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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The $\dot{q}$ term in the Euler-Lagrange equation

The Euler-Lagrange equation is about the functional $$ \int_{t_1}^{t_2} L(q, \dot{q}, t ) dt . $$ From a mathematical point of view, a simpler functional might be $$ \int_{t_1}^{t_2} L(q, t ) ...
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1answer
79 views

Derivation in Modern Supersymmetry by Terning

I am trying to do some calculations from Modern Supersymmetry by Terning and I am stuck on how he derived a particular term. Specifically, I am looking at 2.67 on page 27. My current work is below. $$...
5
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1answer
73 views

Electron - neutrino scattering effective Lagrangian

The electron and neutrino can interact through an intermediary Z boson, via the Lagrangian: $$ L= \frac{1}{2} \partial_\mu \phi_Z \partial^\mu \phi_Z - \frac{1}{2} m_Z ^2 \phi_Z ^2 -g_{\nu} \phi_Z \...
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0answers
32 views

Generalise Noether's theorem [closed]

I'm not sure how to generalise Noether's theorem. For this L, I think $B\cdot\dot{x}$ is conserved so I tried to relate F and K to this and try to show that that was conserved but got no where. any ...
0
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1answer
62 views

Checking the basics in QFT notation [closed]

In the very beginning of QFT we face the action (S) as a functional of the Lagrangian. I am still trying to get used to the notation used here, so I would like to check if the following makes sense: $...
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0answers
44 views

Mass term in Maxwell's Lagragian for Electromagnetism

In the scalar field Lagrangian the mass term is given by $$m^2 \phi^2.$$ But the equivalent term in Maxwell's Lagrangian for electromagnetism is $$m^2A_{\mu}A^{\mu}.$$ But I don't know why the ...
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0answers
42 views

Reason behind $L = T - V$ (Lagrangian formalism) [duplicate]

I've been learning about the Lagrangian formulation recently, and while I'm with the process, I am still struggling somewhat with the theory behind it. As I (rather poorly) understand it, the ...
4
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1answer
100 views

Lagrangian density for Lorentz force of continuous charge distribution in external field?

It's frequently an exercise to derive the Lorentz force law for a particle with charge $q$ in an external electromagnetic field given by the following Lagrangian: $$L = -mc^2\sqrt{1-\frac{\dot{r}^2}{...
1
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1answer
81 views

Magnetic monopoles in field theory

In standard QED, we couple the electron to electromagnetism by replacing $$\partial_\mu \to \partial_\mu + i e A_\mu.$$ Upon taking the classical limit, we find that this gives electrons an electric ...
1
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1answer
49 views

Lagrangian in a system with a specific velocity dependent potential

I have a system of a particle moving under the generalized central potential $$ V= \frac{1}{r}(1+\dot{r}^2) \tag{1} $$ The general Euler-Lagrange equations for such type of potentials are: $$ \frac{...
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2answers
57 views

Virtual Work- Is the presentation in Cornelius Lanczos wrong?

Book: The Variational Principles of Mechanics by Cornelius Lanczos Edition: 4th Chapter: 3, The Principle of Virtual Work I am on the second page of the 3rd chapter (pg 75; it has the Eqn. 31.1). ...
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1answer
86 views

Why the Lagrangian $L$ is KE - PE? Why not KE + PE!

With Lagrangian, is there any way to intuitively grasp why total energy equals the difference between the kinetic and potential energy? Seems counter-intuitive - whereas Hamiltonian calculation (sum ...
3
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3answers
69 views

Why do we need $SU(2)\times U(1)$ invariant mass terms if the symmetry will be broken anyway?

In the SM we can not add fermionic mass terms like $m \overline{e}_R e_L$ to the Lagrangian since these terms are not invariant under $SU(2)\times U(1)_Y$. After introducing the Higgs in the unitary ...
8
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1answer
121 views

Action of a massive free point-particle in relativistic mechanics

I was reading about the formulation of mechanics in special relativity and found that the action for a massive free point-particle as $$ S = -mc\int_a^b ds $$ So, I did a few observations, ie. $$ S =...
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0answers
30 views

Lagrangian Interaction Type and Spin-Dependence

So I'm transitioning from reading particle physics books to the literature, specifically as it pertains to dark matter models. In this case I'm talking about t-channel DM-nucleon scattering. They ...
3
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1answer
54 views

Problem obtaining string equations from Polyakov action [closed]

I am trying to obtain the string equations of motion from the Polyakov action in the conformal gauge, i.e.: $$ S=T\int{d\tau d\sigma (\dot{x}^2-x^{'2})}\equiv\int{d\tau d\sigma \mathcal{L}} $$ where ...
0
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1answer
38 views

The components of applied forces on the masses of a Trebuchet

Background information: I was following an online mechanics document in order to learn how to derive the equations of motion for a trebuchet (shown below) using Lagrangian mechanics. At some point ...
0
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1answer
86 views

Number of degrees of freedom in the Standard Model Lagrangian

Consider a Lagrangian $L$ which depends on a number of fields $F_1$, $\cdots$, $F_N$ and their (spacetime) derivatives. Each of those fields $F_n$ is valued in $\mathbb{R}^{k_n}$. Is the Standard ...
0
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1answer
48 views

Lorentz transformation and symmetries of the Lagrangian [duplicate]

Since the Lagrangian of our quantum field theories is covariant under Lorentz transformations I'm asking myself if there is any link to some symmetries (like that we get from gauge transformations ...
3
votes
2answers
81 views

Gauge field and covariant derivative

To make the kinetic term in the Lagrangian for quantum field theories (for example qed) inveriant under local phase transformations we introduce the covariant derivative $D_{\mu} = \partial _{\mu} + ...
0
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0answers
36 views

Variation with respect to the metric and other tensors

When varying an action with respect to tensors and the metric, I'm afraid I get confused as how to one organizes the Lagrangian and then performs the variation. Take for example, the following example ...
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0answers
43 views

What to do when the fields are arranged in a matrix?

I am dealing with a Lagrangian in which the fields are arranged in an $N\times N$ matrix and i have to find the minima of the potential. Usually i would write the Lagrangian in components and then ...
0
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1answer
45 views

Correct Definition of Angular Momentum of a Charged Particle in an Electromagnetic Field? (Classical Mechanics) [duplicate]

What is the more correct definition of angular momentum $\vec{\mathbf{M}}$ in three dimensions? (I.e. classically/Lagrangian/Hamiltonian, not necessarily quantum or relativistic) $$\vec{\mathbf{M}}...
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2answers
86 views

Prove energy conservation using Noether's theorem

I wonder how you prove that energy is conserved under a time translation using Noether's theorem. I've tried myself but without success. What I've come up with so far is that I start by inducing the ...
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0answers
25 views

How to obtain the Klein Gordon equation for DBI action?

The action for DBI field is given by $$S=d^{4}x\,\sqrt{-g}\left[- V(\phi)\sqrt{1-g^{ij}\partial_{i}\phi\partial_{j}\phi}\right]$$ And the required Klein Gordon is given by $$\square \phi+\frac{\...
7
votes
2answers
257 views

Damped oscillator: time-reversal, time-translation and dissipation

The equation of motion of a damped oscillator $$\frac{d^2x}{dt^2}+\gamma\frac{dx}{dt}+\omega_0^2x=0$$ which is invariant under time-translation $t\rightarrow t+a$, but not under time reversal $t\...
0
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0answers
23 views

Partial Differentiation without chain rule in Euler Lagrange Equations [duplicate]

The Euler-Lagrange equations for a bob attached to a spring are $$ \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial v} = \frac{\partial L}{\partial x} $$ But $v$ is a function of $x$. Is it ...
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0answers
26 views

Derivation of Lagrangian [duplicate]

Actually I am beginning to study Lagrangian and Hamiltonian Mechanics. The book that I am considering directly introduces the Lagrangian L=T-V. Though further on the book , the Euler Lagrange ...
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1answer
57 views

Classical dynamics of a matrix

For a system of interacting particles, we can formulate Hamiltonian dynamics in terms of a vector of position coordinates $q$ and a vector of momentum coordinates $p$. Then the Hamiltonian takes the ...
5
votes
1answer
73 views

Coupling a spinor field to a preexisting scalar field?

So I'm not a physicist, but I'm thinking about a mathematical problem where I think physical insight might be useful. We're working on a Riemannian manifold $(M,g)$ (positive definite metric) with a ...
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2answers
53 views

How do I find the linear components of acceleration in a pendulum?

I have managed to derive the equation of motion of a simple pendulum under the influence of gravity using the Lagrangian, but since that only tells me what the angular acceleration is, I now want to ...
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1answer
65 views

Why is angular acceleration of a pendulum always negative?

I was trying to derive using the Lagrangian the equations of motion of a simple pendulum under the influence of gravity. Eventually, I was brought to this conclusion: $$\alpha = -(g\sinθ)/l$$ where $\...
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1answer
57 views

Understanding standard model and symmetry

I just want to know whether my understanding regarding standard model and symmetry is correct or utter nonsense. The standard model is the (yet incomplete) Lagrangian of the universe. The ...
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2answers
41 views

Galilean invariance and the Lagrangian

My textbook says that in a time invariant space with translational and rotational symmetry the Lagrangian only depends on the magnitude of the velocity. The galilean invariance says that a Lagrangian ...
0
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0answers
32 views

Variables in the Dirac Equation Lagrangian [duplicate]

(Warning: I'm a student of mathematics with no training in physics.) In derivations of the Dirac equation from an action principle, one encounters the action $$S= \displaystyle\int\,d^4x \,\bar\psi(x)...
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2answers
134 views

When is stress-energy tensor defined as variation of action with respect to metric conserved?

In General Relativity Einstein's equation implies that stress-energy tensor on its RHS is conserved (has vanishing divergence), due to the Bianchi identity. Considering variational principles leading ...
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1answer
56 views

Force and energy relation: in case of time dependent force

The equivalent problems are also found in Marion problem 7-22, and other formal classical mechanics textbook. Here what i want to know why instructor solution and some websites gives this kinds of ...
3
votes
1answer
56 views

Why does the 'Jacobian of at least one combination of $n$ functions shall be different from zero'?

I've started reading The Variational Principles of Mechanics by Cornelius Lanczos; here is the concerned excerpt from p. 11: The generalized coordinates $q_1,q_2,\ldots, q_n$ may or may not have a ...
11
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1answer
197 views

Physical Interpretation of the Graph of the Legendre Transform?

See Making Sense of the Legendre Transform and Legendre Transforms for Dummies. Look at the following diagram from the first link: I was trying to think of the simplest example to interpret this ...
3
votes
1answer
114 views

Lagrangian Equations of Motion, Conservative Forces

I'm new to this topic so please bear with me. Here on wikipedia we have the Lagrangian equations of motion: $$ \frac{d}{dt}\left(\frac{\partial T}{\partial \dot{q}}\right) - \frac{\partial T}{\...
3
votes
1answer
44 views

Problem books for concept building in applications of Riemannian and other geometries to mechanics

As a student of physics I have learned solving Euler equations for rigid bodies by solving examples and exercises in self-contained books rather than understanding the proofs of Euler equations (I ...
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0answers
24 views

Are time-$t$ maps of a Hamiltonian system with 1 degree of freedom typically twist?

If we take a typical Hamiltonian system $H(q,p)$ with one degree of freedom, and look at its time-$1$ map $(q(0),p(0)) \mapsto (q(t),p(t))$, will it generically satisfy the twist property, e.g. $\...
0
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1answer
76 views

How are Lagrangians in QFT constructed?

Various particle equations (like the K-G equation, the Dirac equation, the Proca equation etc.) in QFT are derived by applying the Euler-Lagrange equations to the Lagrangian density. But how are these ...
3
votes
2answers
46 views

Variation of Lagrangian density $\mathcal{L}$ w.r.t $x_\mu$

If a function $f(x(t),y(t))$ has no explicit dependence on the variable $t$, then $\frac{\partial f}{\partial t}=0$. In quantum field theory, the Lagrangian density $\mathcal{L}(\phi,\partial_\mu\phi)...
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2answers
59 views

Is there any general theorem which specifies conditions where the critical solution of an action is unique (for given boundary conditions)? [duplicate]

Consider a classical mechanical system with generalized coordinates $q_i$, $i \in \{1,\dots\,n\}$. And Lagrangian $L$. Given a path $\gamma$ (with coordinates $\gamma_i$) and two times $t_1$ and $t_2$ ...
3
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2answers
93 views

Lagrange Multipliers and Virtual Work: Are Joos & Freeman wrong?

I have come to suspect that the treatment of virtual work in configuration space using Lagrange multipliers given here "Theoretical Physics, by Georg Joos & Ira M. Freeman, pg 114" is not correct. ...
4
votes
2answers
86 views

Total derivatives in GR

Without gravity we can easily switch between terms in a Lagrangian, such as $\partial\phi\partial\bar{\phi}$ and $\phi\Box\bar{\phi}$, since total derivative vanishes. But in GR we have additional $e\...
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0answers
28 views

How to get the canonical momentum from the velocity when doing a Legendre transform?

For a Lagrangian $$L=\frac{1}{2}m\dot{q}^2-\frac{1}{2}m\omega^2 q^2$$ the Hamiltonian is defined as $$H=p\dot{q}-L$$ where $p$ is the canonical momentum, which is defined as $p=\frac{\partial L}{\...
2
votes
2answers
74 views

Where is the BRST symmetry?

When quantizing YM we start from the gauge fixed path integral (to remove redundancy of integrating over Gauge symmetric configurations) $$\int \mathcal{D}A \delta(G(A)) \text{det} \Delta_{FP}e^{i\int ...
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1answer
58 views

Classical Yang Mills vacuum

What is the vacuum of classical Yang Mills theory $$\mathcal{L} = - \frac14 F^{a \mu \nu} F^a_{\mu \nu}~?$$ Is it simply $A^a_\mu=0$ for all its components?