Tagged Questions
2
votes
2answers
100 views
Principle of Least Action via Finite-Difference Method
I have to be honest, the principle of least action seems to me more of a religious claim one takes on complete faith, though of course I'm hoping this is just because I don't understand it. I tried to ...
0
votes
1answer
36 views
Kinetic rotational energy of a bar hooked to a coil
I have solved an exercise and I'd like to know if my proceeding about finding kinetic energy is correct or not, because this is the first time that I "meet" a situation like this.
"A bar has mass $M$ ...
0
votes
1answer
23 views
Doubt about coordinates and point of equilibrium
I'm solving an exercise about small oscillations and I have a doubt about coordinates that I have to use.
This is the text of the exercise:
"A bar has mass M and lenght l. Its extremity A is hooked ...
0
votes
1answer
67 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 $v$.
2
votes
3answers
150 views
Lagrangian mechanics and time derivative on general coordinates
I am reading a book on analytical mechanics on Lagrangian. I get a bit idea on the method: we can use any coordinates and write down the kinetic energy $T$ and potential $V$ in terms of the general ...
2
votes
2answers
98 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 ...
-1
votes
1answer
116 views
Lagrangian formulation for relativistic case
Lagrangian for a real scalar field:
$$\mathcal{L}=\frac{1}{2}\eta^{\mu \nu}\partial_{\mu}\phi\partial_{\nu}\phi-\frac{1}{2}m^2\phi^2 $$
Can someone simply drive me how can I write it from ...
1
vote
3answers
179 views
Must the Lagrangian always be known for the Euler-Lagrange equations to be of any use?
When studying classical mechanics using the Euler-Lagrange equations for the first time, my initial impression was that the Lagrangian was something that needed to be determined through integration of ...
0
votes
0answers
97 views
Small oscillations: diagonal matrix [closed]
I'm solving an exercise about small oscillations.
I name $T$ the kinetic matrix and $H$ the hessian matrix of potential.
The matrix $\omega^2 T- H$ is diagonal and so find the auto-frequencies is ...
4
votes
2answers
181 views
Why is the Lagrangian quadratic in $\dot{q}$? [duplicate]
My teacher said we only consider Lagrangians which are quadratic in $\dot{q}$, and we don't take other Lagrangians. I couldn't understand why. Can anyone please explain this?
3
votes
3answers
141 views
Virtual differentials approach to Euler-Lagrange equation - necessary?
I'm currently teaching myself intermediate mechanics & am really struggling with the d'Alembert-based virtual differentials derivation for the Euler-Lagrange equation. The whole notion of, and ...
20
votes
4answers
772 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 ...
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 ...
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
0answers
159 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
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 ...
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 ...
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 ...
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
107 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 ...
1
vote
1answer
146 views
Clarification on a Goldstein formula steps (classical mechanics)
At page 20 of Classical Mechanics' Goldstein (Third edition), there are these two steps given between eqs. (1.51) and (1.52):
$$\sum_i m_i \ddot {\bf r}_i \cdot \frac{\partial {\bf r_i}}{ \partial ...
4
votes
2answers
191 views
Landau Mechanics: why does adding Lagrangians remove the indefiniteness of multiplying each Lagrangian by a different constant?
In Landau Mechanics (third edition page 4), why does adding Lagrangians of two non interacting parts remove the indefiniteness of multiplying each Lagrangian by a different constant?
If both systems ...
3
votes
1answer
177 views
Can I find a potential function in the usual way if the central field contains $t$ in its magnitude?
I'm working on a classical mechanics problem in which the problem states that a particle of mass $m$ moves in a central field of attractive force of magnitude:
$$F(r, t) = \frac{k}{r^2}e^{-at}$$
...
6
votes
2answers
217 views
What are the reasons for leaving the dissipative energy term out of the Hamiltonian when writing the Lyapunov function?
I have a problem with one of my study questions for an oral exam:
The Hamiltonian of a nonlinear mechanical system, i.e. the sum of the kinetic and potential energies, is often used as a Lyapunov ...
1
vote
2answers
220 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 ...
14
votes
4answers
473 views
Is the principle of least action a boundary value or initial condition problem?
Here is a question that's been bothering me since I was a sophomore in university, and should have probably asked before graduating:
In analytic (Lagrangian) mechanics, the derivation of the ...
8
votes
2answers
448 views
Deriving the action and the Lagrangian for a free point particle in Special Relativity
My question relates to
Landau & Lifshitz, Classical Theory of Field, Chapter 2: Relativistic Mechanics, Paragraph 8: The principle of least action.
As stated there, to determine the action ...
4
votes
1answer
174 views
speed of sound and the potential energy of an ideal gas; Goldstein derivation
I am looking the derivation of the speed of sound in Goldstein's Classical Mechanics (sec. 11-3, pp. 356-358, 1st ed). In order to write down the Lagrangian, he needs the kinetic and potential ...
3
votes
1answer
292 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 ...
5
votes
1answer
114 views
Elementary derivation of the motion equations for an inverted pendulum on a cart
Consider a cart of mass $M$ constrained to move on the horizontal axis. A massless rod is attached to the midpoint of the cart, having a mass $m$ on its endpoint. See wikipedia for a picture and for a ...
2
votes
2answers
220 views
What is the significance of action?
What is the physical interpretation of
$$ \int_{t_1}^{t_2} (T -V) dt $$
where, $T$ is Kinetic Energy and $V$ is potential energy.
How does it give trajectory?
3
votes
1answer
182 views
Question about units in Lagrangian dynamics (inertia matrix)
I have a 3 degree of freedom system and my equation of motion is like this:
$$M(q)q_{dd} + C(q,q_d)q_d+G(q)~=~0$$
$M(q)$: inertia matrix
$C(q,q_d)$: Coriolis-centrifugal matrix
$G(q)$: potential ...
3
votes
2answers
1k views
Lagrangian of two particles connected with a spring, free to rotate
Two particles of different masses $m_1$ and $m_2$ are connected by a massless spring of spring constant $k$ and equilibrium length $d$. The system rests on a frictionless table and may both oscillate ...
5
votes
2answers
634 views
Hamilton-Jacobi Equation
In the Hamilton-Jacobi equation, we take the partial time derivative of the action. But the action comes from integrating the Lagrangian over time, so time seems to just be a dummy variable here and ...
2
votes
4answers
301 views
Why are coordinates and velocities sufficient to completely determine the state and determine the subsequent motion of a mechanical system?
I am a Physics undergraduate, so provide references with your responses.
Landau & Lifshitz write in page one of their mechanics textbook:
If all the co-ordinates and velocities are ...
6
votes
2answers
596 views
Can a force in an explicitly time dependent classical system be conservative?
If I consider equations of motion derived from the pinciple of least action for an explicilty time dependend Lagrangian
$$\delta S[L[q(\text{t}),q'(\text{t}),{\bf t}]]=0,$$
under what ...
3
votes
1answer
294 views
The form of Lagrangian for a free particle
I've just registred here, and I'm very glad that finally I have found such a place for questions.
I have small question about Classical Mechanics, Lagrangian of a free particle. I just read Deriving ...
12
votes
2answers
1k views
Deriving the Lagrangian for a free particle
I'm a newbie in physics. Sorry, if the following questions are dumb. I began reading "Mechanics" by Landau and Lifshitz recently and hit a few roadblocks right away.
Proving that a free particle ...
3
votes
1answer
319 views
A Question about Virtual Work related to Newton's Third Law
In describing D'Alembert's principle, the lecture note I was provided with states that the total force $\mathbb F_l$ acting on a particle can be taken as,
$$\mathbb F_l=F_l+\sum_mf_{ml}+C_l,$$
...
3
votes
2answers
135 views
Does locality emerge from (classical) Lagrangian mechanics?
Consider a (classical) system of several interacting particles. Can it be shown that, if the Lagrangian of such a system is Lorenz invariant, there cannot be any space-like influences between the ...
2
votes
3answers
2k views
Finding Lagrangian of a Spring Pendulum
I'm trying to understand Morin's example of a spring pendulum. What I don't get is his expression for $T$. I can understand the $\dot x^2$ term in the brackets. But I don't understand the $(l + ...
5
votes
4answers
755 views
Why can't any term which is added to the Lagrangian be written as a total derivative (or divergence)?
All right, I know there must be an elementary proof of this, but I am not sure why I never came across it before.
Adding a total time derivative to the Lagrangian (or a 4D divergence of some 4 ...
4
votes
2answers
277 views
Derivation of the Lagrangian method using discretized time axis
I'm watching this video lecture by Leonard Susskind of Stanford:
http://www.youtube.com/watch?v=3apIZCpmdls
After some preliminaries, at 34 minutes he jumps into a discretization of the time axis ...
3
votes
1answer
399 views
When is the principle of virtual work valid?
The principle of virtual work says that forces of constraint don't do net work under virtual displacements that are consistent with constraints.
Goldstein says something I don't understand. He says ...
2
votes
3answers
209 views
Why can't we ascribe a (possibly velocity dependent) potential to a dissipative force?
Sorry if this is a silly question but I cant get my head around it.
4
votes
2answers
233 views
How do I show that there exists variational/action principle for a given classical system?
We see variational principles coming into play in different places such as Classical Mechanics (Hamilton's principle which gives rise to the Euler-Lagrange equations), Optics (in the form of Fermat's ...
3
votes
3answers
574 views
Is there a valid Lagrangian formulation for all classical systems?
Can one use the Lagrangian formalism for all classical systems, i.e. systems with a set of trajectories $\vec{x}_i(t)$ describing paths?
On the wikipedia page of Lagrangian mechanics, there is an ...
2
votes
1answer
322 views
Origins of the principle of least time in classical mechanics
Is it possible to derive the principle of least time from the principle of least action in lagrangian or hamiltonian mechanics? Or is Fermat's principle more fundamental than the principle of least ...
