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Questions tagged [lagrangian-formalism]

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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Difference between kinematic momentum and conjugated momentum in purely mechanical setup

I don't know much about physics, but I wanted to understand what was the difference between the "kinematic momentum" and the conjugated momentum. As I understand it, kinematic momentum is mass times ...
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29 views

On the definition of Lagrange's equation

Is it really Newton's third law eq (1.5) ? Isn't the 2nd? I found it on "Abers Ernest Quantum mechanics".
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Time-independent Schrödinger equation Lagrangian derivation

Recently I was taking a calculus of variations class and our professor casually obtained the time-independent Schrödinger equation for a free particle from the integral (constants dropped) and it's ...
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341 views

Applying the Euler-Lagrange equations to Maxwell's Theory

In Prof. David Tong's notes, specifically on page 10, he gives the Lagrangian of Maxwell's theory to be $$ \mathcal{L} = -\frac{1}{2}(\partial_\mu A_\nu)(\partial^\mu A^\nu) + \frac{1}{2}(\partial_\...
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1answer
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Chern-Simons equation of motion

How do I get the equation of motion of the Chern-Simons Lagrangian below? Is there the product rule at work? Do I have to sum over the indices?
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Energy momentum tensor in polarizable media

I am trying to follow Soper's derivation of the energy-momentum tensor for polarizable media in his book 'Classical field theory'. Given a Lagrangian density $\mathcal{L} = \mathcal{L}(\phi_A, \...
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25 views

How general is the Lagrangian formulation? [duplicate]

Haven't seriously tackled this problem myself because it's been awhile since I've done any hard mathematics and I'm a bit rusty. However, you needn't spare the math in your answers. I've been ...
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3answers
124 views

Maxwell Tensor Identity [duplicate]

In Schawrtz, Page 116, formula 8.23, he seems to suggest that the square of the Maxwell tensor can be expanded out as follows: $$-\frac{1}{4}F_{\mu \nu}^{2}=\frac{1}{2}A_{\mu}\square A_{\mu}-\frac{1}{...
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Euler-Lagrange equations from a complex Lagrangian

I'm looking for generalizations of the Euler-Lagrange equations that would be derived from a complex-valued Lagrangian density. I realize that “minimum” and “maximum” don't have obvious meaning for a ...
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Is there a scalar field that gives rise to the energy-momentum tensor of a perfect fluid?

If I understand correctly, Sean Carroll's Spacetime and Gravity says that the energy-momentum tensor for a perfect fluid $$T^{\mu\nu} = (\rho + p)U^\mu U^\nu + pg^{\mu\nu}$$ can be obtained as the ...
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Can we also Legendre transform the Lagrangian using $\frac{\partial L}{\partial q}$ instead of $\frac{\partial L}{\partial \dot q}$? [duplicate]

We calculate the Hamiltonian as the Legendre transform of the Lagrangian $$H(q,p,t) = p \dot q - L(q,\dot q,t), $$ where $p$ is the slope function $$p \equiv \frac{\partial L}{\partial \dot q} .$$ ...
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Is there a higher dimension analogue of Noether's theorem?

So I have recently read the proof of Noether's theorem from the book variation calculus by Gelfand. Basically, what I have already seen is that for any single integral functional, if we have a ...
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3answers
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How do you actually find the path of least action?

Reading up on Lagrangian mechanics, it's fascinating. Entirely different view, one single rule, a complete alternative to Newton's laws. But how do you actually find the path of least action? Let's ...
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Are there non-holonomic constraints that cannot be removed, other than conservation of energy?

I'm learning about Lagrangian mechanics at the moment. The idea that $ L(\mathbf{q}(t),\mathbf{\dot{q}}(t), t) = 0$ fascinates me. It strikes me as this could be viewed as a 1st order non-holonomic ...
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Why is Dirac's remark that inspired Feynman's path integrals approach to quantum mechanics regarded as mysterious? [on hold]

In his book Dirac remarked a relation between the transformation function connecting two different representations and the classical action, stated as: $\langle q_t | q_T \rangle$ corresponds to $exp [...
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1answer
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Is every classical field theory with dimensionless couplings conformally invariant?

I'm trying to learn conformal field theory and getting rather frustrated, because I can't find any source that gives decent examples or straightforward logic. In most sources I have found, conformal ...
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Euler applies the principle of least action to projectile motion

I was reading my physics book and became interested in the origin of the Principle of least action. After some historical research I arrived at Euler's historical text A method for finding curved ...
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1answer
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Spring-Mass system motion and eigenvalues [closed]

I'm trying to solve Lagrange differential equations of motion; where $L=T-U$. Also how can I use the output solution for each $x1, x2, x3,,,, xn$? Assume N=100. Here is my Code ...
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Constructing gauge symmetry for $SU(2)\times U(1)$

I'm currently reading the book "Classical Theory of Gauge Fields" by Rubakov, therefore I will use his convention in this question. In the following we assume that: $\phi$ comprises columns ...
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Intuitive meaning of Maxwell action in terms of geometry (differential form formulation)

For the electromagnetic field strength, $F^{(2)}$, which is an exact $2$-form, i.e. $F^{(2)}=dA^{(1)}$ for a $1$-form $A^{(1)}$, we can define its Hodge dual, $*F^{(2)}$ and then define the action $$\...
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Unitary gauge for spontaneous symmetry breaking

I'm given a lagrangian $$ \mathcal{L} = \partial_{\mu} \Phi^{\dagger} \partial^{\mu} \Phi + m^2 \Phi^{\dagger} \Phi - \lambda (\Phi^{\dagger} \Phi)^2 $$ where $m^2 > 0, \lambda > 0$. This ...
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Is Action Always “Locally” Least?

In general, I know it's true that the Principle of Least Action is more properly called the Principle of "Stationary" Action. However, there are results which seem to suggest that for sufficiently ...
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2answers
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Lagrangian two body gravitational conserved quantities

I have the Lagrangian for two gravitationally attracting bodies: $$ L ={\frac{1}{2}}M\dot{R}^2 +\frac{1}{2}{\mu}\dot{r}^2 + \frac{Gm_1m_2}{|r|}$$ Where M is the total mass, mu the reduced mass and r ...
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Why are the Klein-Gordon equations warranted from the conservation of the energy-momentum tensor?

If we have an action with a scalar field non-minimally coupled to the gravity: $$\int dx^4 \sqrt{-g}(-\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{1}{2}\zeta R\phi^2-V(\phi)).....(1)$$ varying ...
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1answer
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Proving that the Euler-Lagrange Equation has no solution [closed]

I'm trying to show that the Euler - Lagrange equation for the functional $$I(y)=\int_{a}^{b} y\:dx$$ subject to $y(0)=y(1)=0$ has no solutions. The Euler - Lagrange equation states that: $$\frac{d}{...
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1answer
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Boundary terms and Symmetries

Consider Maxwell-Chern-Simons theory in 2+1 dimension, with Lagrangian $$L = -(1/4)F_{\mu v}F^{\mu v} + (m^2/4) \epsilon_{\mu v \rho}A^\mu F^{v \rho},$$ when I make a gauge transformation $A_\mu \to ...
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What is Lagrangian mechanics? And how is it different from Newtonian mechanics? [duplicate]

I am only 15 year old,but I am terribly interested in physics. Once I was watching some kind of interesting lecture,and I came across Lagrangian-mechanics. What is it? How is it different from ...
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2answers
56 views

Change of coordinates of Lagrangian

Consider the system above ($m_1$, $m_2$, and $m_3$ are connected by springs of stiffnesses $k_1$ and $k_2$, respectively. Also, $m_1 \neq m_2 \neq m_3$). The Lagrangian is $$L(x_{1},x_{2},x_{3},\dot ...
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1answer
34 views

Solving Lagrangian given initial and final coordinate

Consider a Lagrangian $$L=L\left(q, \dot{q}\right)$$ I can use the Euler-Lagrange equation to find an expression $$\ddot{q}=A\left(q,\dot{q}\right).$$ Let's assume that the equation can be ...
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How does the Hamiltonian change if $L\to L + \frac{dF}{dt}$? [duplicate]

The Hamiltonian is defined as the Legendre transform of the Lagrangian $$H = p\dot{q} -L .$$ In the Lagrangian formalism we are free to add the total derivative of an arbitrary function $F=F(q,t)$ to ...
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Intuition behind the use of the Principle of Stationary Action in Classical Field Theory [duplicate]

Whilst studying Field Theory and after checking numerous sources it appears that people always just state the action without providing some sort of motivation/intuition as to why we should/can use the ...
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1answer
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Showing that the $\mathcal{N=2}$ SUSY Effective Action is Duality-Invariant

The effective action of the $\mathcal{N}=2$ supersymmetric $SU(2)$ gauge theory contains the following term; $$Im\int d^{4}xd^{2}\theta d^{2}\bar{\theta}\Phi^{\dagger}\mathcal{F}'(\Phi)$$ Where $\...
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1answer
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What is the “special time dependence” that develops in an Ostrogradskian instability?

I've been reading papers that deal with Lagrangians containing second- and higher- order derivatives of field variables. In this paper in Section 3.1, I found this very interesting quote: The ...
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Equations of motion from action variation

I was reading about dilaton gravity in 2D, and I was trying to reproduce the equations of motion of a related theory. If I consider the following action: $$S = \int d^4x \sqrt{-g} e^{-2\phi}(R+4(\...
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1answer
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Lagrangian of Rocket [closed]

While solving equation of rocket motion with Newton's law in 1-d,I pondered to apply Lagrangian method on this. However, I didn't get correct result. Because I can eliminate last 2nd equation using ...
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How to determine whether a set of coordinates are independent and sufficient to determine the system completely?

In Analytical mechanics, when we formulate our principles, in general, it is assumed that we start with a cartesian coordinate system, and then find some generalised coordinates $q_j$s they are all ...
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1answer
72 views

Lagrangian corresponding to these equations of motion [closed]

I have the following equations of motion for a system with two degrees of freedom: $$\ddot{q_1}+q_1^2-q_2^2=0$$ and $$\ddot{q_2}+2q_1q_2=0.$$ I have tried to deduce the Lagragian corresponding ...
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Lattice vibrations via Lagrangian mechanics

When solving the Lagrangian for the one dimensional atomic chain. I get stuck at proving an identity. So my Lagrangian is $ L=\sum_{n=1}^{N}\left[\frac{m}{2} \dot{\phi}_{n}^{2}-\frac{k_{s}}{2}\...
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2answers
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How does Hamilton's Principle give us the path taken?

We defined the action as: $$\mathcal{S}(t)=\int_{t_1}^{t_2}\mathcal{L}(q_i,\dot{q_i},t) dt$$ where $q_i(t_1)$ and $q_i(t_2)$ are known and fixed. Hamilton's principle states that the path that is ...
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1answer
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Does $L=T-V$ still hold when $L$ is NOT time-dependent?

I am aware that the Lagrangian $L=T-V$ where $T$ is the kinetic energy and $V$ is the potential energy when $L$ depends on, for example, $r, \dot{r}, t$. My question is, does this still hold when the ...
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How does the spin connection affect the dynamics of a fermion in curved space?

Consider a massless right-handed Majorana fermion in curved spacetime. Without any other fields present, the Lagrangian density is (I believe) the following: $$ \mathcal{L}_{\psi} = \sqrt{g}i\bar{\...
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1answer
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How can Lagrangian method work whenever the Lagrangian is not convex?

Let $$L(x,\dot x)=\frac{1}{2}m\dot x^2-\frac{1}{2}k(x-x_0)^2-mgx$$ the Lagrangian of a system. Euler Lagrange theorem says that a necessary condition to be a minimizer is to satisfy Euler-Lagrange ...
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System with non-skew symmetric gyroscopic matrix

Lets take the following equations of motions of a two DOF system $$\left[\begin{array}{cc} m & 0\\ 0 & L \end{array}\right]\left[\begin{array}{c} \ddot{u}\\ \ddot{q} \end{array}\right]=-\left[...
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1answer
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Mass attached to another mass hanging from a hole in a table

I've been trying to understand the setup for the Lagrangian for this question from Morin: My issue is why would the mass ever rotate, assuming we release it from rest? Why does the setup of the ...
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1answer
53 views

Effective Lagrangians

I get the impression from reading, e.g., this paper, that the term "effective Lagrangian" refers to a Lagrangian derived from a Taylor series expansion of an arbitrary function of known invariants. ...
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1answer
63 views

Lagrangian of Klein Gordon equation

Consider the following Lagrangian density $$ \mathcal{L}(\Phi,\partial_\mu\Phi)=-\frac{1}{2}\partial_\mu\Phi\partial^\mu\Phi-\frac{m\Phi^2}{2}. $$ I want to calculate the equation of motion using the ...
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1answer
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Lagrange equations in a conservative system, understanding $\nabla_i$

For a system of multiple particles with conservative forces: $\mathbf{F}_i = - \nabla_i V$, with $V \equiv V(\mathbf{r}_1,\dots,\mathbf{r}_N)$ the potential in function of the position of the $N$ ...
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2answers
128 views

History of Principle of Least Action [closed]

I am interested in how Lagrange came up with Principle of Least Action. Is it derived from some experimental data or by mathematical deduction? And how? Either seems hard. I hope this might ...
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Fourier transform of variable in path integral

In Sredinicki's QFT given below, he changed the integration variables in eq(174). This step confuses me. I only know some basics about path integral. In my opinion, when he used fourier transform of ...
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Are all canonical transformations either a point transformation, gauge transformation or a combination of them?

It's regularly argued that in the Hamiltonian formalism, we have more freedom to choose our coordinates and that this is arguably its most important advantage. To quote from two popular textbooks: ...