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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76 views

Conceptual problem with action considered as function of endpoints

I am having some trouble with understanding why it makes sense to consider action in classical mechanics as function of endpoints $q_{initial}, \ q_{final}$ and endtimes $t_{initial}, \ t_{final}$. ...
2
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1answer
51 views

Why is the potential independent of the generalized velocity?

In Goldstein, Classical Mechanics, Chap. 1.4 we derive Lagrange's equations from D'Alembert's Principle. My question is regarding the last part of the derivation, specifically the part where he ...
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2answers
58 views

Invariance under local diffeomorphisms

In the context of the Polyakov action, the action for a relativistic point particle $$ S_P = \frac{1}{2} \int \mathrm{d}\tau \, e(\tau) \left(\frac{1}{e^2(\tau)}\left(\frac{\mathrm{d} X^\mu(\tau)}{\...
2
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1answer
127 views

Why does a system have to be holonomic?

So I'm doing some work from Taylor's mechanics book. He says for the problems in the book, we require the system to be holonomic - that is the number of generalized coordinates = number of Deg. of ...
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2answers
97 views

What can be inferred about this particle from a Lagrangian?

If Lagrangian, $\mathscr L = \dot{q}^2 - q \dot{q}$. Then what can be inferred about the particle? Simply that it is a free particle or something else?
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3answers
110 views

Hamilton's Principle - achieving Hamilton equations

Consider the action function: $$\mathcal{S}(t)=\int_{t_1}^{t_2}\mathcal{L}(q_i,\dot{q_i},t) dt$$ where $\mathcal{L}$ is the Lagrangian of the system. The Hamiltonian is defined by the following ...
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1answer
126 views

Classical spin viewed as $SU(2)$

In which sense is the configuration variable of a classical spin $SU(2)$? I can view a classical spin as a unit vector in $\mathbb{S}^2$ (2-dim. sphere), but it seems it is really given by a matrix $U$...
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1answer
105 views

Invariance of Fermionic action under Lorentz transformations

Suppose I have an Lagrangian $$\mathcal{L} = \frac{1}{2}g_{ab} \bar{\psi}^a \Gamma^k \partial_k \psi^b $$ and I want to show it's invariance under the infinitesimal Lorentz transformations $$\delta \...
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1answer
134 views

Hamiltonian field equations constraints

Let's consider the Lagrangian $$\mathcal{L}~=~-\frac{1}{2}(\partial_\mu\phi^\nu)^2+\frac{1}{2}(\partial_\mu\phi^\mu)^2+\frac{1}{2}m^2\phi_\mu \phi^\mu,$$ with Minkowski metric $\eta_{\mu\nu}={\rm ...
2
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1answer
83 views

Non-linearity and self-coupling of gravity

I have heard that non-linearity of Einstein's field equations has to do with the fact that gravity self-couples. What does non-linearity have to do with self-coupling?
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2answers
150 views

“Find the Lagrangian of the theory”

I've heard a few of my professors throw around the term "finding the Lagrangian of a theory". What exactly is this referring to. From what I understand it seems that you determine invariances (...
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1answer
159 views

What assumptions about the action do we make or give up in transitioning from classical mechanics to quantum mechanics to quantum field theory?

I am reading Quantum Field Theory for the Gifted Amateur and I feel I don't have a good grasp as to how the Lagrangian and the action are used differently in (1) classical mechanics (2) quantum ...
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1answer
30 views

Is the term “Lagrangian density” specific to spacetime?

Wikipedia talks about Lagrangian densities here. But they never actually say whether they're just applying the concept to spacetime or that Lagrangian density is the analog for Lagrangians but for ...
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2answers
129 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 convention)...
2
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1answer
404 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
269 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 \...
2
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1answer
209 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
153 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?
2
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1answer
328 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 a}+\bar{\psi}\left(i\gamma^{\mu}D_{\mu}-m\right)\psi-\frac{\xi}{2}B^{a}B^{a}+B^{a}\partial^{\...
2
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2answers
325 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 A^...
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1answer
523 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: $$L=(1/2)m(\dot{r}^{2}+r^{2}\dot{\...
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4answers
803 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
204 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
635 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
401 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 ...
2
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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 $\delta \phi = \alpha \phi$, ...
2
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1answer
428 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 Euler ...
2
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1answer
271 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
264 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 $a r^2/ M$...
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
790 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.
2
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1answer
544 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 ...
2
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1answer
32 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 ...
2
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2answers
61 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
66 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
75 views

A question regarding $f(R)$ Lagrangians

Consider the class of Lagrangian known as $f(R)$ Lagrangians where the Lagrangian is some function $f(R)$, \begin{equation} S=\int\sqrt{g}d^4x\ f(R) \end{equation} assuming there are no (or ignoring)...
2
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1answer
82 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 $\{\cdot,\cdot\}_\...
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2answers
91 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
113 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 $L(y,\dot{y})=\...
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1answer
115 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 ...
2
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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 ...
2
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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
1k 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
115 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 quantization....
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1answer
104 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} X_L=\sum^M_{j=1}\bigg(v^j\frac{...
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1answer
337 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 $(+,...
2
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1answer
189 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
193 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
163 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 ...