# Tagged Questions

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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### 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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### 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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### Locality defined in terms of the Lagrangian density

I've been reading through Matthew Schwartz's book "Quantum Field Theory and the Standard Model" and in chapter 24 there is a section on locality (section 24.4). In it he defines locality in terms of ...
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### 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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### 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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### What is the relationship between local and global symmetries? [on hold]

A global symmetry has a few different meanings. The obvious one is that for $g = e^{-i\theta}$ if $\theta$ is contant in space then a field $\phi$ transforms $\phi' = e^{-i\theta}\phi$ so that ...
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### Determining the Lagrangian of a double pendulum [on hold]

Ok, I'm reading up on Lagrangian mechanics, and there is a problem that I don't really understand: the double pendulum (in this case, without a gravitational field). So, I want to take it step by step ...
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### 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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### Polyakov From Nambu-Goto Directly, for Strings?

The following derivation, for a classical relativistic point particle, of the 'Polyakov' form of the action from the 'Nambu-Goto' form of the action, without any tricks - no equations of motion or ...
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### Determinant for a coupled fluctuation Lagrangian

Lets consider a bosonic physical system in variables $t, x$ and $y(x)$ ($x$ dependent) with a classical Lagrangian $L$. To first order in fluctuations $x \to x+\xi_1$ and $y \to y+\xi_2$ the ...
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### How to find the rank of the matrix $\frac{\partial ^2\mathcal{L}}{\partial \dot{X^\mu} \partial \dot{X^\nu} }$ for the Nambu-Goto string Lagrangian?

In this case $$\mathcal{L}~=~-T\sqrt{-\dot{X^2}X'^2+(\dot{X}\cdot X')^2}.$$ I was reading some books and papers about the constraints in the Nambu-Goto action, and all of them say something like ...
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### 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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### How to find potential energy(PE) of a system in order to write Lagrangian? Is there is any unique way to write PE for different system?

for simple pendulum or spring system its very easy . we use mgl and mgh. What I need is the unique way which will help to write PE to all the systems.
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### 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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### 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, $α = GMm$ ...
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### 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 ...
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### 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 ...
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### 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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### Point of Lagrange multipliers

I am trying to understand how for a constrained system the introduction of Lagrange multipliers facilitates the incorporation of the holonomic constraints. I am using Classical Mechanics by John ...
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### 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 ...
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### The Lagrangian as a metric

My question is, can the (classical) Lagrangian be thought of as a metric? That is, is there a meaningful sense in which we can think of the least-action path from the initial to the final ...
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### What are the equations of motion for the scalar field in the tetrad formalism?

The action of a massless scalar field in curved spacetime is given by: $$S(\phi)=\int d^{4}x \sqrt{-g}\left(g^{\mu\nu}\phi_{,\mu}\phi_{,\nu}\right)$$ Now the action can ...
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### 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 ...