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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26 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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2answers
269 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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0answers
19 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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2answers
496 views

Kepler's laws to determine radius of circular orbit

"In nonrelativistic limit of general relativity there is a correction to the Newtonian gravitational potential energy $−h/r^3$ with $h = αL^2/(mc)^2$, where $c$ is the speed of light, $α = GMm$ and $...
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1answer
135 views

Is $\phi^4$ theory in 4d conformally invariant at the classial level?

I used to believe the three following statements to be true (at the classical level only): From scale invariance full conformal invariance follows. Scale invariance is present if there are no ...
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1answer
70 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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37 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
200 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
194 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 ...
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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
1k views

Is the Lagrangian density in field theory real?

As the Lagrangian in classical mechanics corresponds to energy, it must be real. But is that the case in quantum field theory? I mean, it should still correspond to some sort of energy, but what about ...
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2answers
850 views

Stress-energy tensor for a fermionic Lagrangian in curved spacetime - which one appears in the EFE?

So, suppose I have an action of the type: $$ S =\int \text{d}^4 x\sqrt{-g}( \frac{i}{2} (\bar{\psi} \gamma_\mu \nabla^\mu\psi - \nabla^\mu\bar{\psi} \gamma_\mu \psi) +\alpha \bar{\psi} \gamma_\mu \...
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1answer
46 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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1answer
230 views

Degrees of freedom in double Atwood machine?

Why the degree of freedom in double Atwood machine (one block on one side and a pulley with one block in its each side on other side) is 2 and not 1? According to the formula $s=3*n-m$; where $n=$...
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1answer
361 views

Distinguishing mechanical systems from general dynamical systems

In the following let a "mechanical system" be a system of $n$ spatial objects moving in physical space. Consider you are given a function $q:\mathbb{R} \rightarrow \mathcal{M}^n$ with $\mathcal{M}$ a ...
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54 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
120 views

Relativistic action is negative or positive number? [duplicate]

Possible Duplicate: Why lagrangian is negative number? In the special relativistic action for a massive point particle, $$\int_{t_i}^{t_f}\mathcal {L}dt,$$ where the Lagrangian $$\mathcal {L}...
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1answer
25 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 ...
2
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1answer
109 views

Equations of motion for double spherical pendulum simply?

I am attempting to simulate a double spherical pendulum, i.e. a combination of the spherical pendulum and the double pendulum. I understand that the equations of motion can be derived via the ...
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2answers
131 views

Why is the solution of the $\phi^6$ potential not a soliton?

Consider a theory with a $\phi^6$-scalar potential: $$ \mathcal{L} = \frac{1}{2}(\partial_\mu\phi)^2-\phi^2(\phi^2-1)^2. $$ I solved its equation of motion but found that the general form of its ...
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0answers
27 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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6answers
6k views

What is the physical meaning of the action in Lagrangian mechanics?

The action is defined as $S = \int_{t_1}^{t_2}L \, dt$ where $L$ is Lagrangian. I know that using Euler-Lagrange equation, all sorts of formula can be derived, but I remain unsure of the physical ...
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4answers
350 views

What makes an equation an 'equation of motion'?

Every now and then, I find myself reading papers/text talking about how this equation is a constraint but that equation is an equation of motion which satisfies this constraint. For example, in the ...
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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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0answers
49 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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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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2answers
2k views

Can a force in an explicitly time dependent classical system be conservative?

If I consider equations of motion derived from the principle of least action for an explicitly time dependent Lagrangian $$\delta S[L[q(\text{t}),q'(\text{t}),{\bf t}]]=0,$$ under what ...
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0answers
31 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
392 views

How do I treat the Lagrangian in the case of a rigid body?

Here's Exercise 1.11 from Goldstein's Classical Mechanics 3rd edition (the first one after having derived the Lagrangian basically): Exercise 1.11: Consider a uniform thin disk that rolls without ...
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1answer
942 views

Noether current for the Yang-Mills-Higgs Lagrangian

I am trying to calculate the Noether current, more specifically, the energy density of the Yang-Mills-Higgs Lagrangian. Please refer to the equations in the Harvey lectures on Magnetic Monopoles, ...
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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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2answers
252 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 ...
3
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1answer
41 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
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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1answer
73 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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48 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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1answer
40 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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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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75 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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4answers
158 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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0answers
22 views

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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2answers
159 views

Does the conservation of the Wronskian follow from Noether's principle?

Noether's principle is the paradigm that symmetries of Hamiltonian and Lagrangian systems correspond to conservation laws of various kinds. Consider a one-dimensional harmonic oscillator $$\tag{*} \...
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1answer
384 views

Reduction of Nambu Goto action to true degrees of freedom

First consider the particle $$S=m\int\sqrt{-\dot{X}^2}d\tau$$ if you choose the static gauge $\tau=X^0$ and replace it in the action you get $$=m\int\sqrt{1-\dot{X}^j\dot{X}^j}d\tau$$ So now, you ...
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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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4answers
850 views

Why the Hamiltonian and the Lagrangian are used interchangeably in QFT perturbation calculations

Whenever one needs to calculate correlation functions in QFT using perturbations one encounters the following expression: $\langle 0| some\ operators \times \exp(iS_{(t)}) |0\rangle$ where, ...
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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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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 $\...
2
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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 ...