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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2answers
126 views

What is the difference between configuration space and phase space?

What is the difference between configuration space and phase space? In particular, I notices that Lagrangians are defined over configuration space and Hamiltonians over phase space. Liouville's ...
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0answers
35 views

Is the Legendre transformation a unique choice in analytical mechanics?

Consider a Lagrangian $L(q_i, \dot{q_i}, t) = T - V$, for kinetic energy $T$ and generalized potential $V$, on a set of $n$ independent generalized coordinates $\{q_i\}$. Assuming the system is ...
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0answers
51 views

Problems with Maxwell density Lagrangian [on hold]

I want to expand the right-hand side of $$\dfrac{1}{4} F_{\mu\nu}F^{\mu\nu} = \dfrac{1}{4}\left(\partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}\right)\left(\partial^{\mu}A^{\nu} - ...
2
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1answer
120 views

Symmetries of the action of the free classical Klein-Gordon field

I've read that the action for the free classical Klein-Gordon field $$S = \int \mathrm{d}^4x~ \mathcal{L} = \frac{1}{2} \int \mathrm{d}x^4 \left(\partial_\mu \phi(x) \, \partial^\mu \phi(x) - ...
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0answers
47 views

Particle on a sphere - lagrange [on hold]

A particle of mass m is on top of a sphere of radius $R$. A small displacement makes the particle slide frictionlessly down the sphere. a)Set up the lagrange equations of the first kind for ...
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1answer
31 views

Reversing time for a closed system of particles

For a closed system of particles, the lagrangian in classical mechanics is $$L=\sum \frac{1}{2}mv_a^2 - U(\mathbf{r_1},\mathbf{r_2}, \cdots)$$ For an arbitrary position function $x(t)$, to see the ...
9
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0answers
74 views

Degenaracy in mass of $8$ and $27$ reps of $SU(3)$ in Coleman's Aspects of Symmetry

In Coleman's Aspect of symmetry he proposes an amusing problem in the first chapter. It asks us to consider a set of eight pseudo-scalar fields transforming in the adjoint representation of $SU(3)$. ...
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2answers
79 views

What exactly is the Action? (Learning lagrangian)

I have been trying to wrap my head around lagrangian mechanics but I find some parts confusing. For example, what exactly is action and why is it defined by the Kinetic energy minus the potential ...
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1answer
29 views

How to check if some term in the Lagrangian involving gauge bosons is gauge invariant without explicit computations?

Normally (for fermions and scalars) we can simply use the decomposition of tensor products of gauge group representations to find invariant terms that we can write into the Lagrangian. For example ...
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0answers
37 views

Doubts taking the second functional derivative of the Klein Gordon action

I have very little background with functional derivatives and I would like to clarify some issues. I am trying to compute the second functional derivative of the Klein Gordon action expressed in real ...
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0answers
15 views

Reformulating a traditional minimization problem as a stationary action problem [on hold]

A famous problem in financial theory is formulated as follows: $$\min_{\mathbf{w}} \sigma^2 \:\:\:\:\:s.t. \:\:\mathbf{w'}\mathbf{1}=1 \:\:\:\: \mathbf{w'}\mathbf{y}=k$$ Where lower case bold faced ...
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2answers
132 views

In QFT how do you write down the most general interactions?

This past year I took a QFT class and I now feel comfortable solving scattering problems, but I am still a bit perplexed by how physicists write down a Lagrangian in the first place. In particular, ...
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2answers
58 views

First fundamental form in the Gibbons-Hawking-York boundary term

Let me expose my problem, I am trying to perform the explicit variation of the Gibbons-Hawking-York boundary term, $$S_{GH}=\int_{\partial M} d^{n-1}x\sqrt{\left|h\right|}K$$ The problem I have is ...
5
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2answers
285 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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0answers
37 views

Mass point on a circle - lagrange [closed]

A mass point of mass m moves on the circle $x^2+y^2=R^2$ and $z=0$. No external forces are acting. Solve the equation of motions and determine the constraint force with the lagrange equations of ...
2
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1answer
63 views

Analytical mechanics with SR

Is there an analytical mechanics with SR? Of course you can write down the Lagrangian and Hamiltonian of a free particle. What about non-free? Are there any problems? To be specific: what would the ...
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2answers
239 views

Are there other less famous yet accepted formalisms of Classical Mechanics?

I was lately studying about the Lagrange and Hamiltonian Mechanics. This gave me a perspective of looking at classical mechanics different from that of Newton's. I would like to know if there are ...
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1answer
76 views

How to write the Lagrangian in terms of a projection

We know that $$ L=\frac{1}{2}\left(\partial_{\mu} A_{\nu} \partial^{\mu} A^{\nu}-\partial_{\mu} A_{\nu} \partial^{\nu} A^{\mu}\right) $$ But how do we write the Lagrangian in the following way: ...
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2answers
72 views

Classical trajectories that are not a minimum of the action [duplicate]

Are there physically realizable dynamical systems where the true trajectory is not a minumum action trajectory? Formally, Lagrangian mechanics only requires that the trajectory be an extremum (or ...
0
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1answer
61 views

A course in Lagrangian Mechanics [duplicate]

I would like to know: what are some of the best introductory books to Lagrangian Mechanics?
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2answers
3k views

Explicit Variation of Gibbons-Hawking-York Boundary Term

Are there any references that present the explicit variation of the Hilbert-Einstein action plus the Hawking-Gibbons-York boundary term, and demonstrate the cancellation of the normal derivatives of ...
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2answers
212 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, $α = GMm$ ...
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1answer
108 views

Determine path of point mass using the Hamilton's principle

I am very new in this field but I try to solve a problem by using the Hamilton's principle and afterwards I want to compare the solution by solving the same problem using conservation laws. What I ...
3
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1answer
255 views

Least Action Principle (Classical and Quantum Theory)

I) My first question would be "why should classical systems obey the principle of least action ?" When we find out the propagator in quantum physics, we find the amplitude to be equal to the sum over ...
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0answers
23 views

How is the electromagnetic tensor expanded?

The electromagnetic tensor is given by $F_{\mu \nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$, and it appears in the Lagrangian as $L = -\frac{1}{4}F_{\mu\nu}^2 - A_{\mu}J_{\mu}$. The text I'm ...
2
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2answers
162 views

Lagrange's equation implying Newton's 2nd law?

The typical first application of Lagrange's equation is showing that it implies Newton's law for a particle whose Lagrangian is $L=\frac{1}{2}mv^2-V(x)$. Plugging this Lagrangian into Lagrange's ...
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1answer
38 views

Apparent discrepancy between Lagrange field equation and Maxwell equation [closed]

I am deriving Maxwell's equations from a Lagrange field equation and have come across something I can not figure out no matter how hard I try. The problem is in the signs. If we take the Lagrange ...
3
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1answer
71 views

Confusion about imposing constraint in the action

I'm totally confused by one thing. I know that I probably shouldn't be confused about that, but at the moment I don't quite know what fails in the following: Suppose we have a particle of unit mass ...
3
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3answers
208 views

Constructing Lagrangian from the Hamiltonian

Given the Lagrangian $L$ for a system, we can construct the hamiltonian $H$ using the definition $H=\sum\limits_{i}p_i\dot{q}_i-L$ where $p_i=\frac{\partial L}{\partial \dot{q}_i}$. Therefore, to ...
4
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4answers
291 views

Geodesic Equation from variation: Is the squared lagrangian equivalent?

It is well known that geodesics on some manifold $M$, covered by some coordinates ${x_\mu}$, say with a Riemannian metric can be obtained by an action principle . Let $C$ be curve $\mathbb{R} \to M$, ...
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1answer
94 views

Dirac bracket for the Majorana Lagrangian

Note: See update below. Consider the Majorana Lagrangian $$\mathcal{L}=-\psi ^{\mathrm{T}}\mathrm{i}% \gamma ^{0}\left( \gamma ^{\rho }\partial _{\rho }+m\right) \psi ,\tag{1}$$ where $% \psi \in ...
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0answers
74 views

A classically charged point particle interacting with electromagnetism and gravity

Consider a classically charged point particle interacting with electromagnetism and gravity. The relevant dynamical variables are $\chi^\mu (\tau)$ of the particle, the electromangetic potential ...
11
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1answer
895 views

Do an action and its Euler-Lagrange equations have the same symmetries?

Assume a certain action $S$ with certain symmetries, from which according to the Lagrangian formalism, the equations of motion (EOM) of the system are the corresponding Euler-Lagrange equations. Can ...
5
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1answer
101 views

How is the Lagrangian defined in GR?

Reading about the Schwarzschild metric in general relativity I see that sometimes $$L=g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}$$ and sometimes $$L=\sqrt{g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}}.$$ Which is ...
3
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2answers
88 views

Derivation of law of inertia from Lagrangian method (Landau)

I'm reading Landau's Book. He tries to conclude the law of inertia from the Lagrange equations. For that, he argues (by nice suppositions about space and time), that the lagrangian must depend only ...
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6answers
5k views

What is canonical momentum?

What does the canonical momentum $\textbf{p}=m\textbf{v}+e\textbf{A}$ mean? Is it just momentum accounting for electromagnetic effects?
8
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2answers
351 views

For a particle to have physical mass, is it always necessary to have a mass term in the lagrangian?

Since the self-energy adds to the bare mass defined in the Lagrangian, is it possible to create a physical particle mass from the self-energy alone, with no mass terms occuring in the Lagrangian? On ...
4
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1answer
383 views

Sufficient conditions for the energy to be not conserved?

I'm almost embarrased to ask this question because I thought I was by now very confident with classical mechanics. Someone has stated that given a mechanical system with a Lagrangian $L$ s.t. ...
0
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1answer
42 views

Q: Goldstein chapter 1 problem 16: Finding the generalized potential from the force

I have started to work through Herbert Goldstein's, Charles Poole's and John Safko's Classical mechanics, and I am having a bit of trouble with one of the problems (chapter 1 problem 16). The problem ...
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0answers
12 views

Morin's Spring Pendulum gravitational potential [duplicate]

I am trying to do Morin's Lagrangian example of a spring pendulum. I can't quite figure out how he derived the gravitational potential energy as $-mg(l+x)cos \theta$. The closest I could get was $mg( ...
3
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3answers
6k views

Finding Lagrangian of a Spring Pendulum

I'm trying to understand Morin's example of a spring pendulum. What I don't get is his expression for $T$. I can understand the $\dot x^2$ term in the brackets. But I don't understand the $(l + ...
0
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1answer
24 views

Bead on a rotating wire - Conservation of angular momentum, fix points

Lets consider a wire in the x-y plane which rotates with constant angular velocity $\omega$. The coordinates of a bead, which is forced to stay on this wire, can then be expressed as $$x=r \cos(\phi) ...
5
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1answer
304 views

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 ...
1
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1answer
298 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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1answer
51 views

Equations of motion for Polyakov action

In Polchinski 2.1.10 we have the action in terms of complex coordinates $$S = \frac{1}{2\pi \alpha'} \int d^{2}z \partial X^{\mu}\bar{\partial}X_{\mu}\tag{2.1.10}$$ This should be a rather trivial ...
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1answer
51 views

Derivation of an ordinary, Lagrangian/Hamiltonian and action formulation

I am confused as to how the different formulations in physics are derived. In many fields of physics, we usually begin with an ordinary formulation (e.g Newton's Laws in classical mechanics), and ...
2
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2answers
158 views

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 ...
2
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1answer
205 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 ...
2
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1answer
109 views

Null geodesic equation

For a null geodesic curve $X^i$, $$0=g_{ij}V^iV^j.$$ When we derive the geodesic equation from E-L equations, will this affine parametrization cause it to blow up? How is it justified to use the ...
18
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3answers
4k views

Deriving the Lagrangian for a free particle

I'm a newbie in physics. Sorry, if the following questions are dumb. I began reading "Mechanics" by Landau and Lifshitz recently and hit a few roadblocks right away. Proving that a free particle ...