For questions involving the Lagrangian formulation of a dynamical system. Namely, the application of an action principle to a suitably chosen Lagrangian or Lagrangian Density in order to obtain the equations of motion of the system.

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190 views

Derivation of Euler-Lagrange equations for Lagrangian with dependence on second order derivatives

Suppose we have a Lagrangian that depends on second-order derivatives: $$L = L(q, \dot{q}, \ddot{q})$$ If we're working on the variational problem for this Lagrangian, then I know that we'll wind up ...
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3answers
180 views

Lagrangian for relativistic massless particle

For relativistic massive particle, the action is $$S ~=~ -m_0 \int ds ~=~ -m_0 \int d\lambda ~(\dot x ^\mu \dot x_\mu)^{\frac{1}{2}} ~=~ \int d\lambda \ L,$$ where $ds$ is the proper time of the ...
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0answers
83 views

Isn't the Jacobi constant just the Lagrangian times 2?

At this wikipedia page the Jacobi constant is expressed as: $$C_J=2\left(\frac{v^2}{2}-U\right)$$ where $U$ is the potential energy and $v$ is velocity. If kinetic energy $T$ is defined (as it ...
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0answers
84 views

Lagrangian with vanishing conjugate momentum, independent variables

Given a Lagrangian density $\mathcal L(\phi_r,\partial_\mu\phi_r,\phi_n,\partial_\mu\phi_n)$, for which we find out that for some $\phi_n$ its conjugate momentum vanishes: ...
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245 views

How to understand the QED, QCD and standard model Lagrangians? [closed]

How do you read the QED, QCD and standard model Lagrangians? What do all the symbols and tensors represent? And, how can you derive them by yourselves?
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62 views

Setting Lagrangian [closed]

Can you help me to set Lagrangian? I found that $$\vec r_A=b\sin\theta\vec i+b\cos\theta\vec j$$ $$\dot{\vec r_A}=b\dot\theta\cos\theta\vec i-b\dot\theta\sin\theta\vec j$$ For point $G$ I've got ...
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161 views

Angular momentum of particle in dipole magnetic field

Basically I'm just trying to find the expression for the angular momentum of a particle of mass $m$ and charge $q$ in a dipole magnetic field. In cylindrical coordinates, ...
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6answers
1k views

How are constraint forces represented in Lagrangian mechanics?

Suppose we try to obtain the movement equation for a particle sliding on a sphere (no friction, ideal bodies...). The only forces acting on the particle are its weight and - here's my problem - a ...
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1answer
529 views

What gauge is used in the Lagrangian for a non-relativistic point particle in an electromagnetic potential

For the Lagrangian $$L = \frac{1}{2}mu^2 - q(\phi - \frac{\vec{A}}{c}\cdot \vec{u})$$ of a non-relativistic point particle in an electromagnetic potential, what gauge is used for the electromagnetic ...
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2answers
315 views

First integral of relativistic Euler-Lagrange equations

Connsider a pseudo-Riemannian ($4$-dimensional) manifold $M$ with a pseudometric $g_{ab}$. The Lagrangian of a free particle in $M$ (in analogy to the flat case) is $$\mathcal ...
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2answers
333 views

Lorentz invariance of the action for free relativistic particle

I tried to check the Lorentz invariance of the standard special relativity action for free particle directly: ($c=1$) $$ S=\int L dt=-m\int\sqrt{1-v^{2}}dt $$ Lorentz boost: $$ ...
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2answers
585 views

Euler-Lagrange equations and friction forces

We can derive Lagrange equations supposing that the virtual work of a system is zero. $$\delta W=\sum_i (\mathbf{F}_i-\dot {\mathbf{p}_i})\delta \mathbf{r}_i=\sum_i ...
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3answers
352 views

What is the definition of momentum when a mass distribution $\rho(r,t)$ is given?

This question is Edited after recieving comments. What is the definition of momentum when a mass distribution $\rho(r,t)$ is given? Assuming a particle as a point mass we know the definition of ...
2
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1answer
124 views

A Question on Hamilton's Principle

In some literatures, the Hamilton's principle for conservative systems is introduced by this equation: $$\delta \int_{t_1}^{t_2}(T-V) ~\mathrm{d}t~=~0$$ In some others, this principle is introduces ...
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2answers
75 views

Intuition Behind Conservation of Angular Momentum

I'm having a fairly hard time understanding the intuition behind Noether's derivation of the conservation of angular momentum from the rotational invariance of the Lagrangian, though I do understand ...
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1answer
164 views

Is Lagrangian a scalar?

I may be wrong: Lagrangian are scalars. They are NOT invariant under coordinate transformations. The simplest example is when you have a gravitational potential ($V=mgz$) and you translate $z$ by $a$ ...
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2answers
1k views

Expression of kinetic energy in polar coordinates

Expression for kinetic energy in Cartesian coordinate: Expression for kinetic energy in polar coordinate (applying the transformation of coordinates): Why can't we express it in the following ...
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3answers
411 views

Lagrangian mechanics and time derivative on general coordinates

I am reading a book on analytical mechanics on Lagrangian. I get a bit idea on the method: we can use any coordinates and write down the kinetic energy $T$ and potential $V$ in terms of the general ...
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4answers
532 views

Why the Hamiltonian and the Lagrangian are used interchangeably in QFT perturbation calculations

Whenever one needs to calculate correlation functions in QFT using perturbations one encounters the following expression: $\langle 0| some\ operators \times \exp(iS_{(t)}) |0\rangle$ where, ...
2
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2answers
527 views

What is the significance of action?

What is the physical interpretation of $$ \int_{t_1}^{t_2} (T -V) dt $$ where, $T$ is Kinetic Energy and $V$ is potential energy. How does it give trajectory?
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1answer
408 views

Spontaneous symmetry breaking and 't Hooft and Polyakov monopoles

What is spontaneous symmetry breaking from a classical point of view. Could you give some examples, using classical systems.I am studying about the 't Hooft and Polyakov magnetic monopoles solutions, ...
2
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1answer
81 views

Schrödinger evolution for a Klein-Gordon equation

I have a problem with the transition from quantum relativistic wave equations (specifically Klein-Gordon equation) to QFT, since a lot of assumptions seem implicit. For example I have a problem with ...
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2answers
167 views

How can I tell that circular motion is a solution for a particle confined to the surface of a cone?

I'm working on a problem where a particle of mass $m$ is confined to the surface of an inverted half cone (and is circling downwards due to gravity), with the cone's half angle $\alpha$. I chose to ...
2
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1answer
153 views

Lorentz force equation from relativistic Lagrangian

The relativistic Lagrangian is given by $$L = - m_0 c^2 \sqrt{1 - \frac{u^2}{c^2}} + \frac{q}{c} (\vec u \cdot \vec A) - q \Phi $$ I need to derive, $\displaystyle \frac{d\vec p}{dt} = q \left( \vec E ...
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1answer
152 views

Can the Lagrange Multipliers depend on the coordinates?

When dealing with Lagrange multipliers to solve systems with constraints we usually have two ways if the constraints are holonomic: Differentiate the constraint and add the appropiate term to the ...
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1answer
105 views

Expansion of a function

In Landau-Lifschitz, following expansion is given, We have, $$L(v'^2)~=~L(v^2+2\textbf{v}\cdot\epsilon+\epsilon ^2)$$ expanding this in powers of $\epsilon$ and neglecting powers of higher order, ...
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1answer
142 views

Does a Lagrangian imply a well-defined quantum Hamiltonianian with a Hilbert space?

The question is about: (1) whether giving a Lagrangian is sufficient enough to (uniquely) well-define a Hamiltonianian quantum theory with a Hilbert space? The answer should be Yes, or No. If ...
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1answer
309 views

How do you derive Lagrange's equation of motion from a Routhian?

Given a Routhian $R(r,\dot{r},\phi,p_{\phi})$, how do you derive Lagrange's equation for $r$? Do you just solve the following for $r$? $$\frac{d}{dt}\frac{∂R}{∂\dot{\phi}}-\frac{∂R}{∂\phi}=0$$ And ...
2
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1answer
183 views

Motivating the Legendre Transform Mathematically

If I begin with a functional of the form $$J[y] = \int_a^b f(x,y,y')dx$$ and find its Euler-Lagrange equations $$\frac{\partial f}{\partial y} - \frac{d}{dx}\frac{\partial f}{\partial y'} = 0 = ...
2
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1answer
390 views

Retrieving Maxwell's equations from the minimum action principle

I'm currently working at the start of Alexei Tsvelik's book Quantum Field Theory in Condensed Matter Physics. I'm kinda stumped on a few essential steps. Starting with the action: $$S = \int dt \int ...
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2answers
213 views

Does a constant factor matter in the definition of the Noether current?

This is a very basic Lagrangian Field Theory question, it is about a definition convention. It takes much more time to typeset it than answering, but here it is: Consider a field Lagrangian with only ...
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1answer
319 views

How to tell local and non-local in QFT?

I'm taking QFT course in this term. I'm quite curious that in QFT by which part of the mathematical expression can we tell a quantity or a theory is local or non-local?
2
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1answer
345 views

Why has the trace of the energy-momentum tensor to vanish for conserved scaling currents to exist?

In this paper, the authors say that the trace of the energy-momentum tensor has to vanish to allow for the existence of conserved dilatation or scaling currents, as defined on p 10, Eq(22) $$ ...
2
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3answers
361 views

Can the Euler-Lagrange equations be derived from an infinitesimal Principle of Least Action?

The Euler-Lagrange equations can be derived from the Principle of Least Action using integration by parts and the fact that the variation is zero at the end points. This has a mystical air about it, ...
2
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1answer
95 views

Mass term in the Lagrangian

I have read that the mass term appearing in the electroweak Lagrangian stops it (the Lagrangian) from becoming gauge invariance. Can someone explain where and why this term is creating the problem?
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1answer
123 views

Lagrangian depends on second derivative of field

In case of the gauge-fixed Faddeev-Popov Lagrangian: $$ \mathcal{L}=-\frac{1}{4}F_{\mu\nu}\,^{a}F^{\mu\nu ...
2
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1answer
127 views

Curved spacetime point particle Lagrangian density

This is probably trivially related to the question: Action for a point particle in a curved spacetime , but am a bit unsure how to write it as a Lagrangian density. In curved spacetime the action is ...
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4answers
366 views

Are Lagrangians and Hamiltonians used by Engineers?

Analytical Mechanics (Lagrangian and Hamiltonian) are useful in Physics (e.g. in Quantum Mechanics) but are they also used in application, by engineers? For example, are they used in designing bridges ...
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1answer
175 views

Definition of Kinetic energy

In class we had that $ T= \frac{1}{2}T_{ij}v_iv_j$ where we used the Einstein summation convention. Hitherto we only discussed examples where the kinetic energy was dependent of the square of one ...
2
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2answers
174 views

A kind of Noether's theorem for the Hamiltonian

How can I (conveniently?) show that an invariance of the Lagrangian and Hamiltonian (i.e. the kinetic as well as the potential energy are independently invariant) will lead to a conservation law using ...
2
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2answers
299 views

Different approaches to calculating the Christoffel symbols

I would be very grateful to whoever can debug the following calculations... We have the metric for static spacetime: $$ds^2 = -\exp(2U(\vec x))dt^2+h_{ij}(\vec x) d x^i d x^j$$ I want to find the ...
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2answers
253 views

Calculating the (on-shell) action of a free particle

I am having difficulty with the first problem from Feynman and Hibbs' book. For a free particle $L = (m/2)\dot{x}^2$. Show that the (on-shell) action $S_{cl}$ corresponding to the classical ...
2
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3answers
263 views

Is the path of stationary action unique? What are the physical implications of $L_{\dot{x}}=L_x$

Below, for any function $Q$ the notation $Q_x$ means $\frac{\partial Q}{\partial x}$, and $Q_{xx}$ means $\frac{\partial^2 Q}{\partial x^2}$. In physics, the trajectory of a particle is given by the ...
2
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1answer
106 views

Obtaining the conserved current of the Lagrangian making the parameter depending on $x$

To calculate the conserved current due to an internal symmetry of the system (expressed by the Lagrangian density) we can proceed as follows: if it is invariant under $\delta \phi = \alpha \phi$, ...
2
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1answer
274 views

Euler-Lagrange Equation

A particle moving towards the origin has initial conditions $x(t=0) = 1$ and $\dot{x}(t=0)=0$ If the Lagrangian is L:=$\frac{m}{2}\dot{x}^2 -\frac{m}{2}ln|x|$ This should satisfy Euler Lagrange ...
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1answer
171 views

Relationship between local and global scaling (Weyl) symmetry

Theorem 5.1 on page 80 of this paper says that Assuming that the matter fields satisfy their equations of motion, the matter field action is locally Weyl invariant if and only if the corresponding ...
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2answers
201 views

Is the gravitational constant G a minimum value in some sense?

Assume a central body of mass $M$, and call $a$ the acceleration of a test body at a distance $r$ due to any interaction whatsoever with the central body. Is is correct to say that the ratio $a r^2/ ...
2
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1answer
243 views

Do Lagrangian points actually maintain a fixed distance?

I was reading on up Lagrangian points and the restricted three-body problem. From what I was able to tell, the Lagrangian points are 5 points in a two-body system such that a third body would be ...
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3answers
4k views

Finding Lagrangian of a Spring Pendulum

I'm trying to understand Morin's example of a spring pendulum. What I don't get is his expression for $T$. I can understand the $\dot x^2$ term in the brackets. But I don't understand the $(l + ...
2
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1answer
426 views

Origins of the principle of least time in classical mechanics

Is it possible to derive the principle of least time from the principle of least action in lagrangian or hamiltonian mechanics? Or is Fermat's principle more fundamental than the principle of least ...