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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12 views

Center of Instantaneous velocity in two degree of freedom problem

I have a problem like this I have two DoF where the force F is acting on the car and another force could be acting on the coordinate $q_1$. The force acting on $q_2$, the coordinate of the car will ...
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2answers
428 views

Can one write down a Hamiltonian in the absence of a Lagrangian?

How can I define the Hamiltonian independent of the Lagrangian? For instance, let's assume that i have a set of field equations that cannot be integrated to an action. Is there any prescription to ...
3
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1answer
74 views

Fayet-Iliopoulos terms

It is mentioned in first page of this paper by Seiberg and Komargodski that the Lagrangian in superspace of a $U(1)$ gauge SUSY theory with FI terms is not gauge invariant. However, the FI terms in ...
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1answer
107 views

Differential holonomic constraints

Differential holonomic constraint is an integrable homogeneous first order differential equation: $$\sum_{i}\mathcal{E}_{i}(q)\frac{dq_{i}}{d\tau}=0;$$ in which $\sum_{i}\mathcal{E}_{i}(q)dq_{i}$ is ...
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1answer
57 views

Existence of lagrangians at strong coupling

It is well known that some QFT do not admit a lagrangian formulation (like the $(2,0)$ SCFT in $d=6$). Up to my understanding, all the examples that I know of non lagrangian theories are always ...
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41 views

Lagrangian mechanics - small oscillations around equilibrium diagonalization

In my analytical mechanics class, we have been taught that normal modes of small oscillations around equilibrium are given by the solution of $$ p(\omega) = \det(K-\omega^2M) = 0 $$ Where $K_{ij} = ...
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1answer
64 views

Equivalence between principle of least action and minimum potential energy

Are the principle of least action and the principle of minimum potential energy equivalent? How does one show that? Also, are Newton's laws of motion equivalent to the principle of least action? How ...
2
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1answer
83 views

Does this vertex equal 0?

If I have an interaction term in my Lagrangian that looks like: $\mathcal{L}_{int} = (\partial_\mu B_\nu)(A^\mu B^\nu - A^\nu B^\mu)$ where B is a massive spin-1 field. Am I correct in thinking that ...
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31 views

Higher order Lagrangians [duplicate]

Recently I have read some papers in which the authors considered higher order lagrangians. For example, in this paper "A path integral leading to higher-order Lagrangians" by C.Acatrinei the higher ...
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0answers
45 views

Principle of Most Action? [duplicate]

In Landau-Lifshitz - Vol 1. Mechanics, right after the introduction of the principle of leas action, there is the following comment: It should be mentioned that this formulation ($S = ...
2
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1answer
70 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 $t_{initial}, \ t_{final}$. ...
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1answer
38 views

Calculate lapse function from the metric

I have a technical question about the lapse function: Assume I have some given (Lorentzian) metric $g$. I have seen the following definition of the lapse function $\alpha^{-2}=-g(\nabla f, \nabla ...
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44 views

Where does the factor of half appear from in the Klein-Gordon Lagrangian?

The lagangian density of a scalar field or a Klein-Gordon field has the form of $$\begin{align} \mathcal{L} = \frac{1}{2} \partial_\mu \phi \partial^\mu \phi - \frac{1}{2} m^2 \phi^2. \end{align}$$ ...
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2answers
68 views

Variation of a Lagrange density Symmetries

So I am reading Goenner's Spezielle Relativitästheorie and I am currently in chapter §4.9.1 Variation under Inclusion of Coordinates p. 129. So basically we have: $$\delta W_\zeta=\int d^4x' ...
2
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1answer
66 views

Functional Derivative of action

Consider the action of free Klein-Gordon theory $S[\phi]=\frac{1}{2}\displaystyle\int d^4y(\partial_\mu\phi(y)\partial^\mu\phi(y)-m^2\phi^2(y))$ Integrating by parts in the first term gives me ...
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35 views

When the equations of motion are not unique (eg. when they are given by eigenvectors), which will the free particle adhere to?

For this question I think it will be easier to express the usual equation describing the motion of a "free particle,"--viz. $g_{ij}\dot{x}^i\dot{x}^j$--in matrix form as follows: ...
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Why is an angle undergoing forced rotation not a generalized coordinate?

In a textbook it is using the following example. A bead of mass m slides freely on a light wire of parabolic shape, which is forced to rotate with angular velocity ω about a vertical axis. The ...
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1answer
80 views

Are there conserved quantities in field theory which don't arise from Noether's Theorem?

In some QFT texts one writes down the number operator $N$ for free theories, such that when acting on an $n$-particle state $|n\rangle$ we have $$N|n\rangle=n|n\rangle$$ In free theories this is a ...
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1answer
42 views

Lagrangian of a block connected to a circular track [closed]

Could someone help me? I am having trouble with obtaining the same result in part b) for this problem: Using the Lagrange Equation with respect to $\theta$, I obtained ...
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2answers
234 views

Symmetry of the Polyakov action?

Let us look at the Polyakov action for a string moving in a spacetime with metric $g_{\mu \nu}(X)$:$$S_P = -{1\over{4\pi \alpha'}} \int d^2 \sigma \sqrt{-\gamma} \gamma^{ab} \partial_a X^\mu ...
3
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1answer
72 views

Application of Euler-Lagrange equations (Trivial problem, instructive one)

I have some doubt about a really trivial and simple problem in which I have to use ELE. Supposing I have a pendulum, in which the rope is a spring, so it's length may change in time. I have a mass ...
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0answers
30 views

Show that a vector field is a symmetry for a Lagrangian [closed]

Let Lagrange function be $$ L=\frac{1}{2}m(\dot{x_1}^2+\dot{x_2}^2+\dot{x_3}^2)-U((x_1^2+x_2^2,x_3)). $$ Show, that vector field $\vec{Y}(\vec{x})=(-x_2,x_1,0)$ comply $$ ...
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0answers
31 views

Majorana fermions and the continuum limit of the Ising model

In Paolo Moligini's Analyzing the two dimensional Ising model with conformal field theory lecture notes, it is shown at the end of chapter 3 that the Lagrangian of the continuum limit of the Ising ...
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50 views

How does satisfying the Euler-Lagrange equation put a Classical Path on-shell?

I am thinking of what the Euler-Lagrange equation, $$ \frac{d}{dt}\left(\frac{\partial L}{\partial \dot{x}}\right) - \frac{\partial L}{\partial x} = 0 $$ specifically represents in satisfying the ...
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25 views

Sign of elastic potential from a spring [closed]

When writing the Lagrangian of a problem, how do I determine the sign of an elastic potential from a spring?
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26 views

Equations of motion for normal modes

I really need some help understanding how to find normal modes. So I brought the euler-lagrange equation of my probelm to this form: $X'' = -AX.$ Where $X$ is the coordinates vector. So I found the ...
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0answers
46 views

Is the Symmetry factor different in Path integral Formalism?

Is the Symmetry factor different in Path integral Formalism and the Perturbation theory (canonical) formalism? For example, the order-1 4-point cross X diagram in the $\phi^4$ theory has symmetry ...
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2answers
111 views

Principle of least action: $\frac{d S_{cl}}{dt_b} = \frac{\partial S_{cl}}{\partial t_b} + \frac{\partial S_{cl}}{\partial x_b}\dot{x}_b$

Question I cannot see how I can obtain the yellow highlighted section on the RHS from that of the LHS. The following equation can be found in both my lecture notes(*1) (page 9, equation 2.7) and is ...
3
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1answer
76 views

To derive the relation between work function and potential energy

I'm reading "The variational principles of mechanics- Lanczos", The author mentions a relation between Work-Function $U(q_1,q_2,\cdots,q_n,\dot q_1,\dot q_2,\cdots,\dot q_n)$ and the potential ...
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0answers
32 views

How to obtain calculus of variation of Einstein summation?

I have the Lagrange density for Maxwell field, which is $\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$, where $F_{\mu\nu}=\partial_{\mu}A_\nu-\partial_{\nu}A_{\mu}$. How can I obtain ...
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2answers
79 views

Are the generalized coordinates in Lagrangian mechanics really independent?

In Goldstein's Classical Mechanics, Chapter 2.3: Derivation of Lagrange's Equations From Hamilton's Principle part of the derivation involves each of the generalized coordinates being independent. $$ ...
3
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1answer
144 views

Why do we require local gauge invariance

My thought on this are somewhat scattered so I apologise in advance. Maxwell's equations are gauge invariant. The physical Electric and Magnetic fields don't depend on whether we use $A_\mu$ or ...
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0answers
63 views

Liouville's theorem for systems with dissipation described by a single hamiltonian

Following this link, one can treat dissipation in the lagrangian by using a factor $e^{\frac{t \beta}{ m}}$ in addition to the Lagrangian $L_0$ of a system without disspation: $ L_0[q, \dot{q}] = ...
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1answer
86 views

Derivation Of Euler-Lagrange Equation [closed]

I want the proof of this relation in details, $$ \frac{\rm d}{{\rm d}t}\left(\frac{\partial\vec{r}_v}{\partial q_\alpha}\right)=\frac{\partial\vec{\dot{r}_v}}{\partial q_\alpha} $$
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1answer
57 views

Conserved currents from Noether's theorem

I'm not sure if I understand the concept correctly. Given an infinitesimal transformation $$\phi \rightarrow \phi + \alpha \Delta\phi$$ the change in the Lagrangian density ...
0
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1answer
54 views

Euler-Lagrange for simple scalar field (Peskin & Shroeder)

I'm reading Peskin & Schroeder and they give as a simple example the Lagrangian $$\mathcal{L} = \frac{1}{2} (\partial_\mu \phi)^2$$ First of all, I'm guessing that $(\partial_\mu \phi)^2$ is ...
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0answers
49 views

Relation between interaction Lagrangian and interaction Hamiltonian

I work with this interaction Lagrangian density $$\mathcal{L}_{int} = ia\bar{\Psi}\gamma^\mu\Psi Z_\mu +ib(\phi^\dagger\partial_\mu \phi - \partial_\mu\phi^\dagger \phi)Z^\mu,$$ where $Z^\mu$ is an ...
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0answers
26 views

Einstein-Infeld-Hoffman Equations generalized coordinates

There are three (heavy,fast) bodies in a Cartesian coordinate system with position vectors $\{\mathbf{r_1},\mathbf{r_2},\mathbf{r_3}\}$, velocities ...
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0answers
74 views

Transition Amplitude in Field Theory

I am currently reading the "Quantum Field Theory" by Lewis Ryder. In chapter 5 he is talking about path integrals and says that the transition amplitude $ \langle q_f t_f \vert q_i t_i\rangle $ is $$ ...
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0answers
41 views

Variation of Bazanski Lagrangian

The Bazanski Lagrangian is defined as $$ L=g_{\alpha \beta }U^{\alpha }\frac{D\psi ^{\beta }}{Ds} $$ and $$ U^{\alpha }=\frac{\mathrm{d} x^{\alpha }}{\mathrm{d} s} $$ $x^{\alpha }$ is the ...
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0answers
41 views

Generating functional for free and interacting theories [closed]

I'm asking probably a stupid question. We define the generating functional for free theories as $$ Z_0[J] = \int D \psi e^{i\int d^4x \left[ L_0(x) + J_l(x)\psi^l(x) \right]} $$ with $L_0$ the free ...
5
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1answer
154 views

Is there a Maupertuis principle for General Relativity?

The motion of a point particle in classical mechanics is given by Newton's equation, $\mathbf{F}=m\mathbf{a}$. Suppose all forces considered are conservative and we have a constant total energy $h$. ...
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1answer
66 views

How is it possible to vary time without affect the coordinates or their derivatives?

In the context of Noether's theorem , the Hamiltonian is the constant of motion associated with the time-translational invariance of the Lagrangian. Time-translational invariance is equivalent to the ...
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1answer
42 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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0answers
43 views

Deriving Maxwell's equation from the Lagrangian of an electromagnetic field with a charge density $\rho$

I want to derive Maxwell's equation from the Lagrangian of an electromagnetic field with a charge density of $\rho$ The Lagrangian is given by ...
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1answer
85 views

How to proceed (Tough Problem) [closed]

The problem that I am considering is to find the shortest path (or geodesic) on a surface with the equation $z=f(x,y)$. The path is parameterized by $s$ so that the path goes from ...
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1answer
54 views

How to derive the true spatial paths (orbits) from the Jacobi-Maupertuis condition

How can differential equations describing a physical object's true spatial paths (orbits) be derived from the time-independent Jacobi-Maupertuis principle of least action? According to this, it is ...
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0answers
34 views

Why do we sum relativistic intervals in relativistic action of a massive point-particle, and not a function from it?

Relativistic action as follows (which should explain relativistic motion of a classical particle): $$ S = C \Delta s=C\int ds $$ Where $C$ is some constant and $\Delta s$ is relativistic interval. ...
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1answer
95 views

Deriving the equation of motion for a rigid system

I want to derive equation of motion for the system shown in picture. How do I choose a generalized coordinate in order to calculate kinetic and potential energy of the system? I need the ...
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2answers
85 views

Principle of least action and greedy algorithm

Is the principle of least action sort of a greedy algorithm that all mechanical systems follow?, sometimes to minimise and sometimes to maximise the quantity we call action, at each individual step.