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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120 views

Conservation of energy when the Lagrangian includes a potential function

When proving that the homogeneity of time leads to the conservation of energy, (This is the proof from Landau for the case when there is no field present.) (Uses the Einstein's summation ...
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1answer
393 views

Example of Hamilton's Principle to Systems with Constraints (Goldstein)

I'm currently studying Goldstein's Classical Mechanics book and I can't get my head around his reasoning in section 2.4. (Extending Hamilton's principle to systems with constraints). I'd like to ...
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3answers
265 views

Virtual Work: How is the applied force related to the coordinates chosen?

I have a question after reading a section from Goldstein's Classical Mechanics. The question deals with equation 1.43 in the text (given below): $$ \tag{1.43} \sum\limits_{i} {\bf F}_i^{(a)}\cdot ...
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1answer
205 views

Lagrangian and grassmann numbers

Why sometimes we remember that "classical" lagrangians of fermions are constructed from grassmann numbers, while sometimes don't? For example, for Majorana's field in terms of 2-component spinors ...
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1answer
152 views

Mass term in the Lagrangian

I have read that the mass term appearing in the electroweak Lagrangian stops it (the Lagrangian) from becoming gauge invariance. Can someone explain where and why this term is creating the problem?
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1answer
324 views

Lagrangian depends on second derivative of field

In case of the gauge-fixed Faddeev-Popov Lagrangian: $$ \mathcal{L}=-\frac{1}{4}F_{\mu\nu}\,^{a}F^{\mu\nu ...
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2answers
319 views

Curved spacetime point particle Lagrangian density

This is probably trivially related to the question: Action for a point particle in a curved spacetime , but am a bit unsure how to write it as a Lagrangian density. In curved spacetime the action is ...
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3answers
1k views

Does a four-divergence extra term in a Lagrangian density matter to the field equations?

Greiner in his book "Field Quantization" page 173, eq.(7.11) did this calculation: ${\mathcal L}^\prime=-\frac{1}{2}\partial_\mu A_\nu\partial^\mu A^\nu+\frac{1}{2}\partial_\mu A_\nu\partial^\nu ...
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1answer
518 views

Working with a Routhian for a specific system

I asked a more general question earlier about the Routhian, but I'm still having trouble working with it. Here's my specific case. Given the following Lagrangian: ...
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4answers
775 views

Are Lagrangians and Hamiltonians used by Engineers?

Analytical Mechanics (Lagrangian and Hamiltonian) are useful in Physics (e.g. in Quantum Mechanics) but are they also used in application, by engineers? For example, are they used in designing bridges ...
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1answer
203 views

Definition of Kinetic energy

In class we had that $ T= \frac{1}{2}T_{ij}v_iv_j$ where we used the Einstein summation convention. Hitherto we only discussed examples where the kinetic energy was dependent of the square of one ...
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2answers
626 views

Different approaches to calculating the Christoffel symbols

I would be very grateful to whoever can debug the following calculations... We have the metric for static spacetime: $$ds^2 = -\exp(2U(\vec x))dt^2+h_{ij}(\vec x) d x^i d x^j$$ I want to find the ...
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3answers
396 views

Is the path of stationary action unique? What are the physical implications of $L_{\dot{x}}=L_x$

Below, for any function $Q$ the notation $Q_x$ means $\frac{\partial Q}{\partial x}$, and $Q_{xx}$ means $\frac{\partial^2 Q}{\partial x^2}$. In physics, the trajectory of a particle is given by the ...
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1answer
142 views

Obtaining the conserved current of the Lagrangian making the parameter depending on $x$

To calculate the conserved current due to an internal symmetry of the system (expressed by the Lagrangian density) we can proceed as follows: if it is invariant under $\delta \phi = \alpha \phi$, ...
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1answer
426 views

Euler-Lagrange Equation with logarithmic potential

A particle moving towards the origin has initial conditions $x(t=0) = 1$ and $\dot{x}(t=0)=0$. If the Lagrangian is $$L:=\frac{m}{2}\dot{x}^2 -\frac{m}{2}\ln|x|$$ This should satisfy ...
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1answer
267 views

Relationship between local and global scaling (Weyl) symmetry

Theorem 5.1 on page 80 of this paper says that Assuming that the matter fields satisfy their equations of motion, the matter field action is locally Weyl invariant if and only if the corresponding ...
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2answers
263 views

Is the gravitational constant G a minimum value in some sense?

Assume a central body of mass $M$, and call $a$ the acceleration of a test body at a distance $r$ due to any interaction whatsoever with the central body. Is is correct to say that the ratio $a r^2/ ...
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1answer
290 views

Do Lagrangian points actually maintain a fixed distance?

I was reading on up Lagrangian points and the restricted three-body problem. From what I was able to tell, the Lagrangian points are 5 points in a two-body system such that a third body would be ...
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2answers
782 views

M-theory no lagrangian?

Is there any formulated lagrangian (density) for M-theory? If not, why is there no lagrangian? If not, is this related to many vacua existing? Thnx.
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1answer
540 views

Origins of the principle of least time in classical mechanics

Is it possible to derive the principle of least time from the principle of least action in lagrangian or hamiltonian mechanics? Or is Fermat's principle more fundamental than the principle of least ...
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2answers
52 views

Where is the BRST symmetry?

When quantizing YM we start from the gauge fixed path integral (to remove redundancy of integrating over Gauge symmetric configurations) $$\int \mathcal{D}A \delta(G(A)) \text{det} \Delta_{FP}e^{i\int ...
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1answer
65 views

Hamiltonian or free energy corresponding to 2+1D Kuramoto-Sivashinsky model

I am trying to understand if the deterministic 2+1D Kuramoto-Sivashinsky equation $$ \partial_t h = -\nu \nabla^2 h - K \nabla^4 h + \frac{\lambda}{2} (\nabla h)^2, $$ where $\nu$, $K$, $\lambda$ ...
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1answer
77 views

Non-holonomic constraints in Dirac-Bergmann theory

The Dirac-Bergmann algorithm effectively isolates the physical degrees of freedom of a system, by changing from Poisson brackets $\{\cdot,\cdot\}_\mathrm{PB}$ to Dirac brackets ...
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2answers
88 views

Action max, min, or saddle?

It is well known that $\delta S = 0$ lays the foundation for variational mechanics. But I am confused as to whether or not this S is a minimum, a maximum, or a saddle point. Some books address this ...
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2answers
112 views

qualitative explanation of Principle of Least Action (vertical movement)

Consider the following situation I want to understand what the PLA means here from an intuitive and qualitative point of view. I understand the mathematical approach. Combining ...
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1answer
106 views

Do time-invariant Hamiltonians define closed systems?

In classical mechanics, every time-invariant Hamiltonian represents a closed dynamical system? Can every closed dynamical system be represented as a time-invariant Hamiltonian? Or are there closed ...
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1answer
87 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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1answer
106 views

Trivial conserved Noether's current with second derivatives

I'm considering a symmetry transformation on a Lagrangian $$ \delta A = \int L(q +\delta q, \dot{q} + \delta \dot{q} , \ddot{q} + \delta \ddot{q}) dt $$ the general variation takes the form $$ ...
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2answers
906 views

Holonomic constraints and degrees of freedom?

Can we see that a constraint can decrease the degrees of freedom of a system if and only if it is holonomic. Either way please can you explain why?
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1answer
106 views

Under what cases is the Batalin-Vilkovisky (BV) operator nilpotent?

It is understood that when we deal with gauge algebras which close on-shell only after using equations of motion or where the space-time is curved, we can no longer just do away with BRST ...
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1answer
102 views

Lagrangian vector field expression

The Lagrangian vector field $X_L$ on the tangent bundle is given in page 4 of Marco Mazzucchelli's "critical Point Theory for Lagrangian systems" to be; \begin{equation} ...
2
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1answer
319 views

The einbein in the action of a relativistic massive point particles [closed]

The action of a relativistic massive point particle moving in space-time is $$S=-m\int d\tau \sqrt{g _{\nu \rho}\frac{dx^{\nu}}{d\tau}\frac{dx^{\rho}}{d\tau}}$$ [with Minkowski sign convention ...
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1answer
182 views

Given a QFT Hamiltonian, is there a unique Lagrangian?

Consider a QFT in one spatial dimension specified by the following Hamiltonian density: $\mathcal{H} = -i \phi^\dagger \frac{\partial}{\partial x} \phi + V(\phi^\dagger,\phi)$ where $\phi$ is a ...
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2answers
180 views

How to calculate the classical on-shell action for a harmonic oscillator? [closed]

So, short and sweet, I've been reading the path integrals book by Feynman and Hibbs, and one of the elementary problems they ask is to calculate the classical on-shell$^1$ action of a harmonic ...
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1answer
157 views

Solving the Kepler problem

I'm trying to solve the Kepler problem using the Lagrangian, $$L = \frac{1}{2} m (\dot{r}^2 + r^2 \dot{\phi}^2) - U(r) $$ which after quite a bit of fiddling with, by noting that the angular ...
2
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1answer
89 views

Lagrangian under time transformation

Given a Lagrangian $$L(q,\dot{q},t)=\sum_{ij}a_{ij}(q)\dot{q}_i\dot{q}_j-V(q_1,q_2,\cdots,q_f)$$show that under a time transformation $t=\lambda T$ ($\lambda$ = constant), the invariance of ...
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1answer
106 views

Introduction of the vector potential $A_{\mu}$ for the local gauge invariance of the complex scalar field lagrangian [duplicate]

In Ryder, when trying to restore the local $U(1)$ gauge symmetry of the complex scalar field $\phi=\phi_1+i\phi_2$, the final Lagrangian consists of the following four parts: ...
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2answers
441 views

The variation of the Lagrangian density under an infinitesimal Lorentz transformation

I'm trying to introduce myself to QFT following these lectures by David Tong. I've started with lecture 1 (Classical Field Theory) and I'm trying to prove that under an infinitesimal Lorentz ...
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2answers
271 views

Pendulum with changing length over time. What's wrong?

I tried to find the equation of this pendulum, but I think I did something wrong. I know I have to get the Bessel's equation but I can't see it. It's a simple 2-D pendulum, without any dissipation. ...
2
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1answer
604 views

Local versus non-local functionals

I'm new to field theory and I don't understand the difference between a "local" functional and a "non-local" functional. Explanations that I find resort to ambiguous definitions of locality and then ...
2
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1answer
291 views

What is the generating functional for a scalar theory with two different (interacting and real) fields?

My question is specifically about how to use sources? For an interacting theory with one field, one puts a $J(x)\phi(x)$ term in the exponential in the path integral for $W[J]$. I now have two ...
2
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1answer
182 views

Lagrangian to Hamiltonian

I'm having some problems with an assignment where I have to state the Hamiltonian from the kinetic energy $T$ and potential energy $U$. These are as follows: ...
2
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1answer
390 views

Energy-momentum tensor for dust

We all know that the energy-momentum tensor for dust is just $T^{\alpha\beta}=\rho_0v^\alpha v^\beta,$ where $\rho_0$ is the mass density in the dust's rest frame and $v^α$ is the dust's ...
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1answer
126 views

How to find the equillibrium points using Jacobian and Hessian?

Given that I have Jacobian and Hessian matrices of three particles interacting with each other in a harmonic trap through Coulomb's law in a 2D plane, how do I find the equilibrium points of them (I ...
2
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1answer
111 views

Stationary action with maximized action [duplicate]

I would like to ask for an example (a lagrangian) both in classical and quantum level for which the action is maximaized (rather than minimized). What is special in these cases?
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1answer
996 views

Lorentz force from velocity-dependent potential and Lagrangian

There is something i'm missing. I am at page 22-23 of Goldstein Classical Mechanics 3rd ed. Lorentz force can be derived from a potential $$U=q\phi-q\mathbf{A}\cdot\mathbf{v}$$ Where $\phi(t,x,y,z)$ ...
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2answers
659 views

Virtual displacement and generalized coordinates

I have a doubt regarding the expression of a virtual displacement using generalized coordinates. I will state the definitions I'm taking and the problem. The system is composed by $n$ points with ...
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2answers
128 views

Where is the magnetic self energy term in $L$ for a charged particle in an electromagnetic field?

In the Lagrangian for a charged particle in an electromagnetic field $$L = \frac{1}{2}mu^2 - q(\phi - \frac{\vec{A}}{c}\cdot \vec{u})$$ the energy of the particle is contained in the kinetic term, ...
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1answer
522 views

Electrodynamics and the Lagrangian density

Could anyone tell me what equations can I obtain from the Lagrangian density $${\cal L}(\phi,\,\,\phi_{,i},\,\,A_i, \dot A_i,\,\,A_{i,j})~=~\frac{1}{2}|\dot A+\nabla\phi|^2-\frac{1}{2}|\nabla \times ...
2
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1answer
497 views

Lagrangian and Equilibrium Points

I'm wondering whether you can tell quickly just from looking at a Lagrangian whether a given point $q^0$ is an equilibrium point. Obviously all you have to do is verify it satisfies the E-L equations, ...