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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7
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1answer
208 views

Sign in front of QFT kinetic terms

I'd like to know if the sign in front of a kinetic term in QFT important. For the scalar field we conventionally write (in the $ + --- $ metric), \begin{equation} {\cal L} _{ kin} = \frac{1}{2} ...
4
votes
2answers
153 views

Getting the Lagrangian from the action in curved spacetime

Suppose I have this action: $$ S = \int \mathrm d^4 x\sqrt{-g}\times \text{something}$$ where $g$ is the determinant of the metric. Should I take the Lagrangian to be: $$ \mathcal L = \sqrt{-g} ...
8
votes
1answer
131 views

Lagrangian formalism and Contact Bundles

In his Applied Differential Geometry book, William Burke says the following after telling that the action should be the integral of a function $L$: A line integral makes geometric sense only if ...
2
votes
1answer
248 views

Energy-momentum tensor for dust

We all know that the energy-momentum tensor for dust is just $T^{\alpha\beta}=\rho_0v^\alpha v^\beta,$ where $\rho_0$ is the mass density in the dust's rest frame and $v^α$ is the dust's ...
8
votes
1answer
214 views

Caldeira-Leggett Dissipation: frequency shift due to bath coupling

I am trying to understand the Caldeira-Leggett model. It considers the Lagrangian $$L = \frac{1}{2} \left(\dot{Q}^2 - \left(\Omega^2-\Delta \Omega^2\right)Q^2\right) - Q \sum_{i} f_iq_i + ...
7
votes
3answers
900 views

Do we need inertial frames in Lagrangian mechanics?

Do Euler-Lagrange equations hold only for inertial systems? If yes, where is the point in the variational derivation from Hamilton's principle where we made that restriction? My question arose ...
5
votes
1answer
381 views

On a trick to derive Noether current

Suppose, in whatever dimension and theory, the action $S$ is invariant for a global symmetry with a continuous parameter $\epsilon$. The trick to get the Noether current consists in making the ...
5
votes
1answer
222 views

Why isn't $F = \frac{\partial \mathcal{L}}{\partial q}$?

If momentum is, $$p = \frac{\partial \mathcal{L}}{\partial \dot{q}}$$ and force is, $$ F = \frac{dp}{dt}$$ and by Euler-Langrange equations, $$ \frac{d}{dt}\frac{\partial \mathcal{L}}{\partial ...
1
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1answer
96 views

How do I obtain the Lagrangian in standard for using action? [closed]

I have action as shown below $$S=\int \mathrm{d}t \int \mathrm{d}x^3 \bar\psi\left(i\partial_t\psi +\frac1{2m}\bar\nabla^2\psi-V(x)\psi\right)$$ How do I manipulate it to obtain the Lagrangian ...
0
votes
1answer
83 views

Finding the Lagrangian from the derivative of position

I have to find the Lagrangian for a system. In the point of interest I have come up with the following position coordinates: $$x = Rcos(\omega t)+\ell sin(\phi)$$ and $$y = Rsin(\omega t)-\ell ...
3
votes
1answer
126 views

Nonlinear Klein Gordon equation

For the Klein Gordon nonlinear equation, $$ u_{tt}- \Delta u +f(u)=0,$$ how could I use Noether's theorem to prove that there is a conserved quantity? I.e., $$ (\Pi _{k} )_{t} - \rm div(j_{k})=0 $$ ...
3
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0answers
150 views

Parity violating Dirac particle

We normally write down the Dirac Lagrangian as \begin{equation} {\cal L} _D = \bar{\psi} ( i \partial _\mu \gamma ^\mu - m ) \psi \end{equation} but are the Lagrangian's, \begin{equation} ...
5
votes
3answers
528 views

Meaning of kinetic part in the Lagrangian density?

What is the physical meaning of the kinetic term in the classical scalar field Lagrangian $$\mathcal{L}_{kin}~=~\frac{1}{2}(\partial_\mu\phi)(\partial^\mu\phi)~?$$ It gives how does the field change ...
3
votes
0answers
142 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 ...
0
votes
0answers
233 views

Finding moment of inertia from Lagrange equation

I'm getting the following information: Consider a system consisting of two rotating bars of length $\ell$ and with uniform mass density and each with total mass $m$. The bars are attached to a common ...
8
votes
4answers
515 views

Does the action and Lagrangian have identical symmetries and conserved quantities?

From the book Introduction to Classical Mechanics With Problems and Solutions by David Morin, page 236 states: Noether's Theorem: For each symmetry of the Lagrangian, there is a conserved ...
12
votes
3answers
491 views

How are symmetries precisely defined?

How are symmetries precisely defined? In basic physics courses it is usual to see arguments on symmetry to derive some equations. This, however, is done in a kind of sloppy way: "we are calculating ...
2
votes
1answer
207 views

Interpreting Lagrange Multipliers as forces

I am (still) working on getting a good understanding of Lagrange multipliers. I understand their function in an optimization problem that is subject to some constraint. For the specific case of ...
1
vote
1answer
254 views

Optimal selection of generalized coordinates in Lagrangian system

EDITED: The number of bonds is actually 2, not 1 (look at edit history). Fixed for archiving purposes. Problem: The edge A of an homogeneous rod (of length $\ell$ and mass $m$) is performing a smooth ...
5
votes
1answer
180 views

How Hamilton's Principle was found?

Hamilton's principle states that the actual path a particle follows from points $p_1$ and $p_2$ in the configuration space between times $t_1$ and $t_2$ is such that the integral $$S = ...
1
vote
1answer
124 views

What is the neatest way to describe a “non-autonomous” (lagrangian) system?

The configuration space of a system of particles $(m_i,x_i)$, $i=1,\dots,n$, subject to constraints $$\Phi (x)=0,\qquad \Phi\colon \mathbb R^{3n}\to \mathbb R ^{3n-k},\qquad x=(x_1,...,x_n),$$ if the ...
2
votes
1answer
107 views

How to find the equillibrium points using Jacobian and Hessian?

Given that I have Jacobian and Hessian matrices of three particles interacting with each other in a harmonic trap through Coulomb's law in a 2D plane, how do I find the equilibrium points of them (I ...
0
votes
1answer
158 views

Block on cart, equation of motion

Consider a rigid block of $b \times h$ having mass $m$ on cart (as depicted below). The cart is given an acceleration $a$, this leads to overturning of the block. The angle of rotation is indicated by ...
2
votes
1answer
411 views

Is there a better choice of coordinates for a bead on a rotating helical wire?

A bead of mass $m$ is threaded around a smooth spiral wire and slides downwards without friction due to gravity. The $z$-axis points upwards vertically. Suppose the spiral wire is rotated about the ...
2
votes
2answers
2k 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 ...
1
vote
1answer
34 views

Lagrangian Oscilattor

I want to know how to calculate the normal modes from a Lagrangian. I make the T (kinetic energy matrix) and U (potential energy matrix), and then I calculate the determinant of $|T-\omega ^2 U|$ ...
1
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1answer
82 views

Converse of the Lagrangian form-invariance

The form-invariance of the Lagrange equations implies the existence of a function $\ A( q_k, t)$ so that $\ \begin{equation} L' (q_k, v_k, t) -L(q_k, v_k, t) = \frac d {dt} A( q_k, t) ...
3
votes
1answer
160 views

Is there a Lagrangian whose Euler-Lagrange equation is the gradient?

I am trying to recast a problem I am working on in terms of Lagrangian mechanics. I am in the following situation. Suppose I have a function $f:X \rightarrow \mathbb{R}$ (a field). In the its ...
1
vote
1answer
292 views

What is meant by a local Lagrangian density?

What is meant by a local Lagrangian density? How will a non-local Lagrangian look like? What is the problem that we do not consider such Lagrangian densities?
3
votes
1answer
230 views

Clarifying constraint forces in Lagrangian dynamics

In the Lagrangian formulation, the addition of constraint forces that are unknown can be done with Lagrange multipliers, which allows for the forces to be found. Taking $k$ constraints of the form ...
6
votes
5answers
842 views

Noether Theorem and Energy conservation in classical mechanics

I have a problem deriving the conservation of energy from time translation invariance. The invariance of the Lagrangian under infinitesimal time displacements $t \rightarrow t' = t + \epsilon$ can be ...
1
vote
1answer
217 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 $a$ ...
4
votes
2answers
173 views

Landau's argument for dependence of Lagrangian on magnitude of velocity

In chapter 1, of Landau-Lifshitz Mechanics' book, Landau through isotropy and homogeneity of space and homogeneity of time proves that the Lagrangian must depend of magnitude of velocity of the ...
3
votes
1answer
98 views

How to check $\renewcommand{\vec}[1]{\mathbf{#1}} \vec{v'}\cdot\vec{V}$ and $\vec{v}'^2$ are time derivatives of some other functions?

From Landau, Lifshitz Mechanics p.127 $\renewcommand{\vec}[1]{\mathbf{#1}}L'=\frac{1}{2}m(\vec{v}'^2+\vec{v'}\cdot\vec{V}+\vec{V}^2)-U $ He states that "$\vec{V}^2(t)$ can be written as the total ...
1
vote
0answers
980 views

Ball rolling without slipping inside a hollow cylinder

A small ball of radius $r$ performes small oscillations within a hollow cylinder of radius $R$. What would be the angular frequency of the oscillations given that the rolling is without slipping? The ...
0
votes
0answers
28 views

Expansion of $L(v^2 + 2\vec{v}\cdot\vec{\epsilon}+\epsilon^2)$ [duplicate]

How can I find the expansion of the Lagragian (it it only dependent on $v^2$) $L(v^2 + 2\vec{v}\cdot\vec{\epsilon}+\epsilon^2)$ in powers of $\vec{\epsilon}$ ? (From L.Landau, E. Lifshitz, Mechanics , ...
3
votes
1answer
435 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 ...
10
votes
2answers
460 views

Lagrangian Mechanics - Commutativity Rule $\frac{d}{dt}\delta q=\delta \frac{dq}{dt} $

I am reading about Lagrangian mechanics. At some point the difference between the temporal derivative of a variation and variation of the temporal derivative is discussed. The fact that the two are ...
1
vote
1answer
180 views

Shouldn't Quantum Mechanics change in a black hole?

I recently learnt that the conservation laws are a consequence of the symmetries of space and time (the Lagrangian in Newton mechanics). Since space-time change in a black hole wouldn't quantum ...
5
votes
5answers
567 views

What distinguishes time from space in Quantum Field Theory?

Consider the following expression for a general QFT action: $$ S ~=~ \int_0^t\mathrm dt~L ~=~\int_0^t\mathrm dt\int_\mathbb {R^3}\mathrm d^3x~\mathcal L ~=~\int\mathrm d^4x~\mathcal L.$$ Here we ...
0
votes
2answers
717 views

Time dependence of Lagrangian and Hamiltonian?

I am reading a online tutorial about Lagrangian mechanics. In one section, it states that if the kinetic term in Lagrangian has no explicit time dependence, the Hamiltonian does not explicitly depends ...
3
votes
1answer
246 views

Missing terms in Hamiltonian after Legendre transformation of Lagrangian

Short question Given any Lagrangian density of fields one could possibly conceive, is it the case that after one has performed a Legendre transformation, if the Hamiltonian is then expressed in terms ...
1
vote
1answer
74 views

Classical Mechanics & Coordinates [closed]

What is the meaning generalised coordinates in Classical Mechanics? How is Lagrangian formalism different from Hamiltonian formalism? How are they related to Hamilton's Principle? How are they ...
0
votes
1answer
168 views

Can't understand the principle of least action [closed]

I tried many hours to understand the principle of least action, and those hours become days... and I still didn't understand that principle/ and how it relates to Newtonian mechanics? Could someone ...
1
vote
1answer
360 views

Calculate Hamiltonian from Lagrangian for electromagnetic field

I am unable to derive the Hamiltonian for the electromagnetic field, starting out with the Lagrangian $$ \mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\frac{1}{2}\partial_\nu A^\nu \partial_\mu A^\mu ...
1
vote
0answers
125 views

Derivation of equations of motion in Nordstrom's theory of scalar gravity?

Nordstrom's theory of a particle moving in the presence of a scalar field $\varphi (x)$ is given by $$ S = -m\int e^{\varphi (x)}\sqrt{\eta_{\alpha \beta}\frac{dx^{\alpha}}{d ...
2
votes
1answer
94 views

Stationary action with maximized action [duplicate]

I would like to ask for an example (a lagrangian) both in classical and quantum level for which the action is maximaized (rather than minimized). What is special in these cases?
1
vote
2answers
272 views

Equations of motion from the Standard Model

For some time now I have been wondering if you could not derive any sort of equations of motion from the Standard Model: ...
1
vote
1answer
105 views

Transforming a lagrangian to hamiltonian and vice versa

I am not refering to Legendre transform, but to something more simple. In analytical mechanics, the Lagrangian can be described as $L=T-V$, and the Hamiltonian is if the Lagrangian doesn't explicitly ...
0
votes
0answers
64 views

What is the “momentum” referred to in the energy-momentum tensor

What is the "momentum" referred to in the energy momentum tensor from GR? Is it $m\dot{x}$ or is it the canonical momentum $\frac{d}{dt} \left(\frac{\partial L}{\partial \dot{x}}\right)$ Also, I ...