# Principle of Least Action

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I have seen how very useful the principle of least action is, but don't really understand why the integral of the Lagrangian with respect to time is minimized. It seems to say the most efficient way to get from A to B is via a "path" that brings kinetic energy closest to potential energy. (Yes?) If so, then why is that most efficient?

Thanks for any insights.

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Nature is essentially "lazy." If potential energy gets converted into kinetic energy (i.e. the potential does work on the particle) then this is reflected by a change in the Lagrangian: L=T-V. Nature doesn't want to do work, so the Lagrangian should change as little as possible over the particle's trajectory. The action is the changes in the Lagrangian added up over the entire trajectory, i.e. essentially how much work was done over the particle's path. So the path that minimizes the action is the "laziest" possible path.

Edited by elfmotat
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The principle of the "laziest possible path" - I like that.

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The principle of the "laziest possible path" - I like that.

Story of my life.

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Nature is essentially "lazy." If potential energy gets converted into kinetic energy (i.e. the potential does work on the particle) then this is reflected by a change in the Lagrangian: L=T-V. Nature doesn't want to do work, so the Lagrangian should change as little as possible over the particle's trajectory. The action is the changes in the Lagrangian added up over the entire trajectory, i.e. essentially how much work was done over the particle's path. So the path that minimizes the action is the "laziest" possible path.

This argument seems to be true

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Nature is essentially "lazy." If potential energy gets converted into kinetic energy (i.e. the potential does work on the particle) then this is reflected by a change in the Lagrangian: L=T-V. Nature doesn't want to do work, so the Lagrangian should change as little as possible over the particle's trajectory. The action is the changes in the Lagrangian added up over the entire trajectory, i.e. essentially how much work was done over the particle's path. So the path that minimizes the action is the "laziest" possible path.

The existence of the universe itself seems to contradict that "Nature is essentially lazy".

The Big Bang is not a lazy thing.

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The existence of the universe itself seems to contradict that "Nature is essentially lazy".

The Big Bang is not a lazy thing.

The big bang is irrelevant to the conversation and your argument makes absolutely no sense.

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The big bang is irrelevant to the conversation and your argument makes absolutely no sense.

hmm.

The BB is not a natural phenomenon ?

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hmm.

The BB is not a natural phenomenon ?

Your premise that "The Big Bang is not a lazy thing" is ill-defined. How do you figure that? What about the big bang qualifies it as a "non-lazy phenomenon?"

The post was also entirely irrelevant to the conversation. In the context of classical mechanics (which, as the section name suggests, is what we're discussing), saying that nature is "lazy" (where I implied in my previous post that "lazy" is by definition doing as little work as possible) is completely accurate. This is the premise of "the least action principle" (which is what the thread is about), and it is an experimentally confirmed principle. If you don't know what the conversation is about then I suggest you refrain from making posts which may make you look silly.

In any case, this thread has been sufficiently derailed.

**I also see that I've been -rep'd for my first post. I'm not suggesting it was you, but whoever it was is certainly being immature.

Edited by elfmotat
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The Big Bang is not a lazy thing.

In order to define what's lazy and what is not you need to look at how fast something proceeds. That has a factor of time in it. Before the Big Bang, we think there was no time (although this is hotly debated on our forum). My point is that since we either say that time started at the Big Bang, or there may have been an infinite amount of time prior to the Big Bang, the only conclusion can be that the Big Bang was possibly the laziest thing ever.

According to one theory, nature had to invent time to get it to move. How is that not lazy?

According to the other, nature waited an infinite amount of time before the Big Bang. If you arrive infinitely late at work, you're really lazy.

(Please do not take this reply too serious - it's probably so wrong that you would have to write several pages of text to explain all my mistakes)

In any case, this thread has been sufficiently derailed.

I agree. And I made it worse. Sorry.

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Your premise that "The Big Bang is not a lazy thing" is ill-defined. How do you figure that? What about the big bang qualifies it as a "non-lazy phenomenon?"

You wrote:

Nature is essentially "lazy." If potential energy gets converted into kinetic energy (i.e. the potential does work on the particle) then this is reflected by a change in the Lagrangian: L=T-V. Nature doesn't want to do work (...)

What I say is that this constatation is not compatible with the BB theory. A nature that "doesn't want to work" doesn't create billion of billion of galaxies full of burning stars.

In the context of classical mechanics (which, as the section name suggests, is what we're discussing), saying that nature is "lazy" (where I implied in my previous post that "lazy" is by definition doing as little work as possible) is completely accurate. This is the premise of "the least action principle" (which is what the thread is about), and it is an experimentally confirmed principle.

True.

If you don't know what the conversation is about then I suggest you refrain from making posts which may make you look silly.

I don't care looking silly.

**I also see that I've been -rep'd for my first post. I'm not suggesting it was you, but whoever it was is certainly being immature.

That was me! My wife may agree with you.

The neg was not for you, it was against the 2 positives who vote without thinking twice.

The least action principle is the most disturbing observation. Following this principle, nothing would never start.

What we observe in classical physics is a principle that works into a system that already flows. But to make this system flow, you'd need another principle, exactly opposite, that we observe nowhere.

A principle that would say that nature likes to spread energy away, in conformity to the BB hypothesis.

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The least action principle is the most disturbing observation. Following this principle, nothing would never start.

What we observe in classical physics is a principle that works into a system that already flows. But to make this system flow, you'd need another principle, exactly opposite, that we observe nowhere.

A principle that would say that nature likes to spread energy away, in conformity to the BB hypothesis.

The principle of least action does not say that nature will never do work. It says that nature will never do more work than is necessary, which is a minor, but fundamental, difference. (At least from my understanding of it - admittedly, it could be flawed).

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So...we all agree that Nature is a lazy, but efficient, hard working but wasteful bastard?

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In Quantum Field Theory, the principle of least action is nicely explained by the path integral formalism. Basically, the system follows all the possible paths at once, but each is weighted by a complex phase (depending on the action). Adding up all the contributions, they almost all cancel one another out except for the path of least action (because it is an extremum).

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What I say is that this constatation is not compatible with the BB theory. A nature that "doesn't want to work" doesn't create billion of billion of galaxies full of burning stars.

How much work does it take to make a star?

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I have seen how very useful the principle of least action is, but don't really understand why the integral of the Lagrangian with respect to time is minimized. It seems to say the most efficient way to get from A to B is via a "path" that brings kinetic energy closest to potential energy. (Yes?) If so, then why is that most efficient?

Thanks for any insights.

The name Principle of Least Action is a misnomer. Its more accurate to refer to it as the Principle of Stationary Action since all that is required is that the first variation vanish.

Think of the analogy with normal functions. We consider here functions that are defined and continuous on open intervals. If the first derivative of a function is zero at a point then there is either a maximum, minimum or extremum at that point. If the second derivative of the function at that same point is positive then the function is a minimum at that point. If its positive then it’s a maximum and if its zero then its an extremal at that point.

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A variation ( precursor ) that applies to massless particles only, is Fermats principle of least time. I first understood both, least time and least action, by the analysis of the path taken by light as it travels through two mediums with different refractive indices ( like water and air ) and the path taken minimises the travel time by going through the media at different angles. Of course this was thirty years ago, but it may be helpful to people.

Edited by MigL

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