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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Why does Principle for least action hold for classical fields [duplicate]

Let $\mathscr L (\phi(\mathbf x), \partial \phi(\mathbf x))$ denote the Lagrangian density of field $\phi(\mathbf x)$. Then then actual value of the field $\phi(\mathbf x)$ can be computed from the ...
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103 views

Internal potential energy and relative distance of the particle

Today, I read a line in Goldstein Classical mechanics and got confused about one line. To satisfy the strong law of action and reaction, $V_{ij}$ can be a function only of the distance between ...
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2answers
151 views

Derivation of Lagrangian?

I know that the Lagrangian $L$ is defined to be $T-V$, i.e. the difference between kinetic energy and potential energy. Also the Action $S$ is defined to be $\int Ldx$ and from this we can derive ...
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1answer
65 views

Why is the Legendre transformation an application of the duality relationship between points and lines?

When I read the Wiki about Legendre transformation, there is a statement The Legendre transformation is an application of the duality relationship between points and lines. What's the meaning of ...
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1answer
81 views

How the lagrangian density is found?

In Classical Mechanics one usually considers the Lagrangian as $L = K - U$ where $K$ is the kinetic energy of the system and $U$ is the potential energy. One then gets the Euler-Lagrange equations and ...
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110 views

Potential energy of an infinitesimal length of elastic rod

I am having an embarrassingly hard time with the derivation for the potential energy of an infinitesimal element of an elastic rod of area A. The picture shown below is an element of the rod that has ...
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4answers
500 views

Equivalence between Hamiltonian and Lagrangian Mechanics

I'm reading a proof about Langrangian => Hamiltonian and one part of it just doesn't make sense to me. The Lagrangian is written $L(q, \dot q, t)$, and is convex in $\dot q$, and then the Hamiltonian ...
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510 views

The Lagrangian as a metric

My question is, can the (classical) Lagrangian be thought of as a metric? That is, is there a meaningful sense in which we can think of the least-action path from the initial to the final ...
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1answer
223 views

The BRST construction for YM with or without auxiliary field

I'm learning BRST symmetry for Yang-Mills theory and I see that there are two ways of writing BRST differential. In some books (for example Ryder's and Ramond's textbooks) BRST differential acts as ...
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46 views

Translation symmetry and the non-conserved momentum in Viscous fluids

Even though a viscous fluid has a translation symmetry (invariance) for its Lagrangian , it still 'waste' Linear momentum. How come ?, isn't the rule that every symmetry yields a conservation law ?
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118 views

Lagrangian for a moving spring device [closed]

How can I write the proper Lagrangian for such as system as the one shown in picture? Am confused about what is the suitable way to designate the coordinate.
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1answer
289 views

How do I recover the 1D wave equation from the Lagrangian?

Consider small displacements, $y(x,t)$, of an element of a string (circled in red and shown below) from equilibrium. The force balance in the vertical direction yields: $$+\uparrow \Sigma F: ...
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70 views

Alternative symmetries for the Maxwell Lagrangian?

I'm wondering about how to show that $A_a\rightarrow A_a+\alpha\partial_0A_a$, with $\alpha$ infinitesimal, is an infinitesimal symmetry of $\mathcal L=-\frac14F_{ab}F^{ab}$. \begin{equation} ...
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1answer
109 views

Lagrangian to Hamiltonian

I'm having some problems with an assignment where I have to state the Hamiltonian from the kinetic energy $T$ and potential energy $U$. These are as follows: ...
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1answer
274 views

Atwood machine with spring

I'm just beginning to learn about Lagrangian mechanics, and I am asked to find the kinetic energy of this Atwood machine (See figure). I am told, that the kinetic energy should be: ...
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1answer
231 views

Are the Hamiltonian and Lagrangian always convex functions?

The Hamiltonian and Lagrangian are related by a Legendre transform: $$ H(\mathbf{q}, \mathbf{p}, t) = \sum_i \dot q_i p_i - \mathcal{L}(\mathbf{q}, \mathbf{\dot q}, t). $$ For this to be a Legendre ...
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1answer
412 views

Minimal vs. Non-minimal coupling

What is the difference between Minimal vs. Non-minimal coupling in General Relativity? A brief introduction to Minimal Coupling in General Relativity could be useful too.
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207 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 ...
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183 views

Lagrange's Equations for a Tetherball

I'm trying to write down the equations of motion for a tetherball moving around a pole while the string is getting shorter. --- MAJOR EDIT --- I started with Lagrange: $$ x(t)=l(t) \sin (\theta) ...
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120 views

Why friction force is force of constraint?

My understanding about constraint force is that it is a force which limits the geometry of particle's motion. For example, situations such as the particle trapped in a track or limited in domain can ...
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2answers
79 views

How does isotropy of free space imply $L(v^2)$ for a free particle? [duplicate]

From Mechanics; Landau and Lifshitz, it's stated on page 5: Since space is isotropic, the Lagrangian must also be indpendent of the direction of $ \mathbf{v}$, and is therfore a function only of ...
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102 views

Lie algebra of axial charges

Starting from the lagrangian (linear sigma model without symmetry breaking, here $N$ is the nucleon doublet and $\tau_a$ are pauli matrices) $L=\bar Ni\gamma^\mu \partial_\mu N+ \frac{1}{2} ...
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1answer
153 views

Lagrangian approach to spinning thread reel

I am trying to better understand Lagrangian dynamics and am struggling to complete the following question: A reel of thread of mass $m$ and radius $r$ is allowed to unwind under gravity, the upper ...
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1answer
119 views

How to obtain Maxwell's Lagrangian from complex scalar fields?

I've looked in several books and they all show how to obtain electrical interactions by forcing local gauge invariance of any complex scalar field Lagrangian (like Klein-Gordon or Dirac). I manage to ...
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1answer
60 views

The exact definition of conjugate momentum density

After checking various websites, I've seen the conjugate momentum density defined as either: \begin{align*} P_r ~=~ \frac{\partial \mathcal{L}}{\partial \dot{A}_r} \end{align*} or \begin{align*} P_r ...
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1answer
77 views

$\cos^{2}(\phi)$ in the kinetic energy term of the Lagrangian is one?

I'm doing some homework in Classical Mechanics, and is about to write out the Lagrangian of a system. But, when I check the answer from my teacher, something is missing. The kinetic energy I'm using ...
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1answer
123 views

Intuition for actions written as integrals over spacetime

Right now I'm simply looking for an intuitive explaination of actions that integrate over a 4-volume element, $d^4x$ rather than a parameter say $\lambda$. More specifically I'm well versed in action ...
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1answer
130 views

From Lagrangian to equations of motion [closed]

I have a given Lagrangian: $$L= e^{st}\cdot\frac12\cdot(mv_y^2-ky^2)$$ And are asked to identify the equations of motions, the constants of motions and physical system. Without the exp-time-term, ...
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1answer
223 views

Pendulum with a rotating point of support from Landau-Lifschitz

I found this problem in Landau-Lifschitz vol.1 (Mechanics) A simple pendulum of mass $m$, length $l$ whose point of support moves uniformly on a vertical circle with constant frequency $\gamma$. ...
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1answer
162 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} ...
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2answers
127 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} ...
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1answer
129 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 ...
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1answer
130 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 ...
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1answer
146 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 + ...
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3answers
530 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 ...
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1answer
258 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 ...
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1answer
204 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 ...
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1answer
93 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 ...
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1answer
67 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 ...
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1answer
98 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., $$ (\Pi _{k} )_{t} - \rm div(j_{k})=0 $$ ...
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120 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} ...
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3answers
369 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 ...
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97 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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140 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 ...
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4answers
350 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 ...
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3answers
455 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 ...
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1answer
131 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 ...
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1answer
224 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 ...
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1answer
139 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 = ...
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1answer
101 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 ...