Understanding Hamilton Mechanics - What is this “action integral”
Multi tool use
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I am learning about Hamilton Mechanics and am reading the below material about it: 
Additionally, I reference: http://www.unige.ch/~hairer/poly_geoint/week1.pdf
I would like help in understanding where the "action integral comes from in the above image. Why are we talking about this integral? That is, why are we taking the integral of the Lagrange function?
ordinary-differential-equations dynamical-systems
asked Jan 18 at 6:53
NaltNalt
776
$endgroup$
|
show 2 more comments
$begingroup$
I am learning about Hamilton Mechanics and am reading the below material about it:
Additionally, I reference: http://www.unige.ch/~hairer/poly_geoint/week1.pdf
I would like help in understanding where the "action integral comes from in the above image. Why are we talking about this integral? That is, why are we taking the integral of the Lagrange function?
ordinary-differential-equations dynamical-systems
asked Jan 18 at 6:53
NaltNalt
776
$endgroup$
$begingroup$
This is chapter 8.3. Presumably, the action integral would be explained earlier on in the text?
$endgroup$
– Raskolnikov
Jan 18 at 7:02
2
$begingroup$
The non-mathematical motivation for caring about the action integral is "because physics works that way": en.wikipedia.org/wiki/Principle_of_least_action. I don't know if there is a purely mathematical motivation for it.
$endgroup$
– Rahul
Jan 18 at 8:47
1
$begingroup$
The idea is that you want to construct a functional $S:mathcal{P}tomathbb{R}$ where $mathcal{P}$ is the space of all possible paths your system can take through phase space. And then looking for the path that minimizes or maximizes the output value of the functional is equivalent to finding the natural evolution of the system you consider. It's not any more strange than saying that you have to construct a force $vec{F}$ and then the solutions of the equation $vec{F}=mvec{a}$ are equivalent to finding the natural evolution of the system under consideration.
$endgroup$
– Raskolnikov
Jan 18 at 9:06
1
$begingroup$
This chapter of the Feynman lectures has been very helpful for me.
$endgroup$
– Giuseppe Negro
Jan 18 at 9:33
1
$begingroup$
@GiuseppeNegro This looks like it will helpful for me as well. Thank you for sharing.
$endgroup$
– Nalt
Jan 18 at 9:59
|
show 2 more comments
$begingroup$
I am learning about Hamilton Mechanics and am reading the below material about it:
Additionally, I reference: http://www.unige.ch/~hairer/poly_geoint/week1.pdf
I would like help in understanding where the "action integral comes from in the above image. Why are we talking about this integral? That is, why are we taking the integral of the Lagrange function?
ordinary-differential-equations dynamical-systems
asked Jan 18 at 6:53
NaltNalt
776
$endgroup$
I am learning about Hamilton Mechanics and am reading the below material about it:
Additionally, I reference: http://www.unige.ch/~hairer/poly_geoint/week1.pdf
I would like help in understanding where the "action integral comes from in the above image. Why are we talking about this integral? That is, why are we taking the integral of the Lagrange function?
ordinary-differential-equations dynamical-systems
ordinary-differential-equations dynamical-systems
asked Jan 18 at 6:53
NaltNalt
776
asked Jan 18 at 6:53
NaltNalt
776
asked Jan 18 at 6:53
NaltNalt
776
asked Jan 18 at 6:53
NaltNalt
776
asked Jan 18 at 6:53
NaltNalt
776
776
$begingroup$
This is chapter 8.3. Presumably, the action integral would be explained earlier on in the text?
$endgroup$
– Raskolnikov
Jan 18 at 7:02
2
$begingroup$
The non-mathematical motivation for caring about the action integral is "because physics works that way": en.wikipedia.org/wiki/Principle_of_least_action. I don't know if there is a purely mathematical motivation for it.
$endgroup$
– Rahul
Jan 18 at 8:47
1
$begingroup$
The idea is that you want to construct a functional $S:mathcal{P}tomathbb{R}$ where $mathcal{P}$ is the space of all possible paths your system can take through phase space. And then looking for the path that minimizes or maximizes the output value of the functional is equivalent to finding the natural evolution of the system you consider. It's not any more strange than saying that you have to construct a force $vec{F}$ and then the solutions of the equation $vec{F}=mvec{a}$ are equivalent to finding the natural evolution of the system under consideration.
$endgroup$
– Raskolnikov
Jan 18 at 9:06
1
$begingroup$
This chapter of the Feynman lectures has been very helpful for me.
$endgroup$
– Giuseppe Negro
Jan 18 at 9:33
1
$begingroup$
@GiuseppeNegro This looks like it will helpful for me as well. Thank you for sharing.
$endgroup$
– Nalt
Jan 18 at 9:59
|
show 2 more comments
$begingroup$
This is chapter 8.3. Presumably, the action integral would be explained earlier on in the text?
$endgroup$
– Raskolnikov
Jan 18 at 7:02
2
$begingroup$
The non-mathematical motivation for caring about the action integral is "because physics works that way": en.wikipedia.org/wiki/Principle_of_least_action. I don't know if there is a purely mathematical motivation for it.
$endgroup$
– Rahul
Jan 18 at 8:47
1
$begingroup$
The idea is that you want to construct a functional $S:mathcal{P}tomathbb{R}$ where $mathcal{P}$ is the space of all possible paths your system can take through phase space. And then looking for the path that minimizes or maximizes the output value of the functional is equivalent to finding the natural evolution of the system you consider. It's not any more strange than saying that you have to construct a force $vec{F}$ and then the solutions of the equation $vec{F}=mvec{a}$ are equivalent to finding the natural evolution of the system under consideration.
$endgroup$
– Raskolnikov
Jan 18 at 9:06
1
$begingroup$
This chapter of the Feynman lectures has been very helpful for me.
$endgroup$
– Giuseppe Negro
Jan 18 at 9:33
1
$begingroup$
@GiuseppeNegro This looks like it will helpful for me as well. Thank you for sharing.
$endgroup$
– Nalt
Jan 18 at 9:59
$begingroup$
This is chapter 8.3. Presumably, the action integral would be explained earlier on in the text?
$endgroup$
– Raskolnikov
Jan 18 at 7:02
$begingroup$
This is chapter 8.3. Presumably, the action integral would be explained earlier on in the text?
$endgroup$
– Raskolnikov
Jan 18 at 7:02
2
2
$begingroup$
The non-mathematical motivation for caring about the action integral is "because physics works that way": en.wikipedia.org/wiki/Principle_of_least_action. I don't know if there is a purely mathematical motivation for it.
$endgroup$
– Rahul
Jan 18 at 8:47
$begingroup$
The non-mathematical motivation for caring about the action integral is "because physics works that way": en.wikipedia.org/wiki/Principle_of_least_action. I don't know if there is a purely mathematical motivation for it.
$endgroup$
– Rahul
Jan 18 at 8:47
1
1
$begingroup$
The idea is that you want to construct a functional $S:mathcal{P}tomathbb{R}$ where $mathcal{P}$ is the space of all possible paths your system can take through phase space. And then looking for the path that minimizes or maximizes the output value of the functional is equivalent to finding the natural evolution of the system you consider. It's not any more strange than saying that you have to construct a force $vec{F}$ and then the solutions of the equation $vec{F}=mvec{a}$ are equivalent to finding the natural evolution of the system under consideration.
$endgroup$
– Raskolnikov
Jan 18 at 9:06
$begingroup$
The idea is that you want to construct a functional $S:mathcal{P}tomathbb{R}$ where $mathcal{P}$ is the space of all possible paths your system can take through phase space. And then looking for the path that minimizes or maximizes the output value of the functional is equivalent to finding the natural evolution of the system you consider. It's not any more strange than saying that you have to construct a force $vec{F}$ and then the solutions of the equation $vec{F}=mvec{a}$ are equivalent to finding the natural evolution of the system under consideration.
$endgroup$
– Raskolnikov
Jan 18 at 9:06
1
1
$begingroup$
This chapter of the Feynman lectures has been very helpful for me.
$endgroup$
– Giuseppe Negro
Jan 18 at 9:33
$begingroup$
This chapter of the Feynman lectures has been very helpful for me.
$endgroup$
– Giuseppe Negro
Jan 18 at 9:33
1
1
$begingroup$
@GiuseppeNegro This looks like it will helpful for me as well. Thank you for sharing.
$endgroup$
– Nalt
Jan 18 at 9:59
$begingroup$
@GiuseppeNegro This looks like it will helpful for me as well. Thank you for sharing.
$endgroup$
– Nalt
Jan 18 at 9:59
|
show 2 more comments
1 Answer
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The idea is that you want to construct a functional $S:mathcal{P}→mathbb{R}$ where $mathcal{P}$ is the space of all possible paths your system can take through phase space. And then looking for the path that minimizes or maximizes the output value of the functional is equivalent to finding the natural evolution of the system you consider. It's not any more strange than saying that you have to construct a force $vec{F}$ and then the solutions of the equation $vec{F} = mvec{a}$ are equivalent to finding the natural evolution of the system under consideration.
There is no mathematical reason why you would want to pick any of these principles. In mechanics, they very often can be proven to be equivalent mathematically. But practically it turns out that the principle of least action has been easily extended to encompass more phenomena than the method with forces has been. Nowadays, most theories in physics, in particular field theories, are cast within the framework of the principle of least action.
answered Jan 18 at 9:23
RaskolnikovRaskolnikov
12.6k23571
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$begingroup$
The idea is that you want to construct a functional $S:mathcal{P}→mathbb{R}$ where $mathcal{P}$ is the space of all possible paths your system can take through phase space. And then looking for the path that minimizes or maximizes the output value of the functional is equivalent to finding the natural evolution of the system you consider. It's not any more strange than saying that you have to construct a force $vec{F}$ and then the solutions of the equation $vec{F} = mvec{a}$ are equivalent to finding the natural evolution of the system under consideration.
There is no mathematical reason why you would want to pick any of these principles. In mechanics, they very often can be proven to be equivalent mathematically. But practically it turns out that the principle of least action has been easily extended to encompass more phenomena than the method with forces has been. Nowadays, most theories in physics, in particular field theories, are cast within the framework of the principle of least action.
answered Jan 18 at 9:23
RaskolnikovRaskolnikov
12.6k23571
$endgroup$
add a comment |
$begingroup$
The idea is that you want to construct a functional $S:mathcal{P}→mathbb{R}$ where $mathcal{P}$ is the space of all possible paths your system can take through phase space. And then looking for the path that minimizes or maximizes the output value of the functional is equivalent to finding the natural evolution of the system you consider. It's not any more strange than saying that you have to construct a force $vec{F}$ and then the solutions of the equation $vec{F} = mvec{a}$ are equivalent to finding the natural evolution of the system under consideration.
There is no mathematical reason why you would want to pick any of these principles. In mechanics, they very often can be proven to be equivalent mathematically. But practically it turns out that the principle of least action has been easily extended to encompass more phenomena than the method with forces has been. Nowadays, most theories in physics, in particular field theories, are cast within the framework of the principle of least action.
answered Jan 18 at 9:23
RaskolnikovRaskolnikov
12.6k23571
$endgroup$
add a comment |
$begingroup$
The idea is that you want to construct a functional $S:mathcal{P}→mathbb{R}$ where $mathcal{P}$ is the space of all possible paths your system can take through phase space. And then looking for the path that minimizes or maximizes the output value of the functional is equivalent to finding the natural evolution of the system you consider. It's not any more strange than saying that you have to construct a force $vec{F}$ and then the solutions of the equation $vec{F} = mvec{a}$ are equivalent to finding the natural evolution of the system under consideration.
There is no mathematical reason why you would want to pick any of these principles. In mechanics, they very often can be proven to be equivalent mathematically. But practically it turns out that the principle of least action has been easily extended to encompass more phenomena than the method with forces has been. Nowadays, most theories in physics, in particular field theories, are cast within the framework of the principle of least action.
answered Jan 18 at 9:23
RaskolnikovRaskolnikov
12.6k23571
$endgroup$
The idea is that you want to construct a functional $S:mathcal{P}→mathbb{R}$ where $mathcal{P}$ is the space of all possible paths your system can take through phase space. And then looking for the path that minimizes or maximizes the output value of the functional is equivalent to finding the natural evolution of the system you consider. It's not any more strange than saying that you have to construct a force $vec{F}$ and then the solutions of the equation $vec{F} = mvec{a}$ are equivalent to finding the natural evolution of the system under consideration.
There is no mathematical reason why you would want to pick any of these principles. In mechanics, they very often can be proven to be equivalent mathematically. But practically it turns out that the principle of least action has been easily extended to encompass more phenomena than the method with forces has been. Nowadays, most theories in physics, in particular field theories, are cast within the framework of the principle of least action.
answered Jan 18 at 9:23
RaskolnikovRaskolnikov
12.6k23571
answered Jan 18 at 9:23
RaskolnikovRaskolnikov
12.6k23571
answered Jan 18 at 9:23
RaskolnikovRaskolnikov
12.6k23571
answered Jan 18 at 9:23
RaskolnikovRaskolnikov
12.6k23571
12.6k23571
add a comment |
add a comment |
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$begingroup$
This is chapter 8.3. Presumably, the action integral would be explained earlier on in the text?
$endgroup$
– Raskolnikov
Jan 18 at 7:02
2
$begingroup$
The non-mathematical motivation for caring about the action integral is "because physics works that way": en.wikipedia.org/wiki/Principle_of_least_action. I don't know if there is a purely mathematical motivation for it.
$endgroup$
– Rahul
Jan 18 at 8:47
1
$begingroup$
The idea is that you want to construct a functional $S:mathcal{P}tomathbb{R}$ where $mathcal{P}$ is the space of all possible paths your system can take through phase space. And then looking for the path that minimizes or maximizes the output value of the functional is equivalent to finding the natural evolution of the system you consider. It's not any more strange than saying that you have to construct a force $vec{F}$ and then the solutions of the equation $vec{F}=mvec{a}$ are equivalent to finding the natural evolution of the system under consideration.
$endgroup$
– Raskolnikov
Jan 18 at 9:06
1
$begingroup$
This chapter of the Feynman lectures has been very helpful for me.
$endgroup$
– Giuseppe Negro
Jan 18 at 9:33
1
$begingroup$
@GiuseppeNegro This looks like it will helpful for me as well. Thank you for sharing.
$endgroup$
– Nalt
Jan 18 at 9:59