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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3
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2answers
937 views

Where does the mass term come from in the Proca Lagrangian?

There are many good books describing how to construct the Lagrangian for an electromagnetic field in a medium. $$ \mathcal{L}~=~-\frac{1}{16\pi}F^{\mu\nu}F_{\mu\nu}-\frac{1}{c}j^{\nu}A_{\nu} $$ ...
1
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0answers
92 views

relevant 4-dimensional theory with interacting vector field

A simple langragian that gives the simplest interaction is $\mathcal{L}=(\partial\phi)^2+(m\phi)^2$ where $m$ is some constant. Does anyone know of theory in four dimensions which is physically ...
0
votes
1answer
2k views

Hamilton's equations for a simple pendulum

I don't get how to use Hamilton's equations in mechanics, for example let's take the simple pendulum with $$H=\frac{p^2}{2mR^2}+mgR(1-\cos\theta)$$ Now Hamilton's equations will be: $$\dot ...
37
votes
4answers
2k views

Is there a Lagrangian formulation of statistical mechanics?

In statistical mechanics, we usually think in terms of the Hamiltonian formalism. At a particular time $t$, the system is in a particular state, where "state" means the generalised coordinates and ...
11
votes
1answer
839 views

Do an action and its Euler-Lagrange equations have the same symmetries?

Assume a certain action $S$ with certain symmetries, from which according to the Lagrangian formalism, the equations of motion (EOM) of the system are the corresponding Euler-Lagrange equations. Can ...
0
votes
2answers
413 views

Hamiltonian and non conservative force

I have to find the Hamiltonian of a charged particle in a uniform magnetic field; the potential vector is $ \vec {A}= B/2 (-y, x, 0)$. I know that $$H=\sum_i p_i \dot q_i -L$$ where $p_i$ is ...
2
votes
1answer
465 views

Electrodynamics and the Lagrangian density

Could anyone tell me what equations can I obtain from the Lagrangian density $${\cal L}(\phi,\,\,\phi_{,i},\,\,A_i, \dot A_i,\,\,A_{i,j})~=~\frac{1}{2}|\dot A+\nabla\phi|^2-\frac{1}{2}|\nabla \times ...
32
votes
13answers
2k views

Lagrangians not of the form $T-U$

My Physics teacher was reluctant to define Lagrangian as Kinetic Energy minus Potential Energy because he said that there were cases where a system's Lagrangian did not take this form. Are you are ...
3
votes
5answers
1k views

Good book for Analytical Mechanics

What is a good book for Analytical Mechanics? To be more specific, I would prefer a book that: Is written "for mathematicians", i.e. with high mathematics precision (for example, with less emphasis ...
0
votes
0answers
369 views

How to find angular velocity of a point inner a circumference

Let's consider a cicumference that have the center in the origin of axes and rotates around x-axes. Let's stick a bar in a point $A$ of this circumference and at the end of the bar let's stick a mass ...
2
votes
3answers
335 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
votes
1answer
128 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 $\delta \phi = \alpha \phi$, ...
4
votes
3answers
368 views

Writing $\dot{q}$ in terms of $p$ in the Hamiltonian formulation

In the Hamiltonian formulation, we make a Legendre transformation of the Lagrangian and it should be written in terms of the coordinates $q$ and momentum $p$. Can we always write $dq/dt$ in terms of ...
2
votes
6answers
3k 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 ...
4
votes
2answers
2k views

Lorentz invariance of the integration measure

This is regards to the lorentz invariance of a classical scalar field theory. We assume that the action which is $S= \int d^4 x \mathcal{L}$, is invariant under a Lorentz transformation. How do you ...
1
vote
0answers
390 views

Normal modes of oscillation: how to find them

Are normal modes the eigenvectors of the matrix $(\omega ^2 T- V)$ where $T$ is the matrix of kinetic energy and $V$ is the matrix of potential energy? Is it the only way to express them? How can I ...
2
votes
1answer
298 views

Scalar Field Theory Decay/Scattering

I have a few questions related to the following interaction Lagrangian (no use of crossing symmetry in the following) involving the uncharged scalar $\chi$ and the charged scalar $\phi$: ...
7
votes
1answer
582 views

Lagrangian of 2D square lattice of point masses connected by springs

Zee's QFT book mentions the Lagrangian of a square 2D horizontal lattice of point masses, connected by springs, and considering only vertical displacements $q_{i}$, as $ L = \frac{1}{2} ...
9
votes
1answer
282 views

Lagrangian for Goldstone mode + topological excitation

The XY-model Hamiltonian is the following, $${\cal H}~=~-J\sum_{\langle i,j\rangle} \cos (\theta_i -\theta_j).$$ The Goldstone mode corresponds to term $(\nabla \theta)^2$ in the effective ...
2
votes
4answers
680 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, ...
9
votes
4answers
308 views

What makes an equation an 'equation of motion'?

Every now and then, I find myself reading papers/text talking about how this equation is a constraint but that equation is an equation of motion which satisfies this constraint. For example, in the ...
2
votes
1answer
351 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 ...
1
vote
1answer
203 views

A particular case when Lagrange equation is equivalent to equation of motion on a Riemannian manifold

Suppose a particle is moving on a surface of a sphere,then it contains a holonomic constraint and so the three Cartesian co-ordinates are available with a constraint equation(equation of surface in ...
2
votes
1answer
204 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 ...
2
votes
1answer
349 views

Application of Noether's theorem

Consider one parameter transformation: $y = y ( \tilde{y}, \alpha)$ such that lagrangian satisfies: $\tilde{L}(\tilde{y}, \alpha) = L(y ( \tilde{y}, \alpha))$. We say that equation is invariant ...
1
vote
1answer
326 views

What's the motivation behind the action principle? [closed]

What's the motivation behind the action principle? Why does the action principle lead to Newtonian law? If Newton's law of motion is more fundamental so why doesn't one derive Lagrangians and ...
1
vote
2answers
377 views

What is the relativistic action of a massive particle?

all Lorentz observers watching a particle move will compute the same value for the quantity $$ds^2 = -(c \, dt)^2 + dx^2 + dy^2 + dz^2,$$ $$ds^2 = g_{\mu\nu}dx^{\mu}dx^{\nu},$$ and ''ds/c'' is then ...
1
vote
3answers
187 views

Constructing Supersymmetric Lagrangians

It is a very trivial doubt but somehow I am not able to figure it out. While constructing a supersymmetric lagrangian we always even number of fermionic fields. One reason is of course the product ...
12
votes
3answers
5k views

What exactly are Hamiltonian Mechanics (and Lagrangian mechanics)

What exactly are Hamiltonian Mechanics (and Lagrangian mechanics)? I want to self-study QM, and I've heard from most people that Hamiltonian mechanics is a prereq. So I wikipedia'd it and the entry ...
2
votes
1answer
475 views

Two masses with interacting forces and an external force

Two masses in 3d space attract each other with a potential relative to the distance between them. There is also an external force on each particle based on the distance from a origin. I want to find ...
0
votes
1answer
459 views

Generalized momentum conjugate and potential $U(q, \dot q)$

On Goldstein's "Classical Mechanics" (first ed.), I have read that if $q_j$ is a cyclic coordinate, its generalized momentum conjugate $p_j$ is costant. He obtained that starting from Lagrange's ...
2
votes
1answer
434 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?
0
votes
1answer
128 views

Non-relativistic Kepler orbits

Consider the Newtonian gravitational potential at a distance of Sun: $$\varphi \left ( r \right )~=~-\frac{GM}{r}.$$ I write the classical Lagrangian in spherical coordinates for a planet with mass ...
3
votes
1answer
307 views

Find the action from given equations of motion

Is there a systematic procedure to generally obtain an appropriate action that corresponds to any given equations of motion (if I know that it exists)?
1
vote
1answer
150 views

Symmetries of spacetime and objects over it

I guess according to mathematical didactic, we first think of spacetime as a set and we reason about elements of its topology and then it's furthermore equipped with a metric. Appearently it is this ...
5
votes
2answers
359 views

Kugo and Ojima's Canonical Formulation of Yang-Mills using BRST

I am trying to study the canonical formulation of Yang-Mills theories so that I have direct access to the $n$-particle of the theory (i.e. the Hilbert Space). To that end, I am following Kugo and ...
1
vote
1answer
211 views

Clarification on a Goldstein formula steps (classical mechanics)

At page 20 of Classical Mechanics' Goldstein (Third edition), there are these two steps given between eqs. (1.51) and (1.52): $$\sum_i m_i \ddot {\bf r}_i \cdot \frac{\partial {\bf r_i}}{ \partial ...
6
votes
2answers
454 views

Can auxiliary fields be thought of as Lagrange multipliers?

In the BRST formalism of gauge theories, the Lautrup-Nakanishi field $B^a(x)$ appears as an auxiliary variable $$\mathcal{L}_\text{BRST}=-\frac{1}{4}F_{\mu\nu}^a F^{a\,\mu\nu}+\frac{1}{2}\xi B^a B^a + ...
14
votes
5answers
2k views

Why does no physical energy-momentum tensor exist for the gravitational field?

Starting with the Einstein-Hilbert Lagrangian $$ L_{EH} = -\frac{1}{2}(R + 2\Lambda)$$ one can formally calculate a gravitational energy-momentum tensor $$ T_{EH}^{\mu\nu} = -2 \frac{\delta ...
5
votes
2answers
526 views

Landau Mechanics: why does adding Lagrangians remove the indefiniteness of multiplying each Lagrangian by a different constant?

In Landau Mechanics (third edition page 4), why does adding Lagrangians of two non interacting parts remove the indefiniteness of multiplying each Lagrangian by a different constant? If both systems ...
1
vote
2answers
730 views

Partial derivative of Lagrangian density for vector field

The lagrangian density of a massless vector field is $ \mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$, where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ Expanding out gives ...
3
votes
1answer
306 views

Can I find a potential function in the usual way if the central field contains $t$ in its magnitude?

I'm working on a classical mechanics problem in which the problem states that a particle of mass $m$ moves in a central field of attractive force of magnitude: $$F(r, t) = \frac{k}{r^2}e^{-at}$$ ...
4
votes
2answers
651 views

How to apply Noether's theorem

Say I have a point transformation: $$x' ~=~ (1 +\epsilon)x,$$ $$t' ~=~ (1 +\epsilon)^2t,$$ and Lagrangian $$ L ~=~ \frac{1}{2}m\dot{x}^2 - \frac{\alpha}{x^2}.$$ How do I go out about showing ...
2
votes
1answer
448 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) $$ ...
15
votes
6answers
3k views

What is the physical meaning of the action in Lagrangian mechanics?

The action is defined as $S = \int_{t_1}^{t_2}L \, dt$ where $L$ is Lagrangian. I know that using Euler-Lagrange equation, all sorts of formula can be derived, but I remain unsure of the physical ...
0
votes
0answers
207 views

Describing the movement of the object in a particular situation in Lagrangian way

Suppose there is a object M, (sliding motion) moving by the initial speed $v$ and the initial location $x_0$. Otherwise noted, friction is assumed to be nonexistent. It then meets a circular mold ...
1
vote
1answer
471 views

Questions regarding solving the Brachistochrone problem using Lagrangian

brachistochrone problem: Suppose that there is a rollercoaster. There is point 1 ($0,0$) and point 2 ($x_2, y_2)$. Point 1 is at the higher place when compared to the point 2, so the rollercoaster ...
6
votes
2answers
1k views

The Lagrangian in Scalar Field Theory

This is perhaps a naive question, but why do we write down the Lagrangian $$\mathcal{L}=\frac{1}{2}\eta^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi - \frac{1}{2}m^2\phi^2$$ as the simplest ...
6
votes
2answers
455 views

What are the reasons for leaving the dissipative energy term out of the Hamiltonian when writing the Lyapunov function?

I have a problem with one of my study questions for an oral exam: The Hamiltonian of a nonlinear mechanical system, i.e. the sum of the kinetic and potential energies, is often used as a Lyapunov ...
2
votes
3answers
479 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, ...