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.
2
votes
2answers
187 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 ...
0
votes
2answers
151 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 ...
0
votes
1answer
57 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 ...
-1
votes
1answer
81 views
Linear/ non linear Scalar field theory
How do I understand that the action for the free relativistic scalar field theory is non linear? What will be the associated interaction potential of that equation?
2
votes
1answer
118 views
Euler-Lagrange for constrained system
Suppose we have Euler-Lagrange system with generalized coordinate $r_1$ and $r_2$, and input $u_1$ and $u_2$. I know how to prove this system is indeed Euler-Lagrange system.
Suppose now if we have a ...
-1
votes
1answer
149 views
Scalar field lagrangian and potential
This question is a continuation of this Phys.SE post.
Scalar field theory does not have gauge symmetry, and in particular, $\phi\to\phi−1$ is not a gauge transformation. but why?
and
I want see the ...
2
votes
2answers
142 views
Does a constant factor matter in the definition of the Noether current?
This is a very basic Lagrangian Field Theory question, it is about a definition convention. It takes much more time to typeset it than answering, but here it is:
Consider a field Lagrangian with only ...
1
vote
2answers
114 views
In Noether's theorem, what is a “classical solution of the equations of motion”?
I'm reading a book which states that:
for each generator of a global symmetry transformation, there is a
current $j^{\mu}_{a}$ which, when evaluated on a classical solution
of the equations of ...
2
votes
1answer
90 views
Calculating the (on-shell) action of a free particle
I am having difficulty with the first problem from Feynman and Hibbs' book.
For a free particle $L = (m/2)\dot{x}^2$. Show that the (on-shell) action $S_{cl}$ corresponding to the classical ...
1
vote
1answer
213 views
What is the Lagrangian from which the Klein-Gordon equation is derived in QFT?
Is there a well-known Lagrangian that, writing the corresponding eq of motion, gives the Klein-Gordon Equation in QFT? If so, what is it?
What is the canonical conjugate momentum? I derive the same ...
3
votes
2answers
91 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 ...
0
votes
1answer
99 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 ...
1
vote
1answer
160 views
How would you use the Euler-Lagrange equation to predict the motion of projectiles with linear (Stokes) drag (but no wind)?
My first instinct would be to use the force
$$\vec{F} =- \alpha \vec{v}$$
and therefore
$$V(\vec{r}) = \alpha \int_C \vec{v}\cdot d\vec{s} = \alpha \int_C \vec{v}\cdot \vec{v} dt = \alpha \int_C ...
1
vote
2answers
197 views
Why lagrangian is negative number?
In the special relativistic action for a massive point particle,
$$\int_{t_i}^{t_f}\mathcal {L}dt,$$
why is the Lagrangian
$$\mathcal {L}=-E_o\gamma^{-1}$$
a negative number?
2
votes
2answers
172 views
Where does the mass term come from in the Proca Lagrangian?
There are many good books describing how to construct the Lagrangian for an electromagnetic field in a medium.
$$
\mathcal{L}~=~-\frac{1}{16\pi}F^{\mu\nu}F_{\mu\nu}-\frac{1}{c}j^{\nu}A_{\nu}
$$
...
1
vote
0answers
53 views
relevant 4-dimensional theory with interacting vector field
A simple langragian that gives the simplest interaction is $\mathcal{L}=(\partial\phi)^2+(m\phi)^2$ where $m$ is some constant. Does anyone know of theory in four dimensions which is physically ...
0
votes
1answer
329 views
Hamilton's equations for a simple pendulum
I don't get how to use Hamilton's equations in mechanics, for example let's take the simple pendulum with
$$H=\frac{p^2}{2mR^2}+mgR(1-\cos\theta)$$
Now Hamilton's equations will be:
$$\dot ...
20
votes
4answers
774 views
Is there a Lagrangian formulation of statistical mechanics?
In statistical mechanics, we usually think in terms of the Hamiltonian formalism. At a particular time $t$, the system is in a particular state, where "state" means the generalised coordinates and ...
7
votes
1answer
319 views
Do an action and its Euler-Lagrange equations have the same symmetries?
Assume a certain action $S$ with certain symmetries, from which according to the Lagrangian formalism, the equations of motion (EOM) of the system are the corresponding Euler-Lagrange equations.
Can ...
0
votes
2answers
150 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 ...
2
votes
1answer
215 views
Electrodynamics and the Lagrangian density
Could anyone tell me what equations can I obtain from the Lagrangian density
$${\cal L}(\phi,\,\,\phi_{,i},\,\,A_i, \dot A_i,\,\,A_{i,j})~=~\frac{1}{2}|\dot A+\nabla\phi|^2-\frac{1}{2}|\nabla \times ...
10
votes
6answers
479 views
Lagrangian NOT of the form $ T-U$
My Physics teacher was reluctant to define Lagrangian as Kinetic Energy minus Potential Energy because he said that there were cases where a system's Lagrangian did not take this form. Are you are ...
1
vote
2answers
165 views
Good book for Analytical Mechanics
What is a good book for Analytical Mechanics?
To be more specific, I would prefer a book that:
Is written "for mathematicians", i.e. with high mathematics precision (for example, with less emphasis ...
0
votes
0answers
236 views
How to find angular velocity of a point inner a circumference
Let's consider a cicumference that have the center in the origin of axes and rotates around x-axes. Let's stick a bar in a point $A$ of this circumference and at the end of the bar let's stick a mass ...
2
votes
3answers
160 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
votes
1answer
58 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$, ...
3
votes
3answers
214 views
Writing $\dot{q}$ in terms of $p$ in the Hamiltonian formulation
In the Hamiltonian formulation, we make a Legendre transformation of the Lagrangian and it should be written in terms of the coordinates $q$ and momentum $p$. Can we always write $dq/dt$ in terms of ...
2
votes
5answers
356 views
How are constraint forces represented in Lagrangian mechanics?
Suppose we try to obtain the movement equation for a particle sliding on a sphere (no friction, ideal bodies...). The only forces acting on the particle are its weight and - here's my problem - a ...
4
votes
2answers
333 views
Lorentz invariance of the integration measure
This is regards to the lorentz invariance of a classical scalar field theory. We assume that the action which is $S= \int d^4 x \mathcal{L}$, is invariant under a Lorentz transformation. How do you ...
2
votes
0answers
160 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 ...
2
votes
1answer
178 views
Scalar Field Theory Decay/Scattering
I have a few questions related to the following interaction Lagrangian (no use of crossing symmetry in the following) involving the uncharged scalar $\chi$ and the charged scalar $\phi$:
...
7
votes
1answer
240 views
Lagrangian of 2D square lattice of point masses connected by springs
Zee's QFT book mentions the Lagrangian of a square 2D horizontal lattice of point masses, connected by springs, and considering only vertical displacements $q_{i}$, as
$ L = \frac{1}{2} ...
9
votes
1answer
179 views
Lagrangian for Goldstone mode + topological excitation
The XY-model Hamiltonian is the following,
$${\cal H}~=~-J\sum_{\langle i,j\rangle} \cos (\theta_i -\theta_j).$$
The Goldstone mode corresponds to term $(\nabla \theta)^2$ in the effective ...
2
votes
4answers
315 views
Why the Hamiltonian and the Lagrangian are used interchangeably in QFT perturbation calculations
Whenever one needs to calculate correlation functions in QFT using perturbations one encounters the following expression:
$\langle 0| some\ operators \times \exp(iS_{(t)}) |0\rangle$
where, ...
8
votes
4answers
233 views
What makes an equation an 'equation of motion'?
Every now and then, I find myself reading papers/text talking about how this equation is a constraint but that equation is an equation of motion which satisfies this constraint.
For example, in the ...
2
votes
1answer
207 views
Euler-Lagrange Equation
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 Lagrange ...
1
vote
1answer
151 views
A particular case when Lagrange equation is equivalent to equation of motion on a Riemannian manifold
Suppose a particle is moving on a surface of a sphere,then it contains a holonomic constraint and so the three Cartesian co-ordinates are available with a constraint equation(equation of surface in ...
2
votes
1answer
126 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 ...
2
votes
1answer
194 views
Application of Noether's theorem
Consider one parameter transformation: $y = y ( \tilde{y}, \alpha)$ such that lagrangian satisfies: $\tilde{L}(\tilde{y}, \alpha) = L(y ( \tilde{y}, \alpha))$. We say that equation is invariant ...
1
vote
1answer
157 views
What's the motivation behind the action principle? [closed]
What's the motivation behind the action principle?
Why does the action principle lead to Newtonian law?
If Newton's law of motion is more fundamental so why doesn't one derive Lagrangians and ...
1
vote
2answers
162 views
What is the relativistic action of a massive particle?
all Lorentz observers watching a particle move will compute the same value for the quantity
$$ds^2 = -(c \, dt)^2 + dx^2 + dy^2 + dz^2,$$
$$ds^2 = g_{\mu\nu}dx^{\mu}dx^{\nu},$$
and ''ds/c'' is then ...
1
vote
3answers
138 views
Constructing Supersymmetric Lagrangians
It is a very trivial doubt but somehow I am not able to figure it out. While constructing a supersymmetric lagrangian we always even number of fermionic fields.
One reason is of course the product ...
2
votes
3answers
603 views
What exactly are Hamiltonian Mechanics (and Lagrangian mechanics)
What exactly are Hamiltonian Mechanics (and Lagrangian mechanics)?
I want to self-study QM, and I've heard from most people that Hamiltonian mechanics is a prereq. So I wikipedia'd it and the entry ...
2
votes
1answer
212 views
Two masses with interacting forces and an external force
Two masses in 3d space attract each other with a potential relative to the distance between them. There is also an external force on each particle based on the distance from a origin. I want to find ...
0
votes
1answer
143 views
Generalized momentum conjugate and potential $U(q, \dot q)$
On Goldstein's "Classical Mechanics" (first ed.), I have read that
if $q_j$ is a cyclic coordinate, its generalized momentum conjugate $p_j$ is costant.
He obtained that starting from Lagrange's ...
2
votes
1answer
195 views
How to tell local and unlocal in QFT?
I'm taking QFT course in this term. I'm quite curious that in QFT by which part of the mathematical expression can we tell a quantity or a theory is local or unlocal.
0
votes
0answers
181 views
Classical Mechanics: A particle move in one dimension under the influence of two springs [closed]
A particle of mass $m$ can move in one dimension under the influence of two springs connected to fixed points a distance $a$ apart (see figure). The springs obey Hooke’s law and have zero unstretched ...
0
votes
1answer
108 views
Non-relativistic Kepler orbits
Consider the Newtonian gravitational potential at a distance of Sun:
$$\varphi \left ( r \right )~=~-\frac{GM}{r}.$$
I write the classical Lagrangian in spherical coordinates for a planet with mass ...
2
votes
1answer
221 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)?
1
vote
1answer
131 views
Symmetries of spacetime and objects over it
I guess according to mathematical didactic, we first think of spacetime as a set and we reason about elements of its topology and then it's furthermore equipped with a metric. Appearently it is this ...
