, the^{ }form of the action for x^{*}(t) remains unchanged and does^{ }not depend on the parameter a. Thus the change in^{ }action with respect to the displacement a is zero:
^{ }
Note^{ }that the worldline x^{*}(t) is just as valid as the^{ }original one. Therefore the worldline x^{*}(t) also obeys the principle^{ }of least action. In translating from x(t) to x^{*}(t) we^{ }do not need to shift all the events simultaneously. The^{ }same effect is obtained if we first change the position^{ }of event 1 (in Fig. 3 only coordinate x_{1} changes,^{ }which creates the worldline 1^{*}23), then event 3 (only x_{3}^{ }changes, which creates 1^{*}23^{*}) and finally event 2 (only x_{2}^{ }changes, which creates 1^{*}2^{*}3^{*}). The total change in action for^{ }displacement a can be written as:
where S_{11*}, S_{22*}, S_{33*}^{ }denotes the changes in the action after the shifts in^{ }the corresponding events. ^{ }
Equation (6) tells us that _{a}S is^{ }always zero. The final change S_{22*} must also be zero,^{ }from the principle of least action applied to the new^{ }worldline. Hence Eqs. (6) and (7) give
If we now^{ }calculate the changes in the action in Eq. (8), we^{ }obtain the conservation law of momentum. Because the displacement a^{ }is infinitesimal, we can write:
If we substitute Eq. (9)^{ }into Eq. (8) and use the fact that the fixed^{ }infinitesimal displacement a is arbitrary, we have^{12}
^{ }
The application of^{ }the derivatives in Eq. (10) to the expression for the^{ }action in Eq. (4) yields the identical result for a^{ }free particle as Eq. (5), but this time as a^{ }result of spatial translation of the entire incremental worldline segment.^{ }Thus the lefthand side of Eq. (10) can also be^{ }interpreted as the momentum at event 1 and the righthand^{ }side as the momentum at event 3. ^{ }
The preceding considerations^{ }can be applied to the entire worldline x(t). We did^{ }not specify the location of the segments A and B.^{ }Therefore, an arbitrary number of additional segments can be added^{ }between them. Then we shift the segments as before (see^{ }Fig. 4). By the same analysis we conclude that the^{ }momentum for segment A (effectively the momentum at event 1)^{ }has the same value as for segment B (effectively at^{ }event 3). The arbitrariness of position of these segments on^{ }the worldline means that the value of the momentum remains^{ }constant at every event on the worldline. Thus, in classical^{ }mechanics, the symmetry of spatial translation means that momentum is^{ }conserved for a free particle. ^{ }
Figure 4. The invariance of the action^{ }with respect to translation in space is also called the^{ }homogeneity of space, which means that all points in space^{ }are equivalent. In other words, it does not matter where^{ }an experiment is performed. Therefore, we can state that the^{ }law of momentum conservation results from the homogeneity of space.^{ }^{ }
B. Translation in time
It is easy to envision the symmetry^{ }related to translation in time. Repeating an experiment on identical^{ }initial systems yields the same result when the two experiments^{ }are separated by a lapse of time. Our conclusion is^{ }that physical laws should not change with translation in time.^{ }^{ }
Again we will show the relation of translation in time^{ }symmetry to a relevant conservation law. We start with an^{ }expression for the action of a particle moving in the^{ }x direction along an infinitesimally small worldline segment in a^{ }potential field described by PE(x). As in Sec. III A the^{ }action for this segment can be written according to Eq.^{ }(1b) as
where the potential energy is evaluated at the^{ }average position along the segment. Now suppose that we translate^{ }the time t by an amount . It is easy^{ }to see that the action will not change, because only^{ }the difference of the time, t_{2} + –(t_{1} + ) = t_{2}–t_{1}, appears in the equation^{ }for the action. So the action is symmetrical with respect^{ }to a fixed displacement of time t. What conservation law^{ }is related to this time symmetry? We will show that^{ }it is conservation of energy. ^{ }
We follow the same line^{ }of reasoning as for the case of translation in space,^{ }but now we fix all position and time coordinates with^{ }the exception of t_{2}. Think of a particle that moves^{ }along the x axis in the potential field with potential^{ }energy PE(x). To simplify the algebra, we denote space and^{ }time differences by
According to Eqs. (11) and (12), the^{ }values of the actions S(A) and S(B) for segments A^{ }and B are equal to
The principle of least action^{ }leads to the following condition for the total action S:^{ }
If we substitute Eq. (13) into Eq. (14), differentiate, and^{ }rearrange the terms, we obtain
The expressions on both sides^{ }of Eq. (15) are sums of average kinetic and potential^{ }energies. For infinitesimally close events, Eq. (15) gives an equality^{ }for the instantaneous values (1/2)mv + PE_{A} = (1/2)mv + PE_{B}, and expresses the conservation of^{ }mechanical energy. ^{ }
Next we carry out an argument that translates^{ }all three times t_{1}, t_{2}, and t_{3} by the same^{ }amount , similar to the way we translated positions for^{ }the momentum case. Equations (6,7,8) apply to the present case^{ }as well, and also Eq. (9) when the derivatives are^{ }taken with respect to time rather than position. Then the^{ }result of the temporal translation is an equation similar to^{ }Eq. (10):
which yields Eq. (15) multiplied by (–1). We^{ }again obtain conservation of energy, but this time as a^{ }result of symmetry under time translation. For infinitesimally close events,^{ }the lefthand side of Eq. (16) also can be interpreted^{ }as the negative of the total energy at event 1^{ }and the righthand side as the negative of the energy^{ }at event 3. The energy is a constant of the^{ }motion for the entire worldline x(t). Similar to the last^{ }paragraph of Sec. III A, we can say that the symmetry^{ }of translation in time, or in other words the homogeneity^{ }of time, implies conservation of energy. ^{ }
C. Rotation through a^{ }fixed angle
We now trace the implications of another symmetry, symmetry^{ }under rotation in space. If we rotate an experimental setup^{ }through a fixed angle, the experiment will yield the same^{ }result. If this symmetry were not true, a laboratory in^{ }New York would not be able to verify what is^{ }measured in another laboratory in Los Angeles. Indeed, repeating the^{ }experiment in New York must lead to the same results^{ }as the earth rotates. So physical laws should remain invariant^{ }with respect to rotation. ^{ }
We use polar coordinates to determine^{ }which conservation law corresponds to this symmetry and consider the^{ }planar motion of a particle in a spherically symmetric potential^{ }field of energy PE(r). As before, we consider the expression^{ }for the action along the infinitesimal segment. The definition (1b)^{ }shows that the action is equal to
The increment s^{ }is the length of a path segment traveled by the^{ }particle during the time interval t and r_{av} is the^{ }average position of the particle on this segment. ^{ }
Consider three^{ }infinitesimally close points on the real path of a particle^{ }and approximate the real path by a broken line consisting^{ }of two infinitesimally small segments A and B (Fig. 5).^{ }(In this case we do not display a worldline because^{ }it would require curves in threedimensional space–time.) To find the^{ }required expression for the action in polar coordinates, we use^{ }the Pythagorean theorem. The infinitesimal lengths s_{A} and s_{B} of^{ }segments A and B are
where r_{A} = r_{2}–r_{1}, _{A} = _{2}–_{1}, r_{B} = r_{3}–r_{2}, and^{ }_{B} = _{3}–_{2}. If we substitute Eq. (18) into Eq. (17), we^{ }find values of the action for segments A, B:
Once^{ }again, note that the action for these two segments depends^{ }only on the difference in the coordinate, and not^{ }on the coordinate itself. As before, we conclude that^{ }neither S(A) nor S(B) will change as we increase all^{ } coordinates by a fixed angle , because _{2} + –(_{1} + ) = _{2}–_{1}. As^{ }a result, the motion of the particle is symmetrical with^{ }respect to a fixed change in angle . Conservation of^{ }angular momentum which arises from this symmetry is derived as^{ }follows. ^{ }
Figure 5. The condition of stationary action S is expressed as:^{ }
We substitute Eq. (19) into Eq. (20), differentiate and do^{ }some rearrangement and obtain:
Equation (21) represents conservation of angular^{ }momentum L, so L_{A} = L_{B}. The rate of change of the^{ }angle is the angular velocity . Thus Eq. (21) can^{ }be expressed as mr_{A} = mr_{B}. ^{ }
A derivation dealing with the fixed^{ }change in the angle coordinate for all three events, similar^{ }to those of the previous cases of translations in space^{ }and time, yields
which immediately implies conservation of angular momentum^{ }(21). Moreover, the lefthand side of Eq. (22) can be^{ }interpreted as the angular momentum at point 1 and the^{ }righthand side as the angular momentum at point 3. Angular^{ }momentum is conserved for the entire path. The result is^{ }that symmetry under rotation through a fixed angle implies conservation^{ }of angular momentum. ^{ }
The condition that physical laws remain invariant^{ }with respect to rotation through a fixed angle is called^{ }the isotropy of space. That is, space has the same^{ }properties in every direction. Therefore conservation of angular momentum results^{ }from the isotropy of space. ^{ }
D. Galilean transformation
Finally, we present^{ }a simple example of an interesting and very important symmetry:^{ }symmetry under uniform linear motion, known in classical mechanics as^{ }Galileo's principle of relativity. We will be surprised to find^{ }that the classical action is not invariant under a Galilean^{ }transformation. ^{ }
Consider again a free particle moving along the x^{ }axis between closely adjacent events 1 and 2 as observed^{ }in a laboratory frame, where the action takes the form^{ }(2). The (slowly moving) rocket observer, moving with a velocity^{ }v_{rel} with respect to the laboratory, calculates the particle's action^{ }given by the same equation
Here we use primes for^{ }rocket coordinates, not for the derivative. If we apply the^{ }Galilean transformation
for the rocket coordinates to Eq. (23), we^{ }obtain the following form of the action S^{} in the^{ }laboratory frame:
This form of action is not the same^{ }as Eq. (2). The action is not invariant under a^{ }Galilean transformation. Which action, S in Eq. (2) or S^{}^{ }in Eq. (25), governs the motion of the particle in^{ }the laboratory? Or is the Galilean transformation incorrect? According to^{ }Appendix A everything is consistent. The two actions S and^{ }S^{} differ by a function that depends only on the^{ }coordinates of a given event, F(x,t) = –v_{rel}mx+mvt, so the mechanical laws^{ }are the same as determined by using S as they^{ }are by using S^{}. ^{ }
If we use slightly more general^{ }considerations, but reasoning similar to that employed previously,^{13} we can^{ }demonstrate that the corresponding conservation law to Galilean transformation (24)^{ }is related to the uniform motion of the center of^{ }mass. ^{ }
IV. SYMMETRY AND CONSERVATION LAWS IN RELATIVITY
A.^{ }Action in relativity
We have shown that the classical action is^{ }not symmetrical with respect to uniform linear motion, but all^{ }laws of motion remain unchanged under a Galilean transformation. We^{ }believe that this asymmetry for the principle of least action^{ }is not accidental, but rather results from the fact that^{ }the Galilean transformation and Newton's laws are only approximate laws^{ }of motion. Symmetry under uniform linear motion is a basic^{ }assumption of Einstein's special relativity. ^{ }
We consider the same free^{ }particle, but now we use the special theory of relativity.^{ }The action for linear segment between 1 and 2 has^{ }the form:^{14}
where c is the velocity of light, t = t_{1}–t_{2},^{ }and v = (x_{2}–x_{1})/(t_{2}–t_{1}). It can be seen from Eq. (26) that^{ }Newtonian mechanics is a special case of relativistic mechanics in^{ }the lowvelocity limit (vc):
^{ }
According to Appendix A, if we^{ }take F(x,t) = –mc^{2}t, Eq. (27) will give the same laws of^{ }motion for a free particle as the classical Newtonian action^{ }in Eq. (2). ^{ }
B. Lorentz transformation
Now we outline the symmetry^{ }argument connected to the relativistic Lorentz transformation which has the^{ }form (c = 1):
where = 1/(1–v)^{1/2}. Here v_{rel} has the same meaning^{ }as in Sec. III D. We express the action (26) along^{ }a segment of the worldline:
The expression in the square^{ }root is the particle's proper time (wristwatch time) between the^{ }two events, which is easily verified to be an invariant^{ }under the Lorentz transformation. Hence the relativistic action is symmetrical^{ }under a transformation connected to uniform linear motion. ^{ }
Noether's theorem^{ }can be used also in relativity. The same procedure used^{ }in Sec. III can be repeated in special relativity to^{ }yield the laws of conservation of relativistic energy, momentum, and^{ }angular momentum:
where is the particle's proper time. As^{ }for the Lorentz transformation, there also exists a corresponding conservation^{ }law, but its derivation goes beyond the scope of this^{ }paper.^{15} ^{ }
We see that the theory of relativity eliminates the^{ }asymmetry of the action under translation. The invariance of the^{ }action under all the transformations we have considered makes the^{ }theory of relativity a more beautiful and elegant theory than^{ }the Newtonian theory of classical mechanics. ^{ }
If one uses the^{ }correct expression for the action (or proper time), the constants^{ }of motion also can be derived for general relativity without^{ }complicated or advanced mathematics.^{16} ^{ }
V. SUMMARY
We^{ }have discussed the connection between symmetries and conservation laws provided^{ }by Noether's theorem using only elementary calculus. This approach can^{ }be used to help familiarize students with the powerful consequences^{ }of symmetry in the physical world. In addition, students can^{ }see a unified and systematic approach to all the conservation^{ }laws, mediated by Noether's theorem and the principle of least^{ }action. ^{ }
All our considerations can be easily generalized to three^{ }dimensions. We note that all symmetries in this paper are^{ }oneparameter transformations, which provide the central conservation laws using the^{ }most common form of Noether's theorem related to the invariance^{ }of the Lagrangian (see Appendix B). Reference 17 and the^{ }pedagogically oriented Refs. 18 and 19 give clear, elegant, and^{ }more mathematically precise (but much more mathematically oriented) applications of^{ }Noether's theorem to particle dynamics. ^{ }
ACKNOWLEDGMENTS
This paper was written after we^{ }read Taylor and Wheeler's general relativity book^{16} sent to us^{ }along with other materials by Taylor, who also made very^{ }helpful suggestions for this paper. The authors also wish to^{ }thank Nilo C. Bobillo Ares for helpful advice. Slavko Chalupka^{ }provided important discussion and encouragement and gave us the opportunity^{ }to teach an experimental course in quantum mechanics using some^{ }of these ideas.^{20} ^{ }
APPENDIX A: THE ADDITION OF CERTAIN TERMS TO THE ACTION^{ }HAS NO EFFECT ON THE LAWS OF MOTION
Think of two^{ }expressions for the action S(12) and S^{*}(12) for a given^{ }worldline between any two events 1 and 2 in space–time.^{ }Suppose that these two expressions are related to each other^{ }as
where F is an arbitrary function that depends only^{ }on the space and time coordinates of a given event.^{ }For example, F(1) could be the value of F at^{ }the event 1. Then laws of motion are the same^{ }for both forms of action. Why? ^{ }
We answer this question^{ }by repeating the same procedure as for earlier symmetries, starting^{ }with three events 1, 2, and 3. If we apply^{ }Eq. (31), we obtain the following equations relating action S^{ }and S^{*} for segment 1–2 and 2–3:
The total action^{ }S^{*}(123) is the sum of (32a) and (32b):
The two^{ }total actions S^{*} and S in Eq. (33) differ only^{ }in the difference in F at the fixed events 3^{ }and 1. If we change the space or time coordinate^{ }(generally u_{2}) of the middle event 2, this difference remains^{ }constant. So the minima of S and S^{*} yield the^{ }same position of event 2, or in other words, the^{ }first derivatives of S and S^{*} with respect to u_{2}^{ }are the same (all other variables being fixed):
^{ }
According to^{ }Eq. (34), the principle of least action for S^{*} gives^{ }the same particle's path as in the case of S.^{ }The laws of motion are unchanged if an additive constant^{ }(the difference in an arbitrary function between final position and^{ }initial position of a particle) is added to the action.^{21}^{ }^{ }
APPENDIX B: NOETHER'S THEOREM AND THE LAGRANGIAN
Noether's theorem^{ }determines the connection between constants of the motion and conditions^{ }of invariance of the action under different kinds of symmetry.^{ }The function KEPE in Newtonian mechanics is called the Lagrangian^{ }and is denoted by the symbol L. So we can^{ }write S_{for segment}S = Lt. (Do not confuse the symbol L for the^{ }action with the symbol L for angular momentum used in^{ }Sec. III C.) If we discuss symmetry transformations such that time^{ }is transformed identically, t^{*} = t, or transformations involving a uniform time^{ }translation, t^{*} = t + , where t = t^{*}, then the invariance of the Lagrangian^{ }implies the invariance of the action. Therefore, most textbooks state^{ }Noether's theorem as: for each symmetry of the Langrangian, there^{ }is a corresponding conserved quantity. ^{ }
REFERENCES
Citation links [e.g., Phys. Rev. D 40, 2172 (1989)] go to online journal abstracts. Other links (see Reference Information) are available with your current login. Navigation of links may be more efficient using a second browser window.

R. P. Feynman and S. Weinberg, Elementary Particles and the Laws of Physics (Cambridge U.P., Cambridge, 1999), p. 73.
first citation in article

In reality, there are two Noether's theorems and their converses. The
first one refers to the invariance of the action with respect to a
group of symmetries where the symmetry transformations depend
analytically on many arbitrary finite parameters. The second theorem
deals with the invariance of the action with respect to a group for
which the transformations depend on arbitrary functions and their
derivatives instead of on arbitrary parameters. Our paper considers
oneparameter symmetry transformations. Therefore, it is connected with
the first theorem. See E. Noether, "Invariante Variationsprobleme,"
Nachr. v. d. Ges. d. Wiss. zu Göttingen, Mathphys. Klasse, 235–257
(1918);
English translation by M. A. Tavel, "Invariant variation problem," Transport Theory Stat. Mech.1 (3), 183–207 (1971). Both papers are available at http://www.physics.ucla.edu/~cwp/Phase2/Noether,_Amalie_Emmy@861234567.html.
first citation in article

N. Byers, "E. Noether's discovery of the deep connection between
symmetries and conservation laws," Isr. Math. Conf. Proc.12, 67–82 (1999).
first citation in article

R. P. Feynman, R. B. Leighton, and M. Sands, The Feynman Lectures on Physics (Addison–Wesley, Reading, MA, 1963), Vol. I, Chap. 11, p. 111 or Chap. 52, p. 521.
first citation in article

More accurately, the principle says that a particle moves along that
path for which the action has a stationary value. So it is frequently
and correctly called the principle of stationary action. See I. M.
Gelfand and S. V. Fomin, Calculus of Variations (Prentice–Hall, Englewood Cliffs, NJ, 1963), Sec. 32.2 or D. J. Morin, http://www.courses.fas.harvard.edu/~phys16/handouts/textbook/ch5.pdf, Chap. 5.
first citation in article

Generally such a quantity is called a cyclic or ignorable coordinate; H. Goldstein, Classical Mechanics (Addison–Wesley, New York, 1970), p. 48 or Ref. 5.
first citation in article

Every quantity that depends on position coordinates and velocities and
whose value does not change along actual trajectories is called a
constant of the motion. first citation in article

We recommend a more detailed described procedure for introducing action
in J. Hanc, S. Tuleja, and M. Hancova, "Simple derivation of Newtonian
mechanics from the principle of least action," Am. J. Phys. 71, 386–391 (2003). [ISI]
first citation in article

The idea of using computers comes from E. F. Taylor. See E. F. Taylor,
S. Vokos, J. M. O'Meara, and N. S. Thornber, "Teaching Feynman's sum
over paths quantum theory," Comput. Phys. 12 (2), 190–199 (1998)
or E. F. Taylor, Demystifying Quantum Mechanics, http://www.eftaylor.com. Our software is based on Taylor's.
first citation in article

R. P. Feynman, The Character of Physical Law (Random House, New York, 1994), Chap. 4.
first citation in article

Reference 4, Vol. II, Chap. 19, p. 198 or the more detailed discussion in Ref. 8.
first citation in article

Strictly speaking, in these and the following cases we should use the
more traditional notation of partial instead of total derivatives. But
in all cases it is clear which coordinates are variable and which are
fixed. first citation in article

In that case it is necessary to consider the invariance of the action
up to an additive constant (the difference in any arbitrary function
between final position and initial position of a particle), which will
give conservation of motion of the center of mass. See also Refs. 17 or
18. first citation in article

The relativistic formula for the action is given in Ref. 4, Vol. II,
Chap. 19. We use the concept of invariance of mass that is used by E.
F. Taylor and J. A. Wheeler, in Spacetime Physics: Introduction to Special Relativity (W. H. Freeman, New York, 1992), 2nd ed.
first citation in article

The conservation law corresponding to the Lorentz transformation is derived in L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields (Pergamon, London, 1975), Vol. 2, pp. 41–42.
first citation in article

E. F. Taylor and J. A. Wheeler, Exploring Black Holes: An Introduction to General Relativity (Addison–Wesley Longman, New York, 2000), Chaps. 1 and 4; also available at http://www.eftaylor.com. The authors use a very similar, easy, and effective variational method.
first citation in article

P. Havas and J. Stachel, "Invariances of approximately relativistic Lagrangians and the center of mass theorem. I," Phys. Rev. 185 (5), 1636–1647 (1969). [ISI]
first citation in article

N. C. BobilloAres, "Noether's theorem in discrete classical mechanics," Am. J. Phys. 56 (2), 174–177 (1988).
first citation in article

C. M. Giordano and A. R. Plastino, "Noether's theorem, rotating potentials, Jacobi's integral of motion," Am. J. Phys. 66 (11), 989–995 (1998).
first citation in article

The substance of this article was used by the authors as subjects for
student projects dealing with a special topic on the principle of least
action in a semester quantum mechanics course for future teachers of
physics at the Faculty of Science, P. J. Safarik University, Kosice,
Slovakia. To obtain our materials and corresponding software, see http://leastaction.topcities.com (the mirror site http://www.LeastAction.host.sk) or see Edwin Taylor's website: http://www.eftaylor.com/leastaction.html, which also includes our newest, continually updated and expanded materials.
first citation in article

L. D. Landau and E. M. Lifshitz, Mechanics (Butterworth–Heinemann, Oxford, 1976), Sec. 1.2.
first citation in article
FIGURES
Full figure (27 kB)Fig. 1. The use of software helps students study the action^{ }along a worldline for a particle moving vertically in a^{ }gravitational field (as shown) or in other conservative potentials. The^{ }user clicks on events to create a worldline and then^{ }drags the events to minimize the action, which the computer^{ }continuously calculates and displays. The computer also displays a table^{ }of energy, momentum, or other quantities that demonstrate conservation of^{ }these quantities. Students discover that for the worldline of minimum^{ }action, momentum is conserved for the motion of a free^{ }particle and that in a gravitational field total energy is^{ }conserved. First citation in article
Full figure (6 kB)Fig. 2. Segment of the worldline of a particle that passes through^{ }three infinitesimally close events, for which every smooth curve can^{ }be approximated by two connected straight segments. First citation in article
Full figure (5 kB)Fig. 3. Three infinitesimally close events^{ }1, 2, 3 on the actual worldline. We shift this^{ }worldline through a fixed infinitesimal displacement a. An arbitrary displacement^{ }can be composed from a sequence of such infinitesimal displacements. First citation in article
Full figure (7 kB)Fig. 4. Following^{ }the same analysis as before, we conclude that the momentum^{ }at event 1 is the same as at event 3.^{ }The events 1 and 3 can be chosen arbitrarily. The^{ }arbitrariness of position of these events on the worldline implies^{ }the same value of momentum at every point (event) along^{ }the whole worldline of the moving object. First citation in article
Full figure (6 kB)Fig. 5. Path segment of planar^{ }motion with three infinitesimally close points whose positions are described^{ }by polar coordinates. The radius r_{A}(r_{B}) represents the average position^{ }of the particle on segment A(B). All coordinates of the^{ }points 1, 2, 3 are fixed with the exception of^{ }the angle coordinate _{2}, which we vary to satisfy the^{ }principle of least action. First citation in article
FOOTNOTES
^{a}Electronic mail: jozef.hanc@tuke.sk^{ }
^{b}Electronic mail: tuleja@stonline.sk
^{c}Electronic mail: hancova@science.upjs.sk
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