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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Noether's Theorem and scale invariance

Noether's theorem usually considers coordinate/field transformations which leave the Lagrangian invariant up to a divergence term, i.e. $\mathcal{L} \rightarrow \mathcal{L} + \partial_{\mu}f^{\mu}$ ...
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50 views

Lagrangian of a coupled pendulum

I am trying to find the Lagrangian for a coupled pendulum: the two pendulums have the same characteristics (length $l$ and mass $m$) and are attached to the same roof at a distance $d$. In addition, ...
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1answer
46 views

Confusion about virtual displacements

I am self-studying Goldstein's book "Classical Mechanics", and I need some help understanding the part where Goldstein discusses using Hamilton's principle to solve systems with holonomic constraints (...
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61 views

What is difference between variations of the work and virtual work?

I really want to know whether or not both equations are the same mathematically. I think that they are the same, I just want to be sure. (Reference: this website.)
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34 views

Flavour basis to mass basis

I am not really understood why we need to change the basis from flavour basis to mass basis after Spontaneous symmetry breaking applying to Yukawa Lagragian? why we can't take (or not making ...
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0answers
23 views

Non-minimal coupling of the gauge fields to the matter

Does any one know the physical meaning of the following gauge invariant gauge coupling to the spinors? $$\bar \psi F_{\mu \nu} [\gamma^\mu, \gamma^\nu] \psi$$ This coupling is not minimal, as $$\bar \...
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38 views

Is my proof of Proca Lagrangian local gauge invariance correct?

My task was to prove that the first term of the Proca Lagrangian is invariant under local gauge transformations. I’m new to Ricci calculus and think I’ve misinterpreted what I was supposed to do, and ...
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3answers
205 views

Principle of Least Action Question

Let's say we have a particle with no forces on it. The path that this classical particle takes is the one that minimizes the integral $$\frac{1}{2}m\int_{t_i}^{t_f}v^2dt.$$ So if we graph this for ...
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1answer
72 views

How does one write the standard model Lagrangian in other smaller Lagrangian counterparts?

How does one write the standard model Lagrangian in other smaller Lagrangian counterparts? Before electroweak symmetry breaking by the Higgs Mechanism: $L_{EW} = L_{g} + L_{f} + L_{h} + L_{y}$ Where ...
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2answers
282 views

Why are the Nambu-Goto action and Polyakov action equivalent at quantum level?

It's a well known elementary fact that the Nambu-Goto action $$S_{NG} = T \int d \tau d \sigma \sqrt{ (\partial_{\tau} X^{\mu})^2 (\partial_{\sigma} X^{\mu})^2 - (\partial_{\sigma} X^{\mu} \partial_{\...
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1answer
52 views

Legendre transform

How do they obtain this? $$g(x, y, u) = ux − f(x, y)$$ Is in page 3 after eqn 4.4.
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57 views

On the definition of Lagrangian

I have a question about "the definition of Lagrangian" in spacetime manifold. In general relativity, the energy-stress tensor and the vacuum energy stress tensor can be written as below: $$T_{\nu\mu}=...
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1answer
56 views

What is the intuitive concept of the action of a relativistic point particle? [duplicate]

The action of a relativistic point particle is its negative rest energy along its worldline, the parameter being its own proper time. $$ S = - mc^2 \int d\tau $$ (see Wikipedia) Action is energy ...
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23 views

Why the action of a relativistic point particle is considered to be negative? [duplicate]

The action of a relativistic point particle is its negative rest energy along its worldline, the parameter being its own proper time. $$ S = - mc^2 \int d\tau $$ (see Wikipedia) Is there a ...
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1answer
28 views

If a generalized coordinate is not cyclic, can we conclude that the corresponding generalized momentum is not conserved?

This is basically the reverse situation to the normal case: We find a cyclic coordinate in the lagrangian function describing the system and can conclude that the corresponding generalized momentum ...
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0answers
31 views

In d'Alembert's principle, how that the Reverse effective force and Force of constraints are different?

In d'Alembert's principle, how that the Reverse effective force and Force of constraints are different? Both are opposition or restriction on the body but how both be different in real?
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0answers
50 views

Extrinsic Curvature variation

I have seen the post Explicit Variation of Gibbons-Hawking-York Boundary Term on variation of Gibbons-Hawking term, that was really helpful, however, I have problem evaluating $\delta K$ and getting ...
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0answers
34 views

To find the conserved quantities in a lagrangian?

Lagrangian of a particle of mass $m$ is given by $$L= \frac{m}{2}[(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2 + (\frac{dz}{dt})^2]-\frac{V}{2}(x^2+y^2)+ W\sin(\omega t)$$ Is energy conserved here since ...
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1answer
43 views

Derivation of Hamilton's Third Relation. Where is the mistake?

As a sort of follow-up from my previous question, I'd like to point out two derivations of Hamilton's third relation that lead to two different results. Clearly there is a mistake within the process, ...
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1answer
74 views

How is this Lagrangian derived? (Lagrangian with an exponential function)

In the second answer of this post, Euler-Lagrange equations and friction forces I see a normal Lagrangian (T-V) times an exponential function. $${\cal L}=e^{t\gamma/m}\left(\frac{m}{2}\dot{x}^2 -U(t,...
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0answers
91 views

Learning about 4 topics in physics [closed]

This isn't really a question on any of those numerous underlying concepts behind the various sub-disciplines of physics, but hear me out: I'm still in Higher Secondary, but I'd really love to know ...
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1answer
43 views

Question in Lagrangian formalism

In lagrangian mechanics, where $L=T-U$ and the lagrangian formulation is $ \frac{d}{dt}\big( \frac{\partial L}{\partial \dot{q_i}}\big)-\frac{\partial L}{\partial {q_i}}=F_i$, where $F$ is the non-...
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49 views

Matching symmetry factor when a heavy vector field is integrated out

Let us consider the lagrangian $$ \mathcal{L} = \alpha \bar{u}\gamma^\mu u V_\mu + \frac{\beta^2}{2}V_\mu V^\mu $$ there $V_\mu$ is a heavy vector field and $u$ is a massless SU(3)-colored quark. If ...
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0answers
76 views

Why don't we have to go through the Lagrangian in QM? [duplicate]

In classical mechanics, I remember whenever we calculated the Hamiltonian, we'd first have to calculate the Lagrangian, and then we'd get the Hamiltonian through the definition: $$H= \sum\dot q_ip_i-...
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1answer
110 views

Feynman's explanation of virtual work given in his book Feynman's lectures on Physics

In his book Chapter 4 Conservation of Energy, on Gravitational potential energy the discussion goes... "Take now the somewhat more complicated example shown in Fig. 4-6. A rod or bar, 8 feet long, ...
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Examples of multiply-connected compact configuration spaces

I'm a looking for examples of dynamical systems that have multiply-connected compact configuration spaces. Since I'm not a 100% sure about the correct terminology for the systems (I am sure about the ...
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4answers
159 views

How can I tell that a Lagrangian has an $SU(2)\times SU(2)$ symmetry?

this is a very basic question and it probably has a very simple answer. I was reading through some handouts when I came over something that I did not understand. One considered the simple Lagrangian ...
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2answers
113 views

Are the partial derivatives of Lagrangian in the varied action functional derivatives?

In particle mechanics Lagrangian $L$ depends upon position, velocity (and may be explicitly on time), whereas in field theory the Lagrangian density ${\cal L}$ similarly (or analogously) depends upon ...
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2answers
150 views

Beyond Hamiltonian and Lagrangian mechanics

Lagrangian and Hamiltonian formulations are the bedrock of particle and field theories, produce the same equations of motion, and are related through a Legendre transform. Are there more such ...
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0answers
124 views

Geometric point of view of configuration space and Lagrangian mechanics

Even though (I think) I understand the concept of a tangent bundle, I have trouble assimilating the idea of the configuration space being one and in relation to what that is the case. How can I ...
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1answer
74 views

Euler-Lagrange Equation in Quantum Field Theory

The quantum fields are operator valued distributions. In some sloppy books like Peskin and Schroeder the Euler-Lagrange equation are used to get the equations of motion. What does it mean to take a ...
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1answer
35 views

My Hamiltonian for a light ray vanishes

I have the following issue with understanding. A light ray traveling from $q(\tau_1)$ to $q(\tau_2)$ minimizes the integral $\int\limits_{\tau_1}^{\tau_2} n(q(\tau))|\dot{q}(\tau)| d\tau$, so the ...
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0answers
22 views

Change of energy of a shortening simple pendulum (Ehrenfest Pendulum) [closed]

I've been going through Lanczos' Variational Principles of Mechanics and have been struggling with a problem that seemed pretty straightforward: A simple pendulum hangs from a fixed pulley. The ...
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0answers
50 views

Functional Differetiation of a complex functional

Suppose I have a simple functional $$F=\int{dx\;\phi^{*}(x)\phi(x)}\tag{1}.$$ Assuming $\phi(x)$ and $\phi^{*}(x)$ are independent and I take a functional differential with respect to $\phi(x)$ and $\...
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31 views

Functional variation of potential in integral form

I am trying to vary the following action, $$ S=\int_{t_0}^{t_1} \text{d}t\,(v^\mu v^\nu g_{\mu\nu} + V(t)) =\int_{t_0}^{t_1}\text{d}t\,(v^\mu v^\nu g_{\mu\nu} + \int_{t_0}^t\text{d}s T_\mu v^\mu) $$ ...
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1answer
27 views

Derivation of photon propagator from EM Lagrangian

I am following Ryder's Quantum Field Theory. In chapter 7, in order to derive the photon propagator, he first derives eq. 7.4 $$\mathcal{L}=\dfrac{1}{2}A^\mu[g_{\mu\nu}\partial^2-\partial_\mu\partial_\...
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0answers
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Lagrangian of classical electromagnetism without $A_{\mu}$ field [duplicate]

Is there a Lagrangian reproducing Maxwell's equations without the use of the scalar and vector potential? Obviously $\mathcal{L} = -\frac14F_{\mu \nu}F^{\mu \nu}$ doesn't work since in terms of $E$ ...
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2answers
40 views

Lagrangian mechanics not relying on time or independent of time [closed]

If neither the potential energy nor kinetic energy depends on time, then Lagrangian is explicitly independent of time I find this statement a little bit odd because velocity is distance over time or ...
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1answer
54 views

Connection between “classical” Grassmann variables and Heisenberg Equation of motion

I have been reading di Francesco et al's textbook on Conformal Field theory, and am confused by a particular statement they make on pg 22. Let $\{\psi_i\}$ be a set of Grassmann variables. Starting ...
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0answers
47 views

Non-canonical transformation

I would like to know any method to transform a known non-canonical set of variables to a canonical set for a given system. The Lagrangian and Hamiltonian are known in the non-canonical variables. I ...
4
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1answer
64 views

Vary action with respect to velocity

Variation of the action $S$ corresponding to a Lagrangian e.g. $L(x(t),\dot{x}(t))$ gives the Euler-Lagrange equations: $$ \frac{\delta S}{\delta x(t)} = 0 \\ \int du \left ( \frac{\delta L}{\delta x(...
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1answer
135 views

The difference between the forms of the Euler-Lagrange equations

I'm trying to learn Lagrangian mechanics and have been reading a lot of articles on it. But many of the articles write the equations in different ways, probably for different purposes. The Euler-...
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0answers
19 views

Euler-Lagrange equation no fixed endpoints

The usual way, to show the Euler-Lagrange equation is, to find the minimum of the Integral $$ I = \int_a^b L(q, \dot q, t) dt $$ and argue, that it must satisfy the following equation $$ \frac{d}{dt} \...
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0answers
49 views

From gauge invariance to charge conservation in covariant electrodynamics

I tried to solve the equations of motion using the action for the electromagnetic field interacting with a current, like $$ L = F_{\mu\nu}F^{\mu\nu} + A_{\nu}j^{\nu} $$ getting the right Maxwell's ...
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2answers
58 views

Lagrangian of an effective potential

If there is a system, described by an Lagrangian $\mathcal{L}$ of the form $$\mathcal{L} = T-V = \frac{m}{2}\left(\dot{r}^2+r^2\dot{\phi}^2\right) + \frac{k}{r},\tag{1}$$ where $T$ is the kinetic ...
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A question about Euler-lagrange equations

This question passes my mind so often, why do we stop at the first order of expansion of the action to get the Euler-lagrange equations and it turns out they exactly get us the Newtonian equations. ...
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1answer
23 views

Parity tranformation on Lagrangian of free fields

Free lagrangians of scalar, Dirac field and vector fields are always invariant under Parity. I am able to get this result mathematically, but I want to know if there is any obvious reason for it. ...
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3answers
89 views

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. $$...
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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 \...