Conservation of momentum from the Lagrangian formulation

[collectedsoul]

I'm going through John Taylor's book on Classical Mechanics and am having some difficulty understanding his derivation of momentum conservation in the Lagrangian section. Firstly he starts the section off with assuming that all N particles of a system are moved in space by a distance \epsilon. My initial reaction to this was why is he saying ALL particles are moved by the same distance? In a collision between particles of a system each particle would be displaced by a different distance - so why is he generalizing?

Then he says that the velocities remain unchanged by the translation. But if the velocities are unchanged then how is this a case where the particles are colliding? And if they are not colliding then how is this case a parallel of the Newtonian momentum conservation where the particles do collide but their total momentum is conserved.

Basically I'm not getting how this approach of simply translating all particles by a displacement \epsilon is a valid approach to analysing conservation of momentum. What am I missing?

 

[vanhees71]

The conservation of momentum is a special case of Noether's theorem, which says that for any symmetry of the action, describing a dynamical system, there's a conserved quantity. In Hamiltonian formulation the conserved quantity is given by the generator of the canonical transformation, describing the symmetry transformation. Thus also the other way works: Each conserved quantity defines a symmetry transformation of the action.

Now, Newtonian space-time has the Galilei group as its symmetry group, which is a ten-dimensional Lie group, and thus you have 10 conserved quantities for any closed system:

Time-translation invariance: Energy
Space-translation invariance (3 independent parameters, namely the translation in three linearly independent directions of space): momentum vector
Rotations (3 dimensional group SO(3)): angular-momentum vector
Boosts (3 dimensional group): velocity of the center of mass

 

[collectedsoul]

The conservation of momentum is a special case of Noether's theorem, which says that for any symmetry of the action, describing a dynamical system, there's a conserved quantity. In Hamiltonian formulation the conserved quantity is given by the generator of the canonical transformation, describing the symmetry transformation. Thus also the other way works: Each conserved quantity defines a symmetry transformation of the action.

Now, Newtonian space-time has the Galilei group as its symmetry group, which is a ten-dimensional Lie group, and thus you have 10 conserved quantities for any closed system:

Time-translation invariance: Energy
Space-translation invariance (3 independent parameters, namely the translation in three linearly independent directions of space): momentum vector
Rotations (3 dimensional group SO(3)): angular-momentum vector
Boosts (3 dimensional group): velocity of the center of mass

I got most of the first paragraph but you lost me after that. So I have to know Noether's theorem to understand this?

Can't it be explained solely from an understanding of Newtonian mechanics?

 

[vanhees71]

Noether's theorem is not so hard to grasp. Take the translations as an example and consider a Lagrangian that is translation invariant. That means that for any vector, \delta \vec{a},

\delta L=\delta L(\vec{x}_j+\delta \vec{a},\dot{\vec{x}})-\delta L(\vec{x}_j,\dot{\vec{x}})=\delta \vec{a} \sum_{j} \frac{\partial L}{\partial \vec{x}_j}=0.

Since this is valid for any \delta \vec{a}, this means that for the solution of the equations of motion,

\frac{\partial L}{\partial \vec{x}_j}-\frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial L}{\partial \dot{\vec{x}}_j}=0,

the total momentum,

\vec{P}=\sum_j \vec{p}_j=\sum_j \frac{\partial L}{\partial \dot{\vec{x}}_j},

is a conserved quantity.

 

[collectedsoul]

I don't get how the third step follows. Why is the momentum a conserved quantity?

On a different track, I'm studying for the Physics GRE test - am I expected to know Hamiltonian momentum, energy conservation in detail? The syllabus mentions Lagrangian and Hamiltonian but the questions I've seen in the actual exam test only basic knowhow.

 

[vanhees71]

Sum the equations of motion over j, and you get

\frac{\mathrm{d}}{\mathrm{d} t} \vec{P} = \frac{\mathrm{d}}{\mathrm{d} t} \sum_{j} \vec{p}_j=\sum_{j} \frac{\partial L}{\partial \vec{x}_j}.

But from the symmetry under translations, we have just proven before that the latter sum vanishes, and thus you have \mathrm{d} \vec{P}/\mathrm{d}t=0.

 

[Cleonis]

I'm going through John Taylor's book on Classical Mechanics and am having some difficulty understanding his derivation of momentum conservation in the Lagrangian section.

I presume you are referring to section 7.8 in John Taylor's book on Classical Mechanics.

From the text I get the impression that John Taylor is talking about conservation in the sense of Newton's First Law.

What is happening is that physicists use the expression 'conservation of momentum' in two different meanings.
1) A conservation in the sense of Newton's First Law
2) Conservation of momentum for two particles that are interacting with each other. (In effect that is covered by the Third Law)

So, whenever you read a textbook, and the author writes in a particular section about 'conservation of momentum' you need figure out which of the above two meanings the author actually has in mind.


In section 7.8 John Taylor is not considering collisions. He may get to collisions later, but in section 7.8 he has confined himself to discussing the total momentum of a system of particles that aren't colliding with each other (or rather, whether the particles collide amongst each other is outside the scope).

The kind of "displacement" that Johh Taylor is writing about is a displacement of the coordinate system.
Since the coordinate system is uniform you can apply any displacement of the coordinate system

The purpose, I suppose, is to show that the Lagrangian formalism encompasses Newton's first law.
Not that anyone doubted that; I guess it's more a warming-up exercise in handling the formula's of the Lagrangian formalism.

 

[collectedsoul]

Sum the equations of motion over j, and you get

\frac{\mathrm{d}}{\mathrm{d} t} \vec{P} = \frac{\mathrm{d}}{\mathrm{d} t} \sum_{j} \vec{p}_j=\sum_{j} \frac{\partial L}{\partial \vec{x}_j}.

But from the symmetry under translations, we have just proven before that the latter sum vanishes, and thus you have \mathrm{d} \vec{P}/\mathrm{d}t=0.

I went through the book again and combined with your help I realized that the conservation follows as a natural consequence of the Lagrangian formulation. Which is amazing. I have more questions now, I'm wondering how did the person who did this first (Lagrange?) know that translating the system would lead to a statement of conservation of momentum. Or did Noether's theorem come first and this relation follow from her theorem? Basically I read a bit about her theorem and was awed by what little I understood. Symmetry = conservation? Amazing. So did Lagrange read her theorem and then come up with his formalism?

I presume you are referring to section 7.8 in John Taylor's book on Classical Mechanics.

From the text I get the impression that John Taylor is talking about conservation in the sense of Newton's First Law.

What is happening is that physicists use the expression 'conservation of momentum' in two different meanings.
1) A conservation in the sense of Newton's First Law
2) Conservation of momentum for two particles that are interacting with each other. (In effect that is covered by the Third Law)

So, whenever you read a textbook, and the author writes in a particular section about 'conservation of momentum' you need figure out which of the above two meanings the author actually has in mind.


In section 7.8 John Taylor is not considering collisions. He may get to collisions later, but in section 7.8 he has confined himself to discussing the total momentum of a system of particles that aren't colliding with each other (or rather, whether the particles collide amongst each other is outside the scope).

The kind of "displacement" that Johh Taylor is writing about is a displacement of the coordinate system.
Since the coordinate system is uniform you can apply any displacement of the coordinate system

The purpose, I suppose, is to show that the Lagrangian formalism encompasses Newton's first law.
Not that anyone doubted that; I guess it's more a warming-up exercise in handling the formula's of the Lagrangian formalism.

Okay...I get what he (John Taylor) is doing now. Thanks for clearing that up. I had assumed he was including collisions in the formulation. So I get now that the conservation of momentum pretty much follows from the Lagrange formulation - but like you specified only in the sense of Newton's first law where external force is 0. But does it (Lagrangian) also work for cases where there are collisions? And do I need to know this for the GRE exam?

 

[Cleonis]

I realized that the conservation follows as a natural consequence of the Lagrangian formulation.

About the relation between symmetry and conservation:

Here is how I understand the gist of Nöthers theorem (as applied to physics). This is my personal opinion. (I believe my interpretation is the mainstream one, but just to be on the safe side I call it my personal opinion.)

When you formulate a theory of motion you formulate concurrently a coordinate system to represent that motion. Thus you have spatial coordinates and a time coordinate, and then velocity is defined as the time derivative of the position coordinates.

Nöthers theorem is about coordinate systems and equations of motion.

The Newtonian formulation uses a uniform coordinate system. (Obviously you're going to use a uniform coordinate system; using a uniform coordinate system is what makes it possible to formulate equations of motion)

Here, uniform coordinate system is meant in the following sense: If you have an infinite grid, then you cannot tell where you are on that grid. Every shift of position ends at a position that is indistinguishable from the previous position.So you have two definitions in place:
- velocity is defined as the time derivative of position
- the coordinate system is uniform.

Those definitions are already enough to imply Newton's First Law.

Consider: with the above two definitions in place, is there any way to get a motion that does not follow Newton's First Law? I don't think there is.
Now to the other type of conservation of momentum:
What about the total momentum of a system of two objects that are interacting with each other? (Attracting each other, or exerting a repulsive force on each other.)

Definition: acceleration, a, is the time derivative of velocity
Law of motion: a = F/m
Definition: the coordinate system is uniformNow some everyday physics:
You want to open a door, but it's somewhat stuck, so that it takes a bit of force to open that door. There's enough strength in your arm, but if pull that door very slowly you will only pull yourself towards the door. So instead of pulling slowly you pull fast. And then the door comes unstuck.
What happens there is a straight example of F=ma, the Second Law.

By pulling very fast you give yourself a big acceleration (for just a fraction of a second) and the bigger the acceleration you give yourself the bigger the force that you exert upon the door. The same applies for a pair of objects that is exerting a force upon each other. For each object: in order to exert a force upon the other object it must be in acceleration itself (just as you have to accelerate yourself in order to exert a force upon that door)

a= F/m says that the object with the largest mass will accelerate the least.
The acceleration will be in proportion, hence the velocities will be in proportion, hence the position of the common center of mass will not change.

So the following two:
Law of motion: a = F/m
Definition: the coordinate system is uniform

Are already enough to imply that for interacting objects (attracting, repulsing, colliding) the total momentum is conserved
The general form of Nöther's theorem states that if you have a theory of motion (which means you have position, velocity and acceleration defined) then for each symmetry of the equations of motion (as defined on the coordinate system used) there is a corresponding conserved quantity in the theory of motion.

 

[vanhees71]

I went through the book again and combined with your help I realized that the conservation follows as a natural consequence of the Lagrangian formulation. Which is amazing. I have more questions now, I'm wondering how did the person who did this first (Lagrange?) know that translating the system would lead to a statement of conservation of momentum. Or did Noether's theorem come first and this relation follow from her theorem? Basically I read a bit about her theorem and was awed by what little I understood. Symmetry = conservation? Amazing. So did Lagrange read her theorem and then come up with his formalism?

The systematic study of symmetry principles is a rather modern concept, although of course the old masters of mechanics like Newton, Lagrange, Hamilton, Euler et al. knew that a good choice of coordinates helps to get simpler equations. E.g., if you have a particle in a central potential, it's most convenient to introduce spherical coordinates around the center of the potential due to the rotational symmetry of the problem. Then you find a lot of "first integrals", i.e., conservation laws mosre easily than when using some other arbitrary coordinates.

A more systematic approach has been developed by Lie, who thought about how to systematically use symmetry concepts to solve differential equations. In this way he came to investigate groups of transformations of equations that depend differentiably on one or more real parameters. Associated with these Lie groups are the Lie algebras which follow from the study of the group around the group identity. In this way you have infinitesimal transformations, which can be applied repeatedly to build up a large transformation. In physics this has been an unknown technique for quite some time. In this case the mathematicians were much more advanced than the physicists.

In physics, symmetry arguments in the modern sense have been first introduced by Einstein in his famous article "On electrodynamics of moving bodies" (1905). In the first sentences of this paper he makes a typical symmetry argument about Maxwell's Equations of electromagnetism which leads him to introduce a new concept of space and time. In other words, in this approach the space-time structure is governed by symmetry principles of observed facts about nature (in this case the independence of induced electromotive forces in Faradays Law of Induction of whether a wire is moved in a magnetic field or the magnet is moved close to the wire).

When Einsteins tried to describe also gravity, the only other then known fundamental force, within his relativistic space-time model, he had to solve a lot of problems. So it took him and his mathematical collaborators (mostly Marcel Grossmann) about 10 years to come up with what is known as the General Theory of Relativity. Here, a geometrical picture of space-time is the most natural approach: GTR can be interpreted as a dynamical theory about the geometrical structure of space-time, and thus the idea to investigate these space-time structures by symmetry principles is very natural.

At this point Emmy Noether comes into the picture. She has worked with some of the most brilliant mathematicians in Göttingen of her time, with David Hilbert and Felix Klein. Particularly the latter is famous for his "symmetry view" on geometry, i.e., to classify geometries according to their symmetries, and Noether did her PhD about this topic, called "Theory of Invariants" at this time (around 1900).

Now, in General Relativity, there's some problem to define energy, and Noether investigated in a very general way (field-)equations of motion which can be derived from an action principle according to symmetries of this action. So, she came to prove one of the most important theorems in modern physics, i.e., the connection between symmetries of the action and conservation laws, known as Noether's theorem(s). Interestingly enough, later she thought very little about this great achievement. Later in her academic life she became the founding mother of modern abstract algebra, investigating the abstract structure of axiomatically defined systems, and she thought her early work on invariant theory is rather boring, but for physics it's one of the most important developments.

When it comes to quantum theory, Noether's theorem is one of the most important ingredients to systematically build the operator algebras, describing observables: The Lie algebras of the symmetry groups of space time determine the commutation relations of operators representing position, momentum, angular momentum and energy (the Hamiltonian) and thus also the dynamics of the system.

In high-energy particle physics, we have the Standard Model of Elementary Particles, and there also other kinds of symmetries, known as local gauge symmetries, play a very important role. All this goes back to Emmy Noether's early work on symmetries of 1918!

 

[collectedsoul]

The Newtonian formulation uses a uniform coordinate system. (Obviously you're going to use a uniform coordinate system; using a uniform coordinate system is what makes it possible to formulate equations of motion)

Here, uniform coordinate system is meant in the following sense: If you have an infinite grid, then you cannot tell where you are on that grid. Every shift of position ends at a position that is indistinguishable from the previous position.


So you have two definitions in place:
- velocity is defined as the time derivative of position
- the coordinate system is uniform.

Those definitions are already enough to imply Newton's First Law.

Not very clear on what you mean by a uniform coordinate system. If one position is indistinguishable from another then what's the point of a coordinate system?

The same applies for a pair of objects that is exerting a force upon each other. For each object: in order to exert a force upon the other object it must be in acceleration itself (just as you have to accelerate yourself in order to exert a force upon that door)

a= F/m says that the object with the largest mass will accelerate the least.
The acceleration will be in proportion, hence the velocities will be in proportion, hence the position of the common center of mass will not change.

So the following two:
Law of motion: a = F/m
Definition: the coordinate system is uniform

Are already enough to imply that for interacting objects (attracting, repulsing, colliding) the total momentum is conserved

By this are you saying that we don't need Noether's theorem to prove momentum conservation? That the definition of acceleration according to Newton's Second Law when applied to a closed system of objects is enough. At least that's the sense I got. Please correct me if I'm wrong. Still don't get the necessity of a 'uniform coordinate system' though.

The general form of Nöther's theorem states that if you have a theory of motion (which means you have position, velocity and acceleration defined) then for each symmetry of the equations of motion (as defined on the coordinate system used) there is a corresponding conserved quantity in the theory of motion.

I can think of only two things (dimensions?) where Noether's theorem would apply for a mechanical system. The space and the time. So spatial translational invariance (symmetry?) produces momentum conservation and temporal invariance of the system produces energy conservation. Am I right?

The systematic study of symmetry principles is a rather modern concept, although of course the old masters of mechanics like Newton, Lagrange, Hamilton, Euler et al. knew that a good choice of coordinates helps to get simpler equations. E.g., if you have a particle in a central potential, it's most convenient to introduce spherical coordinates around the center of the potential due to the rotational symmetry of the problem. Then you find a lot of "first integrals", i.e., conservation laws mosre easily than when using some other arbitrary coordinates.

A more systematic approach has been developed by Lie, who thought about how to systematically use symmetry concepts to solve differential equations. In this way he came to investigate groups of transformations of equations that depend differentiably on one or more real parameters. Associated with these Lie groups are the Lie algebras which follow from the study of the group around the group identity. In this way you have infinitesimal transformations, which can be applied repeatedly to build up a large transformation. In physics this has been an unknown technique for quite some time. In this case the mathematicians were much more advanced than the physicists.

In physics, symmetry arguments in the modern sense have been first introduced by Einstein in his famous article "On electrodynamics of moving bodies" (1905). In the first sentences of this paper he makes a typical symmetry argument about Maxwell's Equations of electromagnetism which leads him to introduce a new concept of space and time. In other words, in this approach the space-time structure is governed by symmetry principles of observed facts about nature (in this case the independence of induced electromotive forces in Faradays Law of Induction of whether a wire is moved in a magnetic field or the magnet is moved close to the wire).

When Einsteins tried to describe also gravity, the only other then known fundamental force, within his relativistic space-time model, he had to solve a lot of problems. So it took him and his mathematical collaborators (mostly Marcel Grossmann) about 10 years to come up with what is known as the General Theory of Relativity. Here, a geometrical picture of space-time is the most natural approach: GTR can be interpreted as a dynamical theory about the geometrical structure of space-time, and thus the idea to investigate these space-time structures by symmetry principles is very natural.

At this point Emmy Noether comes into the picture. She has worked with some of the most brilliant mathematicians in Göttingen of her time, with David Hilbert and Felix Klein. Particularly the latter is famous for his "symmetry view" on geometry, i.e., to classify geometries according to their symmetries, and Noether did her PhD about this topic, called "Theory of Invariants" at this time (around 1900).

Now, in General Relativity, there's some problem to define energy, and Noether investigated in a very general way (field-)equations of motion which can be derived from an action principle according to symmetries of this action. So, she came to prove one of the most important theorems in modern physics, i.e., the connection between symmetries of the action and conservation laws, known as Noether's theorem(s). Interestingly enough, later she thought very little about this great achievement. Later in her academic life she became the founding mother of modern abstract algebra, investigating the abstract structure of axiomatically defined systems, and she thought her early work on invariant theory is rather boring, but for physics it's one of the most important developments.

When it comes to quantum theory, Noether's theorem is one of the most important ingredients to systematically build the operator algebras, describing observables: The Lie algebras of the symmetry groups of space time determine the commutation relations of operators representing position, momentum, angular momentum and energy (the Hamiltonian) and thus also the dynamics of the system.

In high-energy particle physics, we have the Standard Model of Elementary Particles, and there also other kinds of symmetries, known as local gauge symmetries, play a very important role. All this goes back to Emmy Noether's early work on symmetries of 1918!

That was a really interesting read. I don't know any of the math you mentioned, nor a lot of the physics but the historical progress is very interesting. I recall reading somewhere that Einstein lacked knowledge of linear algebra and matrices and he made a statement to the effect that he would have figured out the general theory of relativity 5 years sooner if he had known the math earlier. Also he said that Emmy Noether was the greatest woman mathematician ever.

 

[Cleonis]

Not very clear on what you mean by a uniform coordinate system. If one position is indistinguishable from another then what's the point of a coordinate system?

You must be joking.


To describe motion we assign spatial coordinate to space. Each point in space is equally suitable for using as the zero point of the coordinate system.

By contrast, we also assign numbers to temperature scale, but the points along the temperature scale are distinguishable. Zero point of temparature is absolute. There is a temperature that we call zero Kelvin that is the true zero point of temperature.


As I said, each point in space is equally suitable as the zero point of the coordinate system used to describe motion in space.

Points in space being indistinguishable does not mean objects can pop in and out of existence everywhere. To get from one point to another point an object will have to travel the distance. It's just that on arrival the local space has exactly the same properties as the local space at the point of departure.

Contrast that with temperature. When you move to a higher temperature you have that when the move is completed it's hotter. But when you move in space you don't get nearer or farther to some absolute reference point because there is no absolute reference point.

 

[Cleonis]

I can think of only two things (dimensions?) where Noether's theorem would apply for a mechanical system. The space and the time. So spatial translational invariance (symmetry?) produces momentum conservation and temporal invariance of the system produces energy conservation. Am I right?

Yes.

That is how Nöther's theorem works out for classical mechanics.
- Translational invariance corresponds to the fact that the equations of motion enforce conservation of momentum.
- Invariance under a shift of time coordinate corresponds to the fact that the equations of motion enforce conservation of energy.

There is a third invariance: you can choose any orientation of your coordinate system. There is invariance under a shift of the orientation of the coordinate system, and corresponding to that the equations of motion enforce conservation of angular momentum.

Nöther's theorem generalizes the above three cases: in any theory of motion you have that for each invariance there exists a corresponding conserved quantity.
As in all of mathematics the generalization moves the description to a more abstract level.

Historically the separate cases are identified first, later a generalization may be found to hold good in all known circumstances

 

[fluidistic]

Just curious about posts of vanshees71 (4 and 6).
I don't really understand why when you derive \vec P=\sum _j \frac{ \partial L}{\partial \dot \vec x_j} with respect to time you get \frac{d}{dt} \vec P=\sum_{j} \frac{\partial L}{\partial \vec x_j}.

 

[collectedsoul]

You must be joking.


To describe motion we assign spatial coordinate to space. Each point in space is equally suitable for using as the zero point of the coordinate system.

By contrast, we also assign numbers to temperature scale, but the points along the temperature scale are distinguishable. Zero point of temparature is absolute. There is a temperature that we call zero Kelvin that is the true zero point of temperature.


As I said, each point in space is equally suitable as the zero point of the coordinate system used to describe motion in space.

Points in space being indistinguishable does not mean objects can pop in and out of existence everywhere. To get from one point to another point an object will have to travel the distance. It's just that on arrival the local space has exactly the same properties as the local space at the point of departure.

Contrast that with temperature. When you move to a higher temperature you have that when the move is completed it's hotter. But when you move in space you don't get nearer or farther to some absolute reference point because there is no absolute reference point.

Okay I understand now. I didn't get the sense in which you meant indistinguishable. I was thinking along the lines of 'once I specify a co-ordinate system each position in space is rendered unique'.

Yes.

There is a third invariance: you can choose any orientation of your coordinate system. There is invariance under a shift of the orientation of the coordinate system, and corresponding to that the equations of motion enforce conservation of angular momentum.

Would it be correct to say that this third thing is an arbitrary specification? It seems merely to depend on choice of co-ordinate system and has not to do with the character of the system itself. I don't doubt its usefulness, just wondering if it isn't an extrinsic sort of allocation to the system.

 

[Cleonis]

[...] It seems merely to depend on choice of co-ordinate system [...]

Yeah, the physical invariance translates to a freedom of choice for the zero point of the coordinate system.

- Translational invariance means every point in space is equally suitable as the zero point of the spatial coordinate system
- Time translational invariance means every point in time is equally suitable as zero point of the time coordinate system
- Orientation invariance means every angle in space is equally suitable as zero point of a system of polar coordinates.