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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Quadratic terms in QED lagrangian density

I recently learned that when we speak about a "free lagrangian", this actually means that the lagrangian is quadratic in the fields. When considering the Lagrangian density describing the coupling to ...
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2answers
101 views

Textbook for mathematical Lagrangian mechanics [duplicate]

I'm looking for a textbook or online notes or a review article etc on a rigerous formulation of Lagrangian mechanics. I'm well aware of the book by Arnold but I would like something to accompany it. ...
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43 views

Nabla Terms in the Energy Density of the Lagrangian for the Massive Spin 1 Field (Schwartz QFT 1st Ed. Eqn. 8.19)

The relevant part starts with a Lagrangian guess of, $$\mathcal{L}=-\frac{1}{2}\partial_{\nu}A_{\mu}\partial_{\nu}A_{\mu}+\frac{1}{2}m^2A_{\mu}^2$$ where the EOM's are, $$(\Box+m^2)A_{\mu}=0$$ The ...
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17 views

Finding the gauge transformtation of a Lagrangian [duplicate]

I am asked to find the gauge symmetry of the following Lagrangian: $L = -\frac{1}{4} F^2_{\mu \nu} + (\partial_{\mu} \phi_1 - m_1 A_{\mu})^2 + (\partial_{\mu} \phi_2 - m_2 A_{\mu})^2$ Then I have to ...
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1answer
49 views

Isotropic of Inertial frame?

My understanding of isotropic is the a particular physics law remain same no matter at what direction I look at it? Now suppose in case of inertial frame, we know that its is homogeneous and ...
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1answer
66 views

Lagrangian is isotropic in space

In Landau & Lifshitz Mechanics, while deriving the properties of Lagrangian of a free particle in inertial frame, he uses the following points $:$ As space is homogeneous in inertial frame, a ...
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1answer
63 views

Lagrangian isn't unique [closed]

If $L$ is a Lagrangian for a system of n degrees of freedom satisfying Lagrange's equations, show by direct substitution that $$L' = L + \frac{\mathrm{d}F(q_1,\dots,q_n,t)}{\mathrm{d}t}$$ also ...
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1answer
196 views

What is a Lagrangian of a photon? [duplicate]

In sense of classical mechanics+special relativity what is lagrangian of a photon? Lagrangian of a relativistic massive particle is as follows: $$ L_{massive}= -mc\sqrt{c^2-v^2} $$ So is it a zero?
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2answers
58 views

Invariance under local diffeomorphisms

In the context of the Polyakov action, the action for a relativistic point particle $$ S_P = \frac{1}{2} \int \mathrm{d}\tau \, e(\tau) \left(\frac{1}{e^2(\tau)}\left(\frac{\mathrm{d} X^\mu(\tau)}{\...
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1answer
95 views

Feynman propagator for arbitrary values of the gauge parameter $\zeta$

For the choice $\zeta = 1$ the Lagrangian can be brought into a particularly simple form upon integration by parts in the action integral. Equation$$\mathcal{L}' = -{1\over4}F_{\mu\nu}F^{\mu\nu} - {1\...
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1answer
109 views

Why does overall action need to have an extremum?

Quoting from Landau's and Lifshitz' Mechanics : The integral ${\int\limits_{t_1}^{t_2}}L(q, \dot{q},t)\,dt$ for the entire path must have an extremum, but not necessarily a minimum. This, ...
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1answer
127 views

Conserved quantities for the system, classical mechanics

In certain situations, particularly one-dimensional systems, it is possible to incorporate frictional effects without introducing the dissipation function. As an example, find the equations of motion ...
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93 views

Canonical Momentum Conjugate vs. Momentum

I stumbled upon this while reading about Legendre Transforms today. So consider an n-particle system. The Lagrangian is a function of $ q_i$'s and $\dot q_i$'s. If you consider the manifold $M$ where ...
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1answer
139 views

Noether's Theorem for Hamiltonians and Lagrangians

Looking around I see one version of Noether's Theorem that creates conserved quantities from symmetries that preserve the Lagrangian (e.g. http://math.ucr.edu/home/baez/noether.html), and another ...
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1answer
63 views

When can I apply Lagrangian mechanics?

I am trying to understand Lagrangian mechanics. I am having trouble capturing all of the nuances in one gulp. I can see the equations, but not necessarily the semantics behind such equations. I ...
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139 views

Lagrange: when a potential force, when a generalized force?

Consider the following case of a drum unrolling a mass that is on a massless string wrapped around the drum: According to my professor, here we must consider the mass $m$ being in a gravitational ...
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1answer
58 views

Strong interaction under $SO(3)$ isospin transformation

I'm given the following strong interaction: $$S = \int d^{4}x [\frac{1}{2} \partial_{\mu} \phi^{a} \partial^{\mu} \phi^{a} - \frac{m^2}{2} \phi^{a} \phi^{a}] ,\qquad a = 1,2,3 \text{.}$$ It is stated ...
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1answer
28 views

Function for which action is the minimum

On page 2 of "Mechanics" Landau & Lifshitz say that $q=q(t)$ is a function for which action is a minimum. Before this they say that at times $t_1$ and $t_2$, the system occupies coordinates $q^{(1)...
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0answers
20 views

What are the prerequisites needed to grasp analytical mechanics? [duplicate]

What are the mathematical prerequisites needed to grasp analytical mechanics conceptually and technically? What textbook is adequate for this purpose for an undergraduate student? To understand the ...
2
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0answers
89 views

Derivative of the action integral [closed]

I need to find the partial derivative of the action $S$ with respect to the generalized coordinate $q(t_f)$ and according to my textbook, it should equal the generalized momentum $p(t_f)$. I have a ...
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1answer
279 views

Lagrangian in rotating reference frame

I have a homework question I am working on, and I'm stuck on the last term in this proof. The problem states: Question: Consider a primed set of axes coincident in origin with an inertial set of axes ...
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138 views

Alternate Formulation of Kinetic Energy for Lagrangian

So I have a question regarding this system: It is supposed to be a simple model of an aircraft with the fuselage idealized as a concentrated mass $M_0$ and the wings modeled as rigid bars carrying ...
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1answer
75 views

State of academic fields of analytic mechanics and Multi Body Dynamics (MBD)

My question essentially contains two parts: one practical question and one philosophical question. Background I am an engineering student working in the field of Multi Body Dynamics (MBD). My ...
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1answer
78 views

Neutrinos and global $U(1)$ symmetry of Weyl fields

My book on QFT says that neutrinos are well described by left-handed Weyl spinor. The classical Lorentz-invariant Lagrangian density for that field is: $$ \mathcal{L} = i\psi^{\dagger}_L\overline\...
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1answer
174 views

Ricci tensor as relativistic Hamiltonian

I am little bit dissapointment with action integral in General relativity. The action integral is: $$ \int Rd^{4}x=\int R_{ij}g^{ij}d^{4}x\tag{1} $$ Where $$ R_{ij}=\frac{\partial\Gamma^{l}_{ij}}{\...
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28 views

Books for field theory [duplicate]

I am a math student, studying P.D.E. of some EM theory. I am new to field theory and I don't even know what a Lagrangian density, Hamiltonian density, or gauge field theory is. Could you please ...
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56 views

Covariant form of non-relativistic free particle

I have two questions about the action of free particle. $$S=\int dt~\frac{m}{2}~(\frac{d \vec{x} }{dt})^2 \tag{1}$$ The Covariant form is: (assume: $m=1$) $$S=\int d\tau \frac{1}{v(\tau)}~\...
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1answer
47 views

Homework central force Lagrangian formulation

For central force $$T = \frac{m}{2}(\dot{r^2} + r^2\dot{\theta^2} + r^2\sin^2{\theta}\dot{\phi^2}$$ Now applying the Lagrangian equation of motion, we get $$\frac{\partial{L}}{\partial{\phi}}=0\...
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2answers
102 views

Action max, min, or saddle?

It is well known that $\delta S = 0$ lays the foundation for variational mechanics. But I am confused as to whether or not this S is a minimum, a maximum, or a saddle point. Some books address this ...
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1answer
118 views

Quantum field theory with constraint: energy-momentum conservation?

Suppose I have a 2-form field $B$ and a Lagrange multiplier field $\lambda$, then the Lagrangian $S = \int (B \wedge \delta B + \lambda \delta B \wedge \delta B)$ with a Lie derivative operator $\...
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96 views

How do you find potential by Lagrangian formalism?

Suppose a ball is falling towards earth and hence by Lagrange equation we can find $T$ and $V$ where $V$ is $mgh$. But we know $V$ only because we know $F = mg$. Now since Lagrange equation doesn't ...
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Is the phrase “coupling constant” interchangable with “ strength of interactions”?

Can I use the terms coupling constant and strength of interactions, interchangeably, or are there more subtleties to the term coupling constant that I am not aware of? Coupling Constants from ...
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32 views

Lagrangian for $\mathcal{N}=4$ SQED in 3D

What is the Lagrangian for $3D$ $\mathcal{N}=4$ supersymmetric QED, with $N_f$ hypermultiplets? In particular, which is the form of the Fayet-Iliopulos terms, and the real mass terms?
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1answer
103 views

Help on understanding a concept in Noether's first theorem

Given a Lie group $G$, whose most general transform depends on $\rho$ parameters, under the action of which an integral $I$ is invariant, there are $\rho$ linearly independent combinations of the ...
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102 views

Why do Lagrange multipliers work in mechanics?

I understand that it is not always simple to find generalized coordinates that satisfy the constraint equations, so we try to find an alternative (more mechanical) method that yields curves that ...
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1answer
106 views

Trouble understanding Landau & Lifshitz writing about Lagrangians and Galilean Relativity [duplicate]

We have two inertial coordinate systems, $K'$ and $K$. $K$ is moving with infinitesimal velocity ${\epsilon}$ relative to $K'$. Using Galilean relativity we can transform this into $v'=v+{\epsilon}$. ...
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2answers
120 views

Derivation of Euler-Lagrange equations in Landau's and Lifshitz's “Mechanics”

There's an integral ${\int\limits_{t_1}^{t_2}}(\frac{\partial{L}}{\partial{q}}{\delta}q+\frac{\partial{L}}{\partial{v}}{\delta}v)dt=0$. [1.] $ {\delta}v={\frac{d{\delta}q}{dt}}$ [2.] I should get $ [...
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2answers
716 views

Estimate for energy dissipated by a damper/dashpot

I have a system with a mass $m$ attached to the end of a cable. The cable mass is assumed negligible. The cable is attached to the ground at the one end while the other, with the attached mass $m$, ...
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0answers
26 views

Negative Kinetic energy of normal modes in an elastic solid?

So I'm pretty stupid, admittedly, but I'm also pretty confused. I also fully realize this question is probably completely idiotic. The background is normal modes in linear elastic solids. Let the ...
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0answers
77 views

Lagrangians not related via a total time derivative lead to same Noether symmetries?

Having answered my initial two questions (v1), I now consider a third possibility. Consider two Lagrangians that both lead to equivalent equations of motion. Suppose that they are not related via a ...
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1answer
172 views

Derivatives with upper and lower indices

I'm studying classical and quantum field theory, but evaluating derivatives of fields (scalar and/or vector) described with upper and lower indices is somewhat new to me. I'm trying to evaluate $$\...
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3answers
533 views

Physical meaning of the Lagrangian function [duplicate]

In Lagrangian mechanics, the function $L=T-V$, called Lagrangian, is introduced, where $T$ is the kinetic energy and $V$ the potential one. I was wondering: is there any reason for this quantity to be ...
2
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2answers
98 views

Is there any loss or gain of information if a physics law is changed from one form to another? [closed]

Is there any loss or gain of information if a physics law is changed from one form to another such that the parameter appearing in them is changed from vector to a scalar? For example, consider the ...
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2answers
355 views

Goldstein's derivation of the 'principle of least action'

I want make an punctual question ands it's about The derivation of the expression $$ \Delta\int_{t_1}^{t_2} Ldt=L(t_2)\Delta t_2-L(t_1)\Delta t_1 + \int_{t_1}^{t_2} \delta L dt. \tag{8.74}$$ You can ...
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1answer
69 views

Is it strictly necessary to require gauge invariance of the action and equations of motion?

When writing down an action for a gauge theory, we require that the action be gauge invariant. This is typically taken to mean that the action must be written explicitly in terms of gauge invariant ...
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2answers
81 views

From Newtonian systems to Lagrange mechanics using Euler - Lagrange equations [closed]

I am reading some notions about Lagrangian Mechanics from Holm's book.(page 12). There is: Every Newtonian system, $$m_i \ddot{q_i}=\frac{\partial V}{\partial q_i}, \mbox{ } i=\overline{1,N},...
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1answer
111 views

Finding the Hamiltonian for sound vibrations in a gas in the momentum representation

I am working on a problem where I have been given the following Lagrangian density for describing sound vibrations in a gas: $\mathcal{L}=\frac{1}{2}[\rho_0\dot{\eta}^2+2P_0\nabla\cdot\eta-\gamma P_0(...
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0answers
37 views

Strain energy density (potential energy of elastic continua)

This question has to do with writing down the potential energy of an elastic body, which obeys a generalized Hooke's law [; \sigma_{ik} = \sum_{klm} \lambda_{iklm} u_{lm} ;] Where $\sigma$ is the ...
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1answer
84 views

Potential Energy of Lagrangian system

I want to clear my basic understanding of Lagrangian system. I am confused about the potential energy for the Lagrangian system. According to this picture, m is suspended from a rigid massless ...
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32 views

The Hamilton-Suslov principle

Building on the introduction given in the following paper, how does the Hamilton-Suslov principle generalise Hamilton's principle to consider contained mechanics? \begin{equation} \int ^{t_1}_{t_0}\...