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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Some strange transformation [closed]

In a lecture (look at the chapter "The fermion determinant in a constant field", p. 5) I found some strange transformation, which is given by eq. 18. How to prove it? Exactly, I don't understand the ...
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78 views

How to analyze this constraint question

Let $\gamma$ be a smooth curve in the plane, and introduce curvilinear coordinates $q_1,q_2$ on a neighborhood of $\gamma$; $q_1$ is the direction of $\gamma$ and $q_2$ is distance from the curve. ...
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104 views

A discrete approach to the catenary

I'm trying to work out a model for the system above, that is, $N$ particles of unitary mass subject to the constraints: $$1=\varphi _i(\mathbf r _1,\mathbf {r}_2,...,\mathbf r _n)=|\mathbf ...
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37 views

Where are the L3, L4, and L5 of a hyperbolic orbit?

Do the L3, L4, L5 points exist in hyperbolic orbits? If yes, then where do they lie?
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77 views

How to introduce generating function in Hamiltonian formalism for field theories?

Let's have hamiltonian $$ H(\psi , P ,\partial_{i}\psi ) = P\partial_{0}\psi - L(\psi , \partial_{\mu}\psi ), \quad P = \frac{\partial L}{\partial (\partial_{0}\psi)}. \qquad (.0) $$ I tried to ...
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224 views

Lagrangian Dynamics Question [closed]

Two equal masses of mass M are glued to a massless hoop of radius R is free to rotate about its center in a vertical plane. The angle between the masses is 2$\theta$. Find the frequency of ...
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116 views

Single particle trajectory in a quadrupole potential

I am wondering if there are any studies of a single (classical) particle trajectory in quadrupole potential: $$ V(x,y,z)=A\sqrt[]{\frac{x^2 + y^2}{a} + \frac{z^2}{b}} $$
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1k views

Equations of motion for a pendulum and spring system

The question is available here: I've modeled the building as a rod on a torsional spring (with a pendulum hanging from the top). $\phi$ is the angle from the centre for the pendulum and $\theta$ ...
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138 views

Proving conservation of angular momentum in an elliptic billiard problem

This is for a course focusing on the connections between Newtonian, Lagrangian and Hamiltonian formalisms. We're given an elliptic billiard table with foci 1 and 2, where $L_1$ and $L_2$ are the ...
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102 views

Lagrangians for non-local equations of motion

Say I have a multicomponent field $X_a(x,t)$ such that I know it Fourier modes satisfy the following equation of motion, $(\delta_{ab} \partial_t + \Omega_{ab}(t))X_b(k,t) = e^t \int \frac{d^3p ...
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87 views

relevant 4-dimensional theory with interacting vector field

A simple langragian that gives the simplest interaction is $\mathcal{L}=(\partial\phi)^2+(m\phi)^2$ where $m$ is some constant. Does anyone know of theory in four dimensions which is physically ...
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301 views

Normal modes of oscillation: how to find them

Are normal modes the eigenvectors of the matrix $(\omega ^2 T- V)$ where $T$ is the matrix of kinetic energy and $V$ is the matrix of potential energy? Is it the only way to express them? How can I ...
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490 views

Partial derivative of Lagrangian density for vector field

The lagrangian density of a massless vector field is $ \mathcal{L} = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$, where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ Expanding out gives ...
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521 views

Charge, velocity-dependent potentials and Lagrangian

Given an electric charge $q$ of mass $m$ moving at a velocity ${\bf v}$ in a region containing both electric field ${\bf E}(t,x,y,z)$ and magnetic field ${\bf B}(t,x,y,z)$ (${\bf B}$ and ${\bf E}$ are ...
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1answer
269 views

How does General Relativity Emerge from Brans-Dicke Gravity with an Infinite Omega Parameter?

The action for the Brans-Dicke-Jordan theory of gravity is $$ \\S =\int d^4x\sqrt{-g} \; \left(\frac{\phi R - \omega\frac{\partial_a\phi\partial^a\phi}{\phi}}{16\pi} + ...
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484 views

Find equations of motion from given Lagrangian density [closed]

Could someone help me solve this probably not very hard problem? Given Lagrangian Density: $\mathcal ...
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359 views

Advice on classes: Theoretical Mechanics vs E&M II

So I'm having a tough time deciding between courses next semester. I'm a rising 3rd year undergrad math major whose goal is to get a solid understanding of theoretical physics through advanced math ...
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180 views

Why does Lagrangian of free particle depend on the square of the velocity ?

Why does Lagrangian of free particle depend on the square of the velocity ? For example, $L(v^4)$ also doesn't depend on direction of $v$.
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118 views

Fast question about Lagrangian

I've seen some problems solved in a weird way, I just want to be sure: the whole kinetic energy has to be in the lagrangian, right? For example, if we have a particle fixed in a plane with spherical ...
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209 views

What's an “Action” and what does the Lagrangian equation mean exactly?

How and why would a particle take the shortest path? $L=KE-PE$? What's the $KE-PE$ mean in English? I understand the 'mechanics' but not the idea itself. Please explain simply, I do know Calculus ...
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1answer
1k views

Hamilton's equations for a simple pendulum

I don't get how to use Hamilton's equations in mechanics, for example let's take the simple pendulum with $$H=\frac{p^2}{2mR^2}+mgR(1-\cos\theta)$$ Now Hamilton's equations will be: $$\dot ...
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336 views

How to think of the harmonic oscillator equation in terms of “acceleration = gradient”

This is related to another question I just asked where I learned that the equation of motion of a harmonic oscillator is expressed as: $$\ddot{x}+kx=0$$ What little physics I grasp centers on ...
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1answer
68 views

What are great circles of 2-sphere?

What exactly are great circles, and how does one derive them? Given that the Lagrangian is: $$ L =\frac {1}{2}(\dot\theta^2 + \sin^2\theta\dot\phi^2)$$ it was written that the great circles were ...
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71 views

Finding equilibrium of mechanical system

A system is described as follows: Consider a system consisting of two rotating bars of length $l$ and with uniform mass density and each with total mass $m$. The bars are attached to a common ...
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1answer
63 views

Does mass equal angular momentum?

At the wikipedia pages for angular momentum ($L$) and moment of inertia ($I$) we find the equations: $$L=I \omega$$ $$I=m r^2$$ where $m$ is mass and $r$ is the distance between said mass and ...
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107 views

Lagrangian for a moving spring device [closed]

How can I write the proper Lagrangian for such as system as the one shown in picture? Am confused about what is the suitable way to designate the coordinate.
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172 views

Atwood machine with spring

I'm just beginning to learn about Lagrangian mechanics, and I am asked to find the kinetic energy of this Atwood machine (See figure). I am told, that the kinetic energy should be: ...
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253 views

Time dependence of Lagrangian and Hamiltonian?

I am reading a online tutorial about Lagrangian mechanics. In one section, it states that if the kinetic term in Lagrangian has no explicit time dependence, the Hamiltonian does not explicitly depends ...
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1answer
227 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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1answer
178 views

Solving differential Equation for the Two-Body Problem

So, I'm following the derivation in D. Morin, Introduction to Classical Mechanics, of the equations for a two-body system. I understand all of it, aside from this one step. When he's talking about ...
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2answers
376 views

Lagrange-Euler equations for a bead moving on a ring

A bead with mass $m$ is free to glide on a ring that rotates about an axis with constant angular velocity. Form the Lagrange-Euler equations for the movement of the bead. Solution: Let us ...
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316 views

Hamiltonian and non conservative force

I have to find the Hamiltonian of a charged particle in a uniform magnetic field; the potential vector is $ \vec {A}= B/2 (-y, x, 0)$. I know that $$H=\sum_i p_i \dot q_i -L$$ where $p_i$ is ...
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125 views

Non-relativistic Kepler orbits

Consider the Newtonian gravitational potential at a distance of Sun: $$\varphi \left ( r \right )~=~-\frac{GM}{r}.$$ I write the classical Lagrangian in spherical coordinates for a planet with mass ...
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426 views

How can we represent the motion of a particle in 2D space using Lagrange's equations?

Can we represent the motion of a particle in 2D space using Lagrange's equations? This is what I tried. Please tell me what is wrong? Consider a particle on a plane have the co-ordinates $(x,y)$ with ...
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1answer
50 views

How can a generalised force be dependent on an angle i.e. not a vector?

I'm currently working through an example question in Patrick Hamill's 'A Student's Guide to Hamiltonians and Lagrangians'. The question I'm having conceptual difficulty with is: A particle is ...
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97 views

Potential energy of an infinitesimal length of elastic rod

I am having an embarrassingly hard time with the derivation for the potential energy of an infinitesimal element of an elastic rod of area A. The picture shown below is an element of the rod that has ...
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57 views

Finding the Lagrangian from the derivative of position

I have to find the Lagrangian for a system. In the point of interest I have come up with the following position coordinates: $$x = Rcos(\omega t)+\ell sin(\phi)$$ and $$y = Rsin(\omega t)-\ell ...
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1answer
135 views

Can't understand the principle of least action [closed]

I tried many hours to understand the principle of least action, and those hours become days... and I still didn't understand that principle/ and how it relates to Newtonian mechanics? Could someone ...
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1answer
294 views

Canonical momentum Velocity dependent Lagrangian

I have a homework problem wich I think I'm on the verge of solving but need help with some relations: Show that if the potential $U$ in the Lagrangian contains velocity-dependent terms, the ...
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2answers
256 views

A small oscillations of a rod on the cylinder

Let's have the next case. A rod (with mass $m$, length $L$ and a momentum of inertia $I$) at the initial time is located on a cylinder (with radius $R$) surface so that it's (rod's) center of mass ...
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1answer
74 views

Strong interaction and the Lagrangian for electromagnetic interaction

The Lagrangian for electromagnetic field has the following expression: $$ L = -\frac{1}{c^{2}}A_{\alpha}j^{\alpha} - \frac{1}{8 \pi c}(\partial_{\alpha} A_{\beta})(\partial^{\alpha}A^{\beta}) $$ (I ...
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1answer
360 views

Generalized momentum conjugate and potential $U(q, \dot q)$

On Goldstein's "Classical Mechanics" (first ed.), I have read that if $q_j$ is a cyclic coordinate, its generalized momentum conjugate $p_j$ is costant. He obtained that starting from Lagrange's ...
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40 views

Treating $\psi$ and $\psi^{*}$ as independent variables when varying the action [duplicate]

Consider the following Lagrangian (deinsity): $${\mathcal{L}} = \partial_{\mu}\psi^{*} \partial^{\mu}\psi - V(|\psi|^2) $$ In my notes it says that " it's easier (and equivalent) to treat $\psi$ and ...
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28 views

Conditions for hyperregularity

(Definitions from Marsden & Ratiu, ``Introduction to Mechanics and Symmetry''): A Lagrangian is regular if the Hessian $\partial^2 L/(\partial \dot{q}^i \partial \dot{q}^j)$ is weakly non ...
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39 views

How can we describe the trajectory of an object projected into a viscous fluid?

Can we quantitatively describe the motion of a particle inside a fluid? Let us consider the case when an object (say a sphere, so that stoke's law holds true) is projected into a fluid of viscosity ...
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35 views

Another Power Counting/ mass dimension question

Are the mass dimension of the Dirac field different from those of the Klein-Gordon field, or is this just another issue of "cannonical normalization?" For instance if $\mathcal{L}_{KG}=\int ...
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64 views

Center of mass coordinates in Lagrangians and Laplacians

Is there a quick nice and easy way to write Lagrangian's and the classical/quantum Laplacian operator in terms of center of mass coordinates? The algebra is so involved and it has me confused about ...
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44 views

Lagrangian Field Entropy?

Has anyone investigated the inclusion of a field's entropy in its Lagrangian? I read from information theory that the entropy $H$ of a scalar field $\phi$ distrusted across space is defined as ...
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2answers
140 views

Derivation of Lagrangian?

I know that the Lagrangian $L$ is defined to be $T-V$, i.e. the difference between kinetic energy and potential energy. Also the Action $S$ is defined to be $\int Ldx$ and from this we can derive ...
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98 views

Why friction force is force of constraint?

My understanding about constraint force is that it is a force which limits the geometry of particle's motion. For example, situations such as the particle trapped in a track or limited in domain can ...