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Elementary argument for conservation laws from symmetries *without* using the Lagrangian formalism
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It is well known from Noether's Theorem how from continuous symmetries in the Lagrangian one gets a conserved charge which corresponds to linear momentum, angular momentum for translational and rotational symmetries and others.
Is there any elementary argument for why linear or angular momentum specifically (and not other conserved quantities) are conserved which does not require knowledge of Lagrangians? By elementary I mean, "if this is not so, then this unreasonable thing occurs".
Of course, we can say "if we want our laws to be the same at a different point in space then linear conservation must be conserved", but can we derive mathematically the expression for the conserved quantity without using the Lagrangian?
I want to explain to a friend why they are conserved but he doesn't have the background to understand the Lagrangian formalism.
angular-momentum momentum conservation-laws symmetry noethers-theorem
edited 1 hour ago
Qmechanic♦
98.9k121781090
asked 4 hours ago
Cristian Em.
588
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up vote
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It is well known from Noether's Theorem how from continuous symmetries in the Lagrangian one gets a conserved charge which corresponds to linear momentum, angular momentum for translational and rotational symmetries and others.
Is there any elementary argument for why linear or angular momentum specifically (and not other conserved quantities) are conserved which does not require knowledge of Lagrangians? By elementary I mean, "if this is not so, then this unreasonable thing occurs".
Of course, we can say "if we want our laws to be the same at a different point in space then linear conservation must be conserved", but can we derive mathematically the expression for the conserved quantity without using the Lagrangian?
I want to explain to a friend why they are conserved but he doesn't have the background to understand the Lagrangian formalism.
angular-momentum momentum conservation-laws symmetry noethers-theorem
edited 1 hour ago
Qmechanic♦
98.9k121781090
asked 4 hours ago
Cristian Em.
588
Have you tried using the symmetry of the metric?
– safesphere
3 hours ago
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up vote
3
down vote
favorite
up vote
3
down vote
favorite
It is well known from Noether's Theorem how from continuous symmetries in the Lagrangian one gets a conserved charge which corresponds to linear momentum, angular momentum for translational and rotational symmetries and others.
Is there any elementary argument for why linear or angular momentum specifically (and not other conserved quantities) are conserved which does not require knowledge of Lagrangians? By elementary I mean, "if this is not so, then this unreasonable thing occurs".
Of course, we can say "if we want our laws to be the same at a different point in space then linear conservation must be conserved", but can we derive mathematically the expression for the conserved quantity without using the Lagrangian?
I want to explain to a friend why they are conserved but he doesn't have the background to understand the Lagrangian formalism.
angular-momentum momentum conservation-laws symmetry noethers-theorem
edited 1 hour ago
Qmechanic♦
98.9k121781090
asked 4 hours ago
Cristian Em.
588
It is well known from Noether's Theorem how from continuous symmetries in the Lagrangian one gets a conserved charge which corresponds to linear momentum, angular momentum for translational and rotational symmetries and others.
Is there any elementary argument for why linear or angular momentum specifically (and not other conserved quantities) are conserved which does not require knowledge of Lagrangians? By elementary I mean, "if this is not so, then this unreasonable thing occurs".
Of course, we can say "if we want our laws to be the same at a different point in space then linear conservation must be conserved", but can we derive mathematically the expression for the conserved quantity without using the Lagrangian?
I want to explain to a friend why they are conserved but he doesn't have the background to understand the Lagrangian formalism.
angular-momentum momentum conservation-laws symmetry noethers-theorem
angular-momentum momentum conservation-laws symmetry noethers-theorem
edited 1 hour ago
Qmechanic♦
98.9k121781090
asked 4 hours ago
Cristian Em.
588
edited 1 hour ago
Qmechanic♦
98.9k121781090
asked 4 hours ago
Cristian Em.
588
edited 1 hour ago
Qmechanic♦
98.9k121781090
edited 1 hour ago
Qmechanic♦
98.9k121781090
edited 1 hour ago
Qmechanic♦
98.9k121781090
98.9k121781090
asked 4 hours ago
Cristian Em.
588
asked 4 hours ago
Cristian Em.
588
asked 4 hours ago
Cristian Em.
588
588
Have you tried using the symmetry of the metric?
– safesphere
3 hours ago
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Have you tried using the symmetry of the metric?
– safesphere
3 hours ago
Have you tried using the symmetry of the metric?
– safesphere
3 hours ago
Have you tried using the symmetry of the metric?
– safesphere
3 hours ago
add a comment |Â
3 Answers
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Try the Hamiltonian formalism: If the symmetry generator $Q$ commutes with the Hamiltonian $[Q,H]=0$ then $Q$ is a conserved quantity.
answered 2 hours ago
Qmechanic♦
98.9k121781090
Sure, but that's QM. What about classical physics?
– FGSUZ
1 hour ago
This also works in classical mechanics where $[cdot,cdot]$ is the Poisson bracket.
– Qmechanic♦
1 hour ago
I guessed so, but since I'm used to the notation... haha. Anyways, you should add this to your answer.
– FGSUZ
1 hour ago
add a comment |Â
up vote
1
down vote
The answer is yes, the essence of Noether's theorem for linear and angular momentum can be understood without using the Lagrangian (or Hamiltonian) formulation, at least if we're willing to focus on models in which the equations of motion have the form
$$
m_nmathbfddot x_n = mathbfF_n(mathbfx_1,mathbfx_2,...)
tag1
$$
where $m_n$ and $mathbfx_n$ are the mass and location of the $n$-th object, overhead dots denote time-derivatives, and $mathbfF_n$ is the force on the $n$-th object, which depends on the locations of all of the objects.
(This answer still uses math, but it doesn't use Lagrangians or Hamiltonians. An answer that doesn't use math is also possible, but it would be wordier and less convincing.)
The inputs to Noether's theorem are the action principle together with a (continuous) symmetry. For a system like (1), the action principle can be expressed like this:
$$
mathbfF_n(mathbfx_1,mathbfx_2,...)
= -nabla_n V(mathbfx_1,mathbfx_2,...).
tag2
$$
The key point of this equation is that the forces are all derived from the same function $V$.
Loosely translated, this says that if the force on object $A$ depends on the location of object $B$, then the force on object $B$ must also depend (in a special way) on the location of object $A$.
First consider linear momentum. Suppose that the model is invariant under translations in space. In the context of Noether's theorem, this is a statement about the function $V$. This is important! If we merely assume that the system of equations (1) is invariant under translations in space, then conservation of momentum would not be implied. (To see this, consider a system with only one object subject to a location-independent force.) What we need to do is assume that $V$ is invariant under translations in space. This means
$$
V(mathbfx_1+mathbfc,mathbfx_2+mathbfc,...) =
V(mathbfx_1,mathbfx_2,...)
tag3
$$
for any $mathbfc$. The same condition may also be expressed like this:
$$
fracpartialpartialmathbfcV(mathbfx_1+mathbfc,mathbfx_2+mathbfc,...) = 0,
tag4
$$
where $partial/partialmathbfc$ denotes the gradient with respect to $mathbfc$. Equation (4), in turn, may also be written like this:
$$
sum_nnabla_n V(mathbfx_1,mathbfx_2,,...) = 0.
tag5
$$
Combine equations (1), (2), and (5) to get
$$
sum_n m_nmathbfddot x_n = 0,
tag6
$$
which can also be written
$$
fracddtsum_n m_nmathbfdot x_n = 0.
$$
This is conservation of (total) linear momentum.
Now consider angular momentum. For this, we need to assume that $V$ is invariant under rotations. To be specific, assume that $V$ is invariant under rotations about the origin; this will lead to conservation of angular momentum about the origin.
The analogue of equation (5) is
$$
sum_nmathbfx_nwedge nabla_n V(mathbfx_1,mathbfx_2,,...) = 0
tag7
$$
where the components of $mathbfxwedgenabla$ are $x_jnabla_k-x_knabla_j$. (For three-dimensional space, this is usually expressed using the "cross product", but I prefer a formulation that works in
any number of dimensions so that it can be applied without hesitation to easier cases like two-dimensional space.) Equation (7) expresses the assumption that $V$ is invariant under rotations about the origin. As before, combine equations (1), (2), and (7) to get
$$
sum_n mathbfx_nwedge m_nmathbfddot x_n = 0,
tag8
$$
and use the trivial identity
$$
mathbfdot x_nwedge mathbfdot x_n = 0
tag9
$$
(because $mathbfawedgemathbfb$ has components $a_jb_k-a_kb_j$)
to see that equation (8) can also be written
$$
fracddtsum_n mathbfx_nwedge m_nmathbfdot x_n = 0.
tag10
$$
This is conservation of (total) angular momentum about the origin.
answered 1 hour ago
Dan Yand
1,10411
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0
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Is there any difference from saying that those are just the laws and there really isn't an explanation, that that's just how nature works because there aren't experiment where we ever observed non conservation of those quantities?
Symmetry under the action of the Poincaré group implies those conservation laws, but I'm not sure you're giving a deeper explanation in this. We impose that physical system have those symmetries because we want something to be conserved, and how do you prove the correctness of the hypothesis of the existence of some symmetries? Performing experiments that show conservation of certain quantities. So we're back at the start, they are conserved because they are conserved, that is, experiments tells us they are conserved.
answered 1 hour ago
Run like hell
717518
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3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Try the Hamiltonian formalism: If the symmetry generator $Q$ commutes with the Hamiltonian $[Q,H]=0$ then $Q$ is a conserved quantity.
answered 2 hours ago
Qmechanic♦
98.9k121781090
Sure, but that's QM. What about classical physics?
– FGSUZ
1 hour ago
This also works in classical mechanics where $[cdot,cdot]$ is the Poisson bracket.
– Qmechanic♦
1 hour ago
I guessed so, but since I'm used to the notation... haha. Anyways, you should add this to your answer.
– FGSUZ
1 hour ago
add a comment |Â
up vote
1
down vote
Try the Hamiltonian formalism: If the symmetry generator $Q$ commutes with the Hamiltonian $[Q,H]=0$ then $Q$ is a conserved quantity.
answered 2 hours ago
Qmechanic♦
98.9k121781090
Sure, but that's QM. What about classical physics?
– FGSUZ
1 hour ago
This also works in classical mechanics where $[cdot,cdot]$ is the Poisson bracket.
– Qmechanic♦
1 hour ago
I guessed so, but since I'm used to the notation... haha. Anyways, you should add this to your answer.
– FGSUZ
1 hour ago
add a comment |Â
up vote
1
down vote
up vote
1
down vote
Try the Hamiltonian formalism: If the symmetry generator $Q$ commutes with the Hamiltonian $[Q,H]=0$ then $Q$ is a conserved quantity.
answered 2 hours ago
Qmechanic♦
98.9k121781090
Try the Hamiltonian formalism: If the symmetry generator $Q$ commutes with the Hamiltonian $[Q,H]=0$ then $Q$ is a conserved quantity.
answered 2 hours ago
Qmechanic♦
98.9k121781090
answered 2 hours ago
Qmechanic♦
98.9k121781090
answered 2 hours ago
Qmechanic♦
98.9k121781090
answered 2 hours ago
Qmechanic♦
98.9k121781090
98.9k121781090
Sure, but that's QM. What about classical physics?
– FGSUZ
1 hour ago
This also works in classical mechanics where $[cdot,cdot]$ is the Poisson bracket.
– Qmechanic♦
1 hour ago
I guessed so, but since I'm used to the notation... haha. Anyways, you should add this to your answer.
– FGSUZ
1 hour ago
add a comment |Â
Sure, but that's QM. What about classical physics?
– FGSUZ
1 hour ago
This also works in classical mechanics where $[cdot,cdot]$ is the Poisson bracket.
– Qmechanic♦
1 hour ago
I guessed so, but since I'm used to the notation... haha. Anyways, you should add this to your answer.
– FGSUZ
1 hour ago
Sure, but that's QM. What about classical physics?
– FGSUZ
1 hour ago
Sure, but that's QM. What about classical physics?
– FGSUZ
1 hour ago
This also works in classical mechanics where $[cdot,cdot]$ is the Poisson bracket.
– Qmechanic♦
1 hour ago
This also works in classical mechanics where $[cdot,cdot]$ is the Poisson bracket.
– Qmechanic♦
1 hour ago
I guessed so, but since I'm used to the notation... haha. Anyways, you should add this to your answer.
– FGSUZ
1 hour ago
I guessed so, but since I'm used to the notation... haha. Anyways, you should add this to your answer.
– FGSUZ
1 hour ago
add a comment |Â
up vote
1
down vote
The answer is yes, the essence of Noether's theorem for linear and angular momentum can be understood without using the Lagrangian (or Hamiltonian) formulation, at least if we're willing to focus on models in which the equations of motion have the form
$$
m_nmathbfddot x_n = mathbfF_n(mathbfx_1,mathbfx_2,...)
tag1
$$
where $m_n$ and $mathbfx_n$ are the mass and location of the $n$-th object, overhead dots denote time-derivatives, and $mathbfF_n$ is the force on the $n$-th object, which depends on the locations of all of the objects.
(This answer still uses math, but it doesn't use Lagrangians or Hamiltonians. An answer that doesn't use math is also possible, but it would be wordier and less convincing.)
The inputs to Noether's theorem are the action principle together with a (continuous) symmetry. For a system like (1), the action principle can be expressed like this:
$$
mathbfF_n(mathbfx_1,mathbfx_2,...)
= -nabla_n V(mathbfx_1,mathbfx_2,...).
tag2
$$
The key point of this equation is that the forces are all derived from the same function $V$.
Loosely translated, this says that if the force on object $A$ depends on the location of object $B$, then the force on object $B$ must also depend (in a special way) on the location of object $A$.
First consider linear momentum. Suppose that the model is invariant under translations in space. In the context of Noether's theorem, this is a statement about the function $V$. This is important! If we merely assume that the system of equations (1) is invariant under translations in space, then conservation of momentum would not be implied. (To see this, consider a system with only one object subject to a location-independent force.) What we need to do is assume that $V$ is invariant under translations in space. This means
$$
V(mathbfx_1+mathbfc,mathbfx_2+mathbfc,...) =
V(mathbfx_1,mathbfx_2,...)
tag3
$$
for any $mathbfc$. The same condition may also be expressed like this:
$$
fracpartialpartialmathbfcV(mathbfx_1+mathbfc,mathbfx_2+mathbfc,...) = 0,
tag4
$$
where $partial/partialmathbfc$ denotes the gradient with respect to $mathbfc$. Equation (4), in turn, may also be written like this:
$$
sum_nnabla_n V(mathbfx_1,mathbfx_2,,...) = 0.
tag5
$$
Combine equations (1), (2), and (5) to get
$$
sum_n m_nmathbfddot x_n = 0,
tag6
$$
which can also be written
$$
fracddtsum_n m_nmathbfdot x_n = 0.
$$
This is conservation of (total) linear momentum.
Now consider angular momentum. For this, we need to assume that $V$ is invariant under rotations. To be specific, assume that $V$ is invariant under rotations about the origin; this will lead to conservation of angular momentum about the origin.
The analogue of equation (5) is
$$
sum_nmathbfx_nwedge nabla_n V(mathbfx_1,mathbfx_2,,...) = 0
tag7
$$
where the components of $mathbfxwedgenabla$ are $x_jnabla_k-x_knabla_j$. (For three-dimensional space, this is usually expressed using the "cross product", but I prefer a formulation that works in
any number of dimensions so that it can be applied without hesitation to easier cases like two-dimensional space.) Equation (7) expresses the assumption that $V$ is invariant under rotations about the origin. As before, combine equations (1), (2), and (7) to get
$$
sum_n mathbfx_nwedge m_nmathbfddot x_n = 0,
tag8
$$
and use the trivial identity
$$
mathbfdot x_nwedge mathbfdot x_n = 0
tag9
$$
(because $mathbfawedgemathbfb$ has components $a_jb_k-a_kb_j$)
to see that equation (8) can also be written
$$
fracddtsum_n mathbfx_nwedge m_nmathbfdot x_n = 0.
tag10
$$
This is conservation of (total) angular momentum about the origin.
answered 1 hour ago
Dan Yand
1,10411
add a comment |Â
up vote
1
down vote
The answer is yes, the essence of Noether's theorem for linear and angular momentum can be understood without using the Lagrangian (or Hamiltonian) formulation, at least if we're willing to focus on models in which the equations of motion have the form
$$
m_nmathbfddot x_n = mathbfF_n(mathbfx_1,mathbfx_2,...)
tag1
$$
where $m_n$ and $mathbfx_n$ are the mass and location of the $n$-th object, overhead dots denote time-derivatives, and $mathbfF_n$ is the force on the $n$-th object, which depends on the locations of all of the objects.
(This answer still uses math, but it doesn't use Lagrangians or Hamiltonians. An answer that doesn't use math is also possible, but it would be wordier and less convincing.)
The inputs to Noether's theorem are the action principle together with a (continuous) symmetry. For a system like (1), the action principle can be expressed like this:
$$
mathbfF_n(mathbfx_1,mathbfx_2,...)
= -nabla_n V(mathbfx_1,mathbfx_2,...).
tag2
$$
The key point of this equation is that the forces are all derived from the same function $V$.
Loosely translated, this says that if the force on object $A$ depends on the location of object $B$, then the force on object $B$ must also depend (in a special way) on the location of object $A$.
First consider linear momentum. Suppose that the model is invariant under translations in space. In the context of Noether's theorem, this is a statement about the function $V$. This is important! If we merely assume that the system of equations (1) is invariant under translations in space, then conservation of momentum would not be implied. (To see this, consider a system with only one object subject to a location-independent force.) What we need to do is assume that $V$ is invariant under translations in space. This means
$$
V(mathbfx_1+mathbfc,mathbfx_2+mathbfc,...) =
V(mathbfx_1,mathbfx_2,...)
tag3
$$
for any $mathbfc$. The same condition may also be expressed like this:
$$
fracpartialpartialmathbfcV(mathbfx_1+mathbfc,mathbfx_2+mathbfc,...) = 0,
tag4
$$
where $partial/partialmathbfc$ denotes the gradient with respect to $mathbfc$. Equation (4), in turn, may also be written like this:
$$
sum_nnabla_n V(mathbfx_1,mathbfx_2,,...) = 0.
tag5
$$
Combine equations (1), (2), and (5) to get
$$
sum_n m_nmathbfddot x_n = 0,
tag6
$$
which can also be written
$$
fracddtsum_n m_nmathbfdot x_n = 0.
$$
This is conservation of (total) linear momentum.
Now consider angular momentum. For this, we need to assume that $V$ is invariant under rotations. To be specific, assume that $V$ is invariant under rotations about the origin; this will lead to conservation of angular momentum about the origin.
The analogue of equation (5) is
$$
sum_nmathbfx_nwedge nabla_n V(mathbfx_1,mathbfx_2,,...) = 0
tag7
$$
where the components of $mathbfxwedgenabla$ are $x_jnabla_k-x_knabla_j$. (For three-dimensional space, this is usually expressed using the "cross product", but I prefer a formulation that works in
any number of dimensions so that it can be applied without hesitation to easier cases like two-dimensional space.) Equation (7) expresses the assumption that $V$ is invariant under rotations about the origin. As before, combine equations (1), (2), and (7) to get
$$
sum_n mathbfx_nwedge m_nmathbfddot x_n = 0,
tag8
$$
and use the trivial identity
$$
mathbfdot x_nwedge mathbfdot x_n = 0
tag9
$$
(because $mathbfawedgemathbfb$ has components $a_jb_k-a_kb_j$)
to see that equation (8) can also be written
$$
fracddtsum_n mathbfx_nwedge m_nmathbfdot x_n = 0.
tag10
$$
This is conservation of (total) angular momentum about the origin.
answered 1 hour ago
Dan Yand
1,10411
add a comment |Â
up vote
1
down vote
up vote
1
down vote
The answer is yes, the essence of Noether's theorem for linear and angular momentum can be understood without using the Lagrangian (or Hamiltonian) formulation, at least if we're willing to focus on models in which the equations of motion have the form
$$
m_nmathbfddot x_n = mathbfF_n(mathbfx_1,mathbfx_2,...)
tag1
$$
where $m_n$ and $mathbfx_n$ are the mass and location of the $n$-th object, overhead dots denote time-derivatives, and $mathbfF_n$ is the force on the $n$-th object, which depends on the locations of all of the objects.
(This answer still uses math, but it doesn't use Lagrangians or Hamiltonians. An answer that doesn't use math is also possible, but it would be wordier and less convincing.)
The inputs to Noether's theorem are the action principle together with a (continuous) symmetry. For a system like (1), the action principle can be expressed like this:
$$
mathbfF_n(mathbfx_1,mathbfx_2,...)
= -nabla_n V(mathbfx_1,mathbfx_2,...).
tag2
$$
The key point of this equation is that the forces are all derived from the same function $V$.
Loosely translated, this says that if the force on object $A$ depends on the location of object $B$, then the force on object $B$ must also depend (in a special way) on the location of object $A$.
First consider linear momentum. Suppose that the model is invariant under translations in space. In the context of Noether's theorem, this is a statement about the function $V$. This is important! If we merely assume that the system of equations (1) is invariant under translations in space, then conservation of momentum would not be implied. (To see this, consider a system with only one object subject to a location-independent force.) What we need to do is assume that $V$ is invariant under translations in space. This means
$$
V(mathbfx_1+mathbfc,mathbfx_2+mathbfc,...) =
V(mathbfx_1,mathbfx_2,...)
tag3
$$
for any $mathbfc$. The same condition may also be expressed like this:
$$
fracpartialpartialmathbfcV(mathbfx_1+mathbfc,mathbfx_2+mathbfc,...) = 0,
tag4
$$
where $partial/partialmathbfc$ denotes the gradient with respect to $mathbfc$. Equation (4), in turn, may also be written like this:
$$
sum_nnabla_n V(mathbfx_1,mathbfx_2,,...) = 0.
tag5
$$
Combine equations (1), (2), and (5) to get
$$
sum_n m_nmathbfddot x_n = 0,
tag6
$$
which can also be written
$$
fracddtsum_n m_nmathbfdot x_n = 0.
$$
This is conservation of (total) linear momentum.
Now consider angular momentum. For this, we need to assume that $V$ is invariant under rotations. To be specific, assume that $V$ is invariant under rotations about the origin; this will lead to conservation of angular momentum about the origin.
The analogue of equation (5) is
$$
sum_nmathbfx_nwedge nabla_n V(mathbfx_1,mathbfx_2,,...) = 0
tag7
$$
where the components of $mathbfxwedgenabla$ are $x_jnabla_k-x_knabla_j$. (For three-dimensional space, this is usually expressed using the "cross product", but I prefer a formulation that works in
any number of dimensions so that it can be applied without hesitation to easier cases like two-dimensional space.) Equation (7) expresses the assumption that $V$ is invariant under rotations about the origin. As before, combine equations (1), (2), and (7) to get
$$
sum_n mathbfx_nwedge m_nmathbfddot x_n = 0,
tag8
$$
and use the trivial identity
$$
mathbfdot x_nwedge mathbfdot x_n = 0
tag9
$$
(because $mathbfawedgemathbfb$ has components $a_jb_k-a_kb_j$)
to see that equation (8) can also be written
$$
fracddtsum_n mathbfx_nwedge m_nmathbfdot x_n = 0.
tag10
$$
This is conservation of (total) angular momentum about the origin.
answered 1 hour ago
Dan Yand
1,10411
The answer is yes, the essence of Noether's theorem for linear and angular momentum can be understood without using the Lagrangian (or Hamiltonian) formulation, at least if we're willing to focus on models in which the equations of motion have the form
$$
m_nmathbfddot x_n = mathbfF_n(mathbfx_1,mathbfx_2,...)
tag1
$$
where $m_n$ and $mathbfx_n$ are the mass and location of the $n$-th object, overhead dots denote time-derivatives, and $mathbfF_n$ is the force on the $n$-th object, which depends on the locations of all of the objects.
(This answer still uses math, but it doesn't use Lagrangians or Hamiltonians. An answer that doesn't use math is also possible, but it would be wordier and less convincing.)
The inputs to Noether's theorem are the action principle together with a (continuous) symmetry. For a system like (1), the action principle can be expressed like this:
$$
mathbfF_n(mathbfx_1,mathbfx_2,...)
= -nabla_n V(mathbfx_1,mathbfx_2,...).
tag2
$$
The key point of this equation is that the forces are all derived from the same function $V$.
Loosely translated, this says that if the force on object $A$ depends on the location of object $B$, then the force on object $B$ must also depend (in a special way) on the location of object $A$.
First consider linear momentum. Suppose that the model is invariant under translations in space. In the context of Noether's theorem, this is a statement about the function $V$. This is important! If we merely assume that the system of equations (1) is invariant under translations in space, then conservation of momentum would not be implied. (To see this, consider a system with only one object subject to a location-independent force.) What we need to do is assume that $V$ is invariant under translations in space. This means
$$
V(mathbfx_1+mathbfc,mathbfx_2+mathbfc,...) =
V(mathbfx_1,mathbfx_2,...)
tag3
$$
for any $mathbfc$. The same condition may also be expressed like this:
$$
fracpartialpartialmathbfcV(mathbfx_1+mathbfc,mathbfx_2+mathbfc,...) = 0,
tag4
$$
where $partial/partialmathbfc$ denotes the gradient with respect to $mathbfc$. Equation (4), in turn, may also be written like this:
$$
sum_nnabla_n V(mathbfx_1,mathbfx_2,,...) = 0.
tag5
$$
Combine equations (1), (2), and (5) to get
$$
sum_n m_nmathbfddot x_n = 0,
tag6
$$
which can also be written
$$
fracddtsum_n m_nmathbfdot x_n = 0.
$$
This is conservation of (total) linear momentum.
Now consider angular momentum. For this, we need to assume that $V$ is invariant under rotations. To be specific, assume that $V$ is invariant under rotations about the origin; this will lead to conservation of angular momentum about the origin.
The analogue of equation (5) is
$$
sum_nmathbfx_nwedge nabla_n V(mathbfx_1,mathbfx_2,,...) = 0
tag7
$$
where the components of $mathbfxwedgenabla$ are $x_jnabla_k-x_knabla_j$. (For three-dimensional space, this is usually expressed using the "cross product", but I prefer a formulation that works in
any number of dimensions so that it can be applied without hesitation to easier cases like two-dimensional space.) Equation (7) expresses the assumption that $V$ is invariant under rotations about the origin. As before, combine equations (1), (2), and (7) to get
$$
sum_n mathbfx_nwedge m_nmathbfddot x_n = 0,
tag8
$$
and use the trivial identity
$$
mathbfdot x_nwedge mathbfdot x_n = 0
tag9
$$
(because $mathbfawedgemathbfb$ has components $a_jb_k-a_kb_j$)
to see that equation (8) can also be written
$$
fracddtsum_n mathbfx_nwedge m_nmathbfdot x_n = 0.
tag10
$$
This is conservation of (total) angular momentum about the origin.
answered 1 hour ago
Dan Yand
1,10411
edited 1 hour ago
answered 1 hour ago
Dan Yand
1,10411
answered 1 hour ago
Dan Yand
1,10411
answered 1 hour ago
Dan Yand
1,10411
1,10411
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Is there any difference from saying that those are just the laws and there really isn't an explanation, that that's just how nature works because there aren't experiment where we ever observed non conservation of those quantities?
Symmetry under the action of the Poincaré group implies those conservation laws, but I'm not sure you're giving a deeper explanation in this. We impose that physical system have those symmetries because we want something to be conserved, and how do you prove the correctness of the hypothesis of the existence of some symmetries? Performing experiments that show conservation of certain quantities. So we're back at the start, they are conserved because they are conserved, that is, experiments tells us they are conserved.
answered 1 hour ago
Run like hell
717518
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up vote
0
down vote
Is there any difference from saying that those are just the laws and there really isn't an explanation, that that's just how nature works because there aren't experiment where we ever observed non conservation of those quantities?
Symmetry under the action of the Poincaré group implies those conservation laws, but I'm not sure you're giving a deeper explanation in this. We impose that physical system have those symmetries because we want something to be conserved, and how do you prove the correctness of the hypothesis of the existence of some symmetries? Performing experiments that show conservation of certain quantities. So we're back at the start, they are conserved because they are conserved, that is, experiments tells us they are conserved.
answered 1 hour ago
Run like hell
717518
add a comment |Â
up vote
0
down vote
up vote
0
down vote
Is there any difference from saying that those are just the laws and there really isn't an explanation, that that's just how nature works because there aren't experiment where we ever observed non conservation of those quantities?
Symmetry under the action of the Poincaré group implies those conservation laws, but I'm not sure you're giving a deeper explanation in this. We impose that physical system have those symmetries because we want something to be conserved, and how do you prove the correctness of the hypothesis of the existence of some symmetries? Performing experiments that show conservation of certain quantities. So we're back at the start, they are conserved because they are conserved, that is, experiments tells us they are conserved.
answered 1 hour ago
Run like hell
717518
Is there any difference from saying that those are just the laws and there really isn't an explanation, that that's just how nature works because there aren't experiment where we ever observed non conservation of those quantities?
Symmetry under the action of the Poincaré group implies those conservation laws, but I'm not sure you're giving a deeper explanation in this. We impose that physical system have those symmetries because we want something to be conserved, and how do you prove the correctness of the hypothesis of the existence of some symmetries? Performing experiments that show conservation of certain quantities. So we're back at the start, they are conserved because they are conserved, that is, experiments tells us they are conserved.
answered 1 hour ago
Run like hell
717518
answered 1 hour ago
Run like hell
717518
answered 1 hour ago
Run like hell
717518
answered 1 hour ago
Run like hell
717518
717518
add a comment |Â
add a comment |Â
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Have you tried using the symmetry of the metric?
– safesphere
3 hours ago