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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Equation of motion of a simple pendulum

Consider a simple pendulum of length $l$ moving in $(x,y)$ plane. Assume its point of attachment is being jiggled by external forces with prescribed acceleration $a(t)=(a_x(t),a_y(t))$. Let $\theta(t)$...
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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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Hamilton's principle and virtual work by constraint forces

I have a question about the following pages(pg 47 and 48) from Goldstein's "Classical Mechanics" I do not understand how (2.34) shows that the virtual work done by forces of constraint is zero. ...
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Use my example to explain why loop diagram will not occur in classical equation of motion?

We always say that tree levels are classical but loop diagrams are quantum. Let's talk about a concrete example: $$\mathcal{L}=\partial_a \phi\partial^a \phi-\frac{g}{4}\phi^4+\phi J$$ where $J$ is ...
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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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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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2k views

Calculating the moment of inertia in bifilar pendulums

I'm an A2 student, and I've been looking into how experimental and theoretical determined mass moments of inertia differ. I came across a method (search Youtube for Measuring Mass Moment of Inertia - ...
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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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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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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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283 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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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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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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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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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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207 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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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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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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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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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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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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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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363 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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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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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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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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131 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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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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111 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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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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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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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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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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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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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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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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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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397 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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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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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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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 ...
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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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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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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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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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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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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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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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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 ...