All Questions
Tagged with classical-mechanics lagrangian-formalism
1,466
questions
3
votes
1
answer
773
views
Euler-Lagrange Equation with logarithmic potential
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 ...
- 751
1
vote
1
answer
2k
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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 ...
- 103
1
vote
1
answer
820
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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 ...
- 105
2
votes
1
answer
1k
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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 ...
- 21
0
votes
1
answer
3k
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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 ...
- 1,133
0
votes
1
answer
191
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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 $...
- 307
2
votes
1
answer
577
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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):
$$\sum_i m_i \ddot {\bf r}_i \cdot \frac{\partial {\bf r_i}}{ \partial ...
- 1,133
8
votes
3
answers
2k
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Landau Mechanics: why does adding Lagrangians remove the indefiniteness of multiplying each Lagrangian by a different constant?
In Landau Mechanics (third edition page 4), why does adding Lagrangians of two non interacting parts remove the indefiniteness of multiplying each Lagrangian by a different constant?
If both systems ...
- 305
3
votes
1
answer
511
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Can I find a potential function in the usual way if the central field contains $t$ in its magnitude?
I'm working on a classical mechanics problem in which the problem states that a particle of mass $m$ moves in a central field of attractive force of magnitude:
$$F(r, t) = \frac{k}{r^2}e^{-at}$$
$r$ ...
- 31
6
votes
2
answers
2k
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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 ...
- 201
3
votes
2
answers
3k
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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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48
votes
5
answers
4k
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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 Euler-...
- 1,350
10
votes
2
answers
4k
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Deriving the action and the Lagrangian for a free massive 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 ...
- 737
7
votes
2
answers
5k
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Centrifugal Force and Polar Coordinates
In Classical Mechanics, both Goldstein and Taylor (authors of different books with the same title) talk about the centrifugal force term when solving the Euler-Lagrange equation for the two body ...
- 115
4
votes
1
answer
945
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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 ...
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