lagrangian Definition and Topics - 109 Discussions

Introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788, Lagrangian mechanics is a formulation of classical mechanics and is founded on the stationary action principle.
Lagrangian mechanics defines a mechanical system to be a pair



(
M
,
L
)


{\displaystyle (M,L)}
of a configuration space



M


{\displaystyle M}
and a smooth function



L
=
L
(
q
,
v
,
t
)


{\displaystyle L=L(q,v,t)}
called Lagrangian. By convention,



L
=
T

V
,


{\displaystyle L=T-V,}
where



T


{\displaystyle T}
and



V


{\displaystyle V}
are the kinetic and potential energy of the system, respectively. Here



q

M
,


{\displaystyle q\in M,}
and



v


{\displaystyle v}
is the velocity vector at



q


{\displaystyle q}




(
v


{\displaystyle (v}
is tangential to



M
)
.


{\displaystyle M).}
(For those familiar with tangent bundles,



L
:
T
M
×


R


t




R

,


{\displaystyle L:TM\times \mathbb {R} _{t}\to \mathbb {R} ,}
and



v


T

q


M
)
.


{\displaystyle v\in T_{q}M).}

Given the time instants




t

1




{\displaystyle t_{1}}
and




t

2


,


{\displaystyle t_{2},}
Lagrangian mechanics postulates that a smooth path




x

0


:
[

t

1


,

t

2


]

M


{\displaystyle x_{0}:[t_{1},t_{2}]\to M}
describes the time evolution of the given system if and only if




x

0




{\displaystyle x_{0}}
is a stationary point of the action functional






S


[
x
]





=


def









t

1





t

2




L
(
x
(
t
)
,



x
˙



(
t
)
,
t
)

d
t
.


{\displaystyle {\cal {S}}[x]\,{\stackrel {\text{def}}{=}}\,\int _{t_{1}}^{t_{2}}L(x(t),{\dot {x}}(t),t)\,dt.}
If



M


{\displaystyle M}
is an open subset of





R


n




{\displaystyle \mathbb {R} ^{n}}
and




t

1


,


{\displaystyle t_{1},}





t

2




{\displaystyle t_{2}}
are finite, then the smooth path




x

0




{\displaystyle x_{0}}
is a stationary point of





S




{\displaystyle {\cal {S}}}
if all its directional derivatives at




x

0




{\displaystyle x_{0}}
vanish, i.e., for every smooth



δ
:
[

t

1


,

t

2


]



R


n


,


{\displaystyle \delta :[t_{1},t_{2}]\to \mathbb {R} ^{n},}





δ


S







=


def







d

d
ε






|



ε
=
0




S



[


x

0


+
ε
δ

]

=
0.


{\displaystyle \delta {\cal {S}}\ {\stackrel {\text{def}}{=}}\ {\frac {d}{d\varepsilon }}{\Biggl |}_{\varepsilon =0}{\cal {S}}\left[x_{0}+\varepsilon \delta \right]=0.}
The function



δ
(
t
)


{\displaystyle \delta (t)}
on the right-hand side is called perturbation or virtual displacement. The directional derivative



δ


S




{\displaystyle \delta {\cal {S}}}
on the left is known as variation in physics and Gateaux derivative in Mathematics.
Lagrangian mechanics has been extended to allow for non-conservative forces.

View More On Wikipedia.org
  1. M

    Optimization: How to find the dual problem?

    Hi, I am working on the following optimization problem, and am confused how to apply the Lagrangian in the following scenario: Question: Let us look at the following problem \min_{x \in \mathbb{R}_{+} ^{n}} \sum_{j=1}^{m} x_j log(x_j) \text{subject to} A \vec{x} \leq b \rightarrow A\vec{x}...
  2. K

    Find the Conserved Quantity of a Lagrangian Using Noether's Theorem

    So Noether's Theorem states that for any invarience that there is an associated conserved quantity being: $$ \frac {\partial L}{\partial \dot{Q}} \frac {\partial Q}{\partial s}$$ Let $$ X \to sx $$ $$\frac {\partial Q}{\partial s} = \frac {\partial X}{\partial s} = \frac {\partial...
  3. K

    A Dissipation function is homogeneous in ##\dot{q}## second degree proof

    We have Rayleigh's dissipation function, defined as ## \mathcal{F}=\frac{1}{2} \sum_{i}\left(k_{x} v_{i x}^{2}+k_{y} v_{i j}^{2}+k_{z} v_{i z}^{2}\right) ## Also we have transformation equations to generalized coordinates as ##\begin{aligned} \mathbf{r}_{1} &=\mathbf{r}_{1}\left(q_{1}, q_{2}...
  4. Istiakshovon

    Find the values of A, B, and C such that the action is a minimum

    > A particle is subjected to the potential V (x) = −F x, where F is a constant. The particle travels from x = 0 to x = a in a time interval t0 . Assume the motion of the particle can be expressed in the form ##x(t) = A + B t + C t^2## . Find the values of A, B, and C such that the action is a...
  5. p1ndol

    I Trouble understanding coordinates for the Lagrangian

    Hello, I'm having some trouble understanding this solution provided in Landau's book on mechanics. I'd like to understand how they arrived at the infinitesimal displacement for the particles m1. I appreciate any kind of help regarding this problem, thank you!
  6. M

    I General relativity - covariant superconductivity, Meissner effect

    I am doing a project where the final scope is to find an extra operator to include in the proca lagrangian. When finding the new version of this lagrangian i'll be able to use the Euler-Lagrange equation to find the laws of motion for a photon accounting for that particular extra operator. I...
  7. Admiralibr123

    Sign of potential term in Lagrangian mechanics

    I have heard many times that it does not matter where you put the zero to calculate the potential energy and then ##L=T-V##. But mostly what we are doing is taking potential energy negative like in an atom for electron or a mass in gravitational field and then effectively adding it to kinetic...
  8. F

    Energy-momentum tensor for a relativistic system of particles

    I think it is quite simple as an exercise, following the two relevant equations, but at the beginning I find myself stuck in going to identify the lagrangian for a relativistic system of non-interacting particles. For a free relativistic particle I know that lagrangian is...
  9. lambdajitsu

    A Lagrangian for straight line in XY-plane (dependent on time)

    https://dst-public.s3-us-west-2.amazonaws.com/lagrangian.png
  10. R

    Particle constrained on a curve

    I tried 1. using the Lagrangian method: From ##y=-kx^2## I got ##\dot y = -2kx \dot x## and ##\ddot y = -2k \dot x^2 - 2 kx \dot x##. (Can I use ##\dot y = g## here due to gravity?) This gives for kinetic energy: $$T = \frac{1}{2} mv^2 = \frac{1}{2} m (\dot x^2 + \dot y^2) = \frac{1}{2} m (\dot...
  11. Pouramat

    Variation principle -- looking for resources to read and understand

    Summary:: Can anyone introduce an informative resource with solved examples for learning variation principle? For example I cannot do the variation for the electromagnetic lagrangian when ##A_\mu J^\mu## added to the free lagrangian and also some other terms which are possible: $$ L =...
  12. B

    The definition of generalised momentum

    Why, in lagrangian mechanics, do we calculate: ##\frac{d}{dt}\frac{\partial T}{\partial \dot{q}}## to get the (generalised) momentum change in time instead of ##\frac{d T}{dq}##? (T - kinetic energy; q - generalised coordinate; p - generalised momentum; for simplicity I assumed that no external...
  13. steve1763

    A Feynman Diagrams associated with the generating function of this free scalar field Lagrangian

    With free part L=-½(∂Φ)^2 -½m^2 Φ^2 and interaction term L=½gΦ^2 Any help would be appreciated, thank you.
  14. H

    Plane pendulum: Lagrangian, Hamiltonian and energy conservation

    Hello! I need some help with this problem. I've solved most of it, but I need some help with the Hamiltonian. I will run through the problem as I've solved it, but it's the Hamiltonian at the end that gives me trouble. To find the Lagrangian, start by finding the x- and y-positions of the...
  15. joneall

    I Gauge theory symmetry breaking in L&B

    I’m reading Lancaster & Blundell, Quantum field theory for the gifted amateur (even tho I”m only an amateur...) and have a problem with their explanation of symmetry breaking from page 242. They start with this Lagrangian: ## \mathcal{L} = (\partial_{\mu} \psi^{\dagger} - iq...
  16. F

    A Demonstration of the Brans-Dicke's Lagrangian

    Helo, The Lagrangian in general relativity is written in the following form: \begin {aligned} \mathcal {L} & = \frac {1} {2} g ^ {\mu \nu} \nabla \mu \phi \nabla \nu \phi-V (\phi) \\ & = R + \dfrac {16 \pi G} {c ^ {4}} \mathcal {L} _ {\mathcal {M}} \end {aligned} with ## g ^ {\mu \nu}: ## the...
  17. S

    I Wilson's RG trajectories, Lagrangians and many worlds?

    In this article [1] we can read an explanation about Wilson's approach to renormalization I have read that Kenneth G Wilson favoured the path integral/many histories interpretation of Feynman in quantum mechanics to explain it. I was wondering if he did also consider that multiple worlds...
  18. T

    Euler Lagrange equation and a varying Lagrangian

    Hello, I have been working on the three-dimensional topological massive gravity (I'm new to this field) and I already faced the first problem concerning the mathematics, after deriving the lagrangian from the action I had a problem in variating it Here is the Lagrangian The first variation...
  19. D

    Simple pendulum of variable mass

    Hello, I've got to rationally analice the form of the solutions for the equations of motion of a simple pendulum with a varying mass hanging from its thread of length ##l## (being this length constant). I approached this with lagrangian mechanics, asumming the positive ##y## direction is...
  20. J

    I Changing spherical coordinates in a Lagrangian

    In order to compute de lagrangian in spherical coordinates, one usually writes the following expression for the kinetic energy: $$T = \dfrac{1}{2} m ( \dot{r}^2 + r^2 \dot{\theta}^2 + r^2 \sin^2 \theta \dot{\phi}^2 )\ ,$$ where ##\theta## is the colatitud or polar angle and ##\phi## is the...
  21. polytheneman

    What is the derivative of velocity with respect to position?

    I'm reading a book on Classical Mechanics (No Nonsense Classical Mechanics) and one particular section has me a bit puzzled. The author is using the Euler-Lagrange equation to calculate the equation of motion for a system which has the Lagrangian shown in figure 1. The process can be seen in...
  22. M

    I Geodesics subject to a restriction

    Hi, I'm trying to solve a differential geometry problem, and maybe someone can give me a hand, at least with the set up of it. There is a particle in a 3-dimensional manifold, and the problem is to find the trajectory with the smallest distance for a time interval ##\Delta t=t_{1}-t_{0}##...
  23. peguerosdc

    Noether's theorem with non-finite transformations

    Hi! I am given the lagrangian: ## L = \dot q_1 \dot q_2 - \omega q_1 q_2 ## (Which corresponds to a 2D harmonic oscillator) And I am given two transformations and I am asked to say if there is a constant of motion associated to each transformation and to find it (if that's the case). I am...
  24. Wrichik Basu

    B How do you find the Lagrangians for different fields?

    I am currently studying QFT from this book. I have progressed to the chapter of QED. In the course, the authors have been writing the Lagrangian for different fields as and when necessary. For example, the Lagrangian for the complex scalar field is $$\mathcal{L} \ = \ (\partial ^\mu...
  25. SamRoss

    I Missing exponent in "Theoretical Minimum"?

    In "The Theoretical Minimum" (the one on classical mechanics), on page 218, the authors write a Lagrangian $$L=\frac m 2 (\dot r^2 +r^2\dot \theta ^2)+\frac {GMm} r$$ They then apply the Euler-Lagrange equation ##\frac d {dt}\frac {dL} {d\dot r}=\frac {dL} {dr}## (I know there should be...
  26. S

    I Interpreting SM Lagrangian "differential particle" terms

    The Standard Model Lagrangian contains terms like these: ##-\partial_\mu \phi^+ \partial_\mu\phi^-## ##-\frac{1}{2}\partial_\nu Z^0_\mu\partial_\nu Z^0_\mu## ##-igc_w\partial_\nu Z^0(W^+_\mu W^-_\nu-W^+_\nu W^-_\mu)## How should one interpret the "derivative particle fields" like...
  27. M

    Is it possible to integrate acceleration?

    Alright so I was just messing around with Lagrangian equation, I just learned about it, and I had gotten to this equation of motion: Mg*sin{α} - 1.5m*x(double dot)=0 I am trying to get velocity, and my first thought was to integrate with dt, but I didn't know how to. And I'm not even sure it's...
  28. filip97

    I The Lagrangian and the second derivative?

    Why Lagrangian not depend of higher derivatives of generalised coordinates ?
  29. S

    A Lagrangian vs pseudo-Lagrangian vs Eulerian

    I'm reading up a series of papers on hydrodynamical simulations for galaxies and cosmology. They keep mentioning things like "Lagrangian" or "pseudo-lagrangian" or "Eulerian". I tried looking it up on the internet, but the answers are either too broad and could mean a huge number of things in...
  30. Cathr

    Question about a derivation: velocity and position

    I started studying Lagrangian mechanics, and the movement equation is like this: d/dt (d/dz') L - d/dz L = 0 if the movement is on the z axis. Now the problem is, let's say L = M(z')2/2 - Mgz. How do we derivate an expression depending of z with respect to z' and also , an expression depending...
Top