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.

learn more… | top users | synonyms (1)

3
votes
1answer
373 views

Point of Lagrange multipliers

I am trying to understand how for a constrained system the introduction of Lagrange multipliers facilitates the incorporation of the holonomic constraints. I am using Classical Mechanics by John ...
3
votes
1answer
208 views

Are there two types of D-term and two types of F-term in SUSY?

I've noticed that one can obtain D-terms either by integrating a vector superfield (the vector multiplet) over superspace or by integrating a Kahler potential over superspace. In both cases we get ...
3
votes
1answer
140 views

Can the Solar System be assumed a single body concentrated in the Sun?

This question springs from a comment against my question posted on the Space SE My questions may seem inane, or obvious to most of you real physics people too ... Any number of sources put the peg ...
3
votes
1answer
331 views

Find the action from given equations of motion

Is there a systematic procedure to generally obtain an appropriate action that corresponds to any given equations of motion (if I know that it exists)?
3
votes
2answers
2k views

Deriving Lagrangian density for electromagnetic field

In considering the (special) relativistic EM field, I understand that assuming a Lagrangian density of the form $$\mathcal{L} =-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + \frac{1}{c}j_\mu A^\mu$$ and ...
3
votes
2answers
89 views

A charged particle moves in a plane subject to the oscillatory potential

A charged particle moves in a plane subject to the oscillatory potential: $U(r)=\frac{m\omega^2 r^2}{2}$ There is also a constant EM-field described by: $\vec{A}=\frac{1}{2}[\vec{B}\times\vec{r}]$ ...
3
votes
1answer
376 views

Constructing the “most general” two-particle spin interaction with $SU(2)$ symmetry

Suppose I want to write down an interaction term for an action for spin 1/2 fermions that is $SU(2)$-symmetric. I start from the most naive general form of such an action: $$S_{int} ~=~ \int_{4321} ...
3
votes
4answers
740 views

Is there any physics that cannot be expressed in terms of Lagrange equations?

A lot of physics, such as classical mechanics, General Relativity, Quantum Mechanics etc can be expressed in terms of Lagrangian Mechanics and Hamiltonian Principles. But sometimes I just can't help ...
3
votes
3answers
112 views

Where does the $(\ell + x)^2\dot\theta^2$ term come from in the Lagrangian of a spring pendulum?

I am reading some notes about Lagrangian mechanics. I don't understand equation 6.9, which gives the Lagrangian for a spring pendulum (a massive particle on one end a spring). $$T = ...
3
votes
1answer
47 views

Exact meaning of locality and its implications on the formulation of a QFT

As far as I understand it, locality in physics is the statement that interactions can only occur between physical objects if the spacetime interval separating them is null or time-like. Thus, if the ...
3
votes
1answer
85 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
votes
1answer
124 views

Why two different Lagrangians to derive geodesic equations?

I'm trying (very early stages) to understand the derivation of the geodesic equation ...
3
votes
1answer
52 views

Lagrangian for second-order system

Given an $n$-dimensional second-order system $$\ddot q^i-\sum_{j=1}^n A^i_j\dot q^j=0,$$ where $A$ is a constant matrix, is it possible to find a Lagrangian such that the above equation is the ...
3
votes
2answers
195 views

Ball Bearing Inside a Hollow, Spinning Rod: where is the logical flaw?

As described in the title, suppose we have a frictionless, hollow rod that is rotating in the $xy$-plane with some fixed angular velocity $\omega$. The rod is pivoting around its midpoint. Suppose we ...
3
votes
2answers
718 views

Geodesic equation from Euler - Lagrange

There are several ways to derive the geodesic equation. One of which is the variational method which I seemed to understand it because it was written in great details. Then it was mentioned that the ...
3
votes
2answers
120 views

What is the “associated scalar equation” of equations of motion?

In an essay I am reading on celestial mechanics the equations of motion for a 2 body problem is given as: $$\mathbf{r}''=\nabla(\frac{\mu}{r})=-\frac{\mu \mathbf{r}}{r^3}$$ Fine. Then it says the ...
3
votes
2answers
149 views

Internal potential energy and relative distance of the particle

Today, I read a line in Goldstein Classical mechanics and got confused about one line. To satisfy the strong law of action and reaction, $V_{ij}$ can be a function only of the distance between ...
3
votes
1answer
228 views

Clarifying constraint forces in Lagrangian dynamics

In the Lagrangian formulation, the addition of constraint forces that are unknown can be done with Lagrange multipliers, which allows for the forces to be found. Taking $k$ constraints of the form ...
3
votes
1answer
93 views

How to check $\renewcommand{\vec}[1]{\mathbf{#1}} \vec{v'}\cdot\vec{V}$ and $\vec{v}'^2$ are time derivatives of some other functions?

From Landau, Lifshitz Mechanics p.127 $\renewcommand{\vec}[1]{\mathbf{#1}}L'=\frac{1}{2}m(\vec{v}'^2+\vec{v'}\cdot\vec{V}+\vec{V}^2)-U $ He states that "$\vec{V}^2(t)$ can be written as the total ...
3
votes
1answer
206 views

Why can we assume independent variables when using Lagrange multipliers in nonholonomic systems?

I'm studying from Goldstein's Classical Mechanics. In section 2.4, he discusses nonholonomic systems. We assume that the constraints can be put in the form $f_\alpha(q, \dot{q}, t) =0$, $\alpha = 1 ...
3
votes
1answer
164 views

Hamilton-Jacobi formalism and on-shell actions

My question is essentially how to extract the canonical momentum out of an on-shell action. The Hamilton-Jacobi formalism tells us that Hamilton's principal function is the on-shell action, which ...
3
votes
1answer
435 views

Non-relativistic limit of complex scalar field

In page 42 of David Tong's lectures on Quantum Field Theory, he says that one can also derive the Schrödinger Lagrangian by taking the non-relativistic limit of the (complex?) scalar field Lagrangian. ...
3
votes
1answer
1k views

Equations of motion for a spherical pendulum in a non-inertial reference frame

Take a spherical pendulum with bob mass $m$, rod length $\ell$ and physical coordinates $\theta$, $\phi$ (spherical angles) and $h$ (the hinge height with respect to the coordinate origin). The rod is ...
3
votes
1answer
437 views

Euler's equations of rigid body motion from least action principle

I would like to derive Euler's equations of rigid body motion from least action principle. Suppose we are in free space so we have no gravity so Lagrangian is equal to kinetic energy. $$ L = T = ...
3
votes
1answer
361 views

How to introduce the electromagnetic field in Quantum Field Theory?

There are many ways to introduce the electromagnrtic field in Quantum Field Theory(QFT), such as canonical quantization method which introduces the creation and annihilation oprators by treating the ...
3
votes
2answers
2k views

Null geodesic given metric

I (desperately) need help with the following: What is the null geodesic for the space time $$ds^2=-x^2 dt^2 +dx^2?$$ I don't know how to transform a metric into a geodesic...! There is no need to ...
3
votes
2answers
1k views

Where does the mass term come from in the Proca Lagrangian?

There are many good books describing how to construct the Lagrangian for an electromagnetic field in a medium. $$ \mathcal{L}~=~-\frac{1}{16\pi}F^{\mu\nu}F_{\mu\nu}-\frac{1}{c}j^{\nu}A_{\nu} $$ ...
3
votes
1answer
309 views

Can I find a potential function in the usual way if the central field contains $t$ in its magnitude?

I'm working on a classical mechanics problem in which the problem states that a particle of mass $m$ moves in a central field of attractive force of magnitude: $$F(r, t) = \frac{k}{r^2}e^{-at}$$ ...
3
votes
1answer
2k views

How do you know if a coordinate is cyclic if its generalized velocity is not present in the Lagrangian?

Goldstein's Classical Mechanics says that a cyclic coordinate is one that doesn't appear in the Lagrangian of the system, even though its generalized velocity may appear in it (emphasis mine). For ...
3
votes
2answers
4k views

Lagrangian of two particles connected with a spring, free to rotate

Two particles of different masses $m_1$ and $m_2$ are connected by a massless spring of spring constant $k$ and equilibrium length $d$. The system rests on a frictionless table and may both oscillate ...
3
votes
2answers
340 views

Elimination of velocities from momenta equations for singular Lagrangian

this doubt is related to Generalized Hamiltonian Dynamics paper by Dirac. Consider the set of $n$ equations : $p_i$ = $∂L/∂v_i$, (where $v_i$ is $q_i$(dot) = $dq_i/dt$, or time derivative of ...
3
votes
1answer
309 views

Showing constraint is nonholonomic

One example of a nonholonomic constraint is a disk rolling around in the cartesian plane that is constrained to not be slipping. These leads to the constraint $dx - a \sin\theta d\phi = 0$ and $dy - ...
3
votes
1answer
299 views

Differentiation of the action functional

In the QFT book by Itzykson and Zuber, the variation of the action functional $I=\int_{t_1}^{t_2}dtL$ is written as: $$\delta I=\int_{t_1}^{t_2}dt\frac{\delta I}{\delta q(t)}\delta q(t)$$ How is ...
3
votes
1answer
323 views

Distinguishing mechanical systems from general dynamical systems

[Remark: I admit that my first attempt on What makes a space a real space? was rather ill-posed and led to some confusion. Sorry for that, but please give me a second try. Part of the confusion arose ...
3
votes
2answers
53 views

Determination of the ground state of a field theory

Consider the Spontaneous symmetry breaking in the theory $$\mathcal{L}=\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{\mu^2}{2}\phi^2+\frac{\lambda}{4!}\phi^4.$$ By the ground state of a ...
3
votes
2answers
96 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 ...
3
votes
1answer
233 views

How to formulate variational principles (Lagrangian/Hamiltonian) for nonlinear, dissipative or initial value problems?

Although this questions is very much math related, I posted it in Physics since it is related to variational (Lagrangian/Hamiltonian) principles for dynamical systems. If I should migrate this ...
3
votes
1answer
60 views

Introducing Randomness into Lagrangian Mechanics

Let's say at $t_o$ we have a ball rolling along a (rigid) tight rope. Is there anyway that we can solve for the trajectory of the ball knowing that at some $ t' $ there will be a random constraint ...
3
votes
1answer
150 views

Using Lagrangian mechanics instead of Newtonian mechanics

When studying advanced classical mechanics, we all study about Lagrangians and the Euler-Lagrange equations and their importance. Of course, the Lagrangian is calculated based on the potential and ...
3
votes
1answer
95 views

Deriving the Canonical Energy Momentum Tensor

In the Mathematics for Physics of Stone and Goldbart the canonical energy momentum tensor is derived by the action principle as follows. To the action of the form $$ S=\int ...
3
votes
2answers
121 views

Hamiltonian for a Lagrangian with coupling

I am dealing with the following Lagrangian density $$\mathscr{L}_{em}= -\frac{1}{2}\rho\omega^2 u^2 +\frac{1}{2}\nabla u:\Sigma :\nabla ...
3
votes
1answer
205 views

Invariance of canonical Hamiltonian equation when adding the total time derivative of a function of $q_i$ and $t$ to the Lagrangian

The following is exercise 8.2 in 3rd edition (and exercise 8.19 in 2nd edition) of Goldstein's Classical Mechanics. Adding the total time derivative of a function of $q_i$ and t to the Lagrangian ...
3
votes
1answer
180 views

Solving electromagnetic vector field using the Lagrangian

Given an action of the form \begin{equation}S=-\frac{1}{4}\int d^4x\eta^{\mu\nu}\eta^{\lambda\rho}F_{\mu\lambda}F_{\nu\rho}\end{equation} where ...
3
votes
1answer
126 views

Nonlinear Klein Gordon equation

For the Klein Gordon nonlinear equation, $$ u_{tt}- \Delta u +f(u)=0,$$ how could I use Noether's theorem to prove that there is a conserved quantity? I.e., $$ (\Pi _{k} )_{t} - \rm div(j_{k})=0 $$ ...
3
votes
1answer
436 views

D'Alembert's principle

Actually I have some troubles to understand what this principle is all about, so I want to use the simple pendulum in order to get the idea. Since I have read a few passages that dealt with this ...
3
votes
2answers
337 views

Higher order covariant Lagrangian

I'm in search of examples of Lagrangian, which are at least second order in the derivatives and are covariant, preferable for field theories. Up to now I could only find first-order (such at ...
3
votes
2answers
167 views

How is the physical Lagrangian related to the constrained minimization Lagrangian?

If we're minimizing an energy $V(q)$ subject to constraints $C(q) = 0$, the Lagrangian is $$L = V(q) + \lambda C(q).$$ I have fairly solid intuition for this Lagrangian, namely that the energy ...
3
votes
1answer
416 views

Question about units in Lagrangian dynamics (inertia matrix)

I have a 3 degree of freedom system and my equation of motion is like this: $$M(q)q_{dd} + C(q,q_d)q_d+G(q)~=~0$$ $M(q)$: inertia matrix $C(q,q_d)$: Coriolis-centrifugal matrix $G(q)$: potential ...
3
votes
2answers
258 views

Using the area element in derivation of geodesic

In the derivation of the geodesic, one starts with the integral of the line element (arclength): $$L(C)=\int_{\tau_1}^{\tau_2}d\tau\sqrt{g_{\mu \nu}\dot{x}^{\mu} \dot{x}^{\nu}}$$ The integrand is ...
3
votes
2answers
90 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 ...