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Then, the Euler-Lagrange equation gives the maxima and minima of the that type of functional. Note that this type of functional is just one very specific type of functional amongst all possible functionals that one might come up with. However, it turns out to be enough to do most of physics, so we are happy with with it.

Given

F

, the Euler-Lagrange equations are a system of ordinary differential equations constructed from that

F

such that the solutions to that system are the maxima/minima.

In the one dimensional case, the system has a single ordinary differential equation:

\frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q} - \frac{d}{d t} \frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q ˙} = 0

(2)

\frac{\partial F}{\partial q}

and

\frac{\partial F}{\partial q ˙}

we simply mean "the partial derivative of

F

with respect to its second and third arguments". The notation is a bit confusing at first, but that's all it means.

Therefore, that expression ends up being at most a second order ordinary differential equation where

q

is the unknown, since:

the term $\frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q}$ is a function of $t, q (t), \overset{q}{˙} (t)$
the term $\frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q ˙}$ is a function of $t, q (t), \overset{q}{˙} (t)$ . And so it's derivative with respect to time will contain only up to $\overset{q}{¨}$

Now let's think about the multi-dimensional case. Instead of having

q : R \to R

, we now have

q : R \to R^{n}

. Think about the Lagrangian mechanics motivation of a double pendulum where for a given time we have two angles.

Let's do the 2-dimensional case then. In that case,

F

is going to be a function of 5 variables

F : R^{5} \to R

rather than 3 as in the one dimensional case, and the functional looks like:

I (q) = \int_{t_{0}}^{t} F (t, q_{1} (t), q_{2} (t), \overset{q_{1}}{˙} (t), \overset{q_{2}}{˙}) (t) d t

(3)

This time, the Euler-Lagrange equations are going to be a system of two ordinary differential equations on two unknown functions

q_{1}

and

q_{2}

of order up to 2 in both variables:

\frac{\partial F ( t , q _{1} ( t ) , q _{2} ( t ) , q _{1} ˙ ( t ) , q _{2} ˙ ( t ))}{\partial q _{1}} - \frac{d}{d t} \frac{\partial F ( t , q _{1} ( t ) , q _{2} ( t ) , q _{1} ˙ ( t ) , q _{2} ˙ ( t )}{\partial q _{1} ˙} = 0 \frac{\partial F ( t , q _{1} ( t ) , q _{2} ( t ) , q _{1} ˙ ( t ) , q _{2} ˙ ( t ))}{\partial q _{2}} - \frac{d}{d t} \frac{\partial F ( t , q _{1} ( t ) , q _{2} ( t ) , q _{1} ˙ ( t ) , q _{2} ˙ ( t )}{\partial q _{2} ˙} = 0

(4)

At this point, notation is getting a bit clunky, so people will often condense the

q_{i}

vector

\frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q _{1}} - \frac{d}{d t} \frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q _{1} ˙} = 0 \frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q _{2}} - \frac{d}{d t} \frac{\partial F ( t , q ( t ) , q ˙ ( t ))}{\partial q _{2} ˙} = 0

(5)

or just omit the arguments of

F

entirely:

\frac{\partial F}{\partial q _{i}} - \frac{d}{d t} \frac{\partial F}{\partial q _{i} ˙} = 0

(6)

Video 1.

Calculus of Variations ft. Flammable Maths by vcubingx (2020)

Source.

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Equations of motion by

Ciro Santilli 40 Updated 2025-07-16

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These are the final equations that you derive from the Lagrangian via the Euler-Lagrange equation which specify how the system evolves with time.

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Lagrangian (field theory) by

Ciro Santilli 40 Updated 2025-07-16

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Basically: S.

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Lagrangian density by

Ciro Santilli 40 Updated 2025-07-16

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When we particles particles, the action is obtained by integrating the Lagrangian over time:

A c t i o n = \int_{t_{0}}^{t} L (x (t), \frac{\partial x ( t )}{\partial t}, t) d t

(1)

In the case of field however, we can expand the Lagrangian out further, to also integrate over the space coordinates and their derivatives.

Since we are now working with something that gets integrated over space to obtain the total action, much like density would be integrated over space to obtain a total mass, the name "Lagrangian density" is fitting.

E.g. for a 2-dimensional field

f (x, y, t)

A c t i o n = \int_{t_{0}}^{t} L (f (x, y, t), \frac{\partial f ( x , y , t )}{\partial x}, \frac{\partial f ( x , y , t )}{\partial y}, \frac{\partial f ( x , y , t )}{\partial t}, x, y, t) d x d y d t

(2)

Of course, if we were to write it like that all the time we would go mad, so we can just write a much more condensed vectorized version using the gradient with

x \in R^{2}

A c t i o n = \int_{t_{0}}^{t} L (f (x, t), \nabla f (x, t), x, t) d x d t

(3)

And in the context of special relativity, people condense that even further by adding

t

to the spacetime Four-vector as well, so you don't even need to write that separate pesky

t

The main point of talking about the Lagrangian density instead of a Lagrangian for fields is likely that it treats space and time in a more uniform way, which is a basic requirement of special relativity: we have to be able to mix them up somehow to do Lorentz transformations. Notably, this is a key ingredient in a/the formulation of quantum field theory.

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Noether's theorem by

Ciro Santilli 40 Updated 2025-07-16

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For every continuous symmetry in the system (Lie group), there is a corresponding conservation law.

Furthermore, given the symmetry, we can calculate the derived conservation law, and vice versa.

As mentioned at buzzard.ups.edu/courses/2017spring/projects/schumann-lie-group-ups-434-2017.pdf, what the symmetry (Lie group) acts on (obviously?!) are the Lagrangian generalized coordinates. And from that, we immediately guess that manifolds are going to be important, because the generalized variables of the Lagrangian can trivially be Non-Euclidean geometry, e.g. the pendulum lives on an infinite cylinder.

Video 1.

The most beautiful idea in physics - Noether's Theorem by Looking Glass Universe (2015)

Source. One sentence stands out: the generated quantities are called the generators of the transforms.

Video 2.

The Biggest Ideas in the Universe | 15. Gauge Theory by Sean Carroll (2020)

Source. This attempts a one hour hand wave explanation of it. It is a noble attempt and gives some key ideas, but it falls a bit short of Ciro's desires (as would anything that fit into one hour?)

Video 3.

The Symmetries of the universe by ScienceClic English (2021)

Source. youtu.be/hF_uHfSoOGA?t=144 explains intuitively why symmetry implies consevation!

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Time invariance implies energy conservation by

Ciro Santilli 40 Updated 2025-07-16

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physics.stackexchange.com/questions/614757/intuitive-explanation-for-why-time-symmetry-implies-conservation-of-energy on Physics Stack Exchange

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Welcome to the OurBigBook Project! Our goal is to create the perfect publishing platform for STEM subjects, and get university-level students to write the best free STEM tutorials ever.

Everyone is welcome to create an account and play with the site: ourbigbook.com/go/register. We belive that students themselves can write amazing tutorials, but teachers are welcome too. You can write about anything you want, it doesn't have to be STEM or even educational. Silly test content is very welcome and you won't be penalized in any way. Just keep it legal!

Video 1.

Intro to OurBigBook

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This feature makes it possible for readers to find better explanations of any topic created by other writers. And it allows writers to create an explanation in a place that readers might actually find it.
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OurBigBook Web topics demo
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