What is Lagrangian dynamics: Definition and 36 Discussions
Introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788, Lagrangian mechanics is a formulation of classical mechanics and is founded on the stationary action principle.
Lagrangian mechanics defines a mechanical system to be a pair
(
M
,
L
)
{\displaystyle (M,L)}
of a configuration space
M
{\displaystyle M}
and a smooth function
L
=
L
(
q
,
v
,
t
)
{\displaystyle L=L(q,v,t)}
called Lagrangian. By convention,
L
=
T
−
V
,
{\displaystyle L=T-V,}
where
T
{\displaystyle T}
and
V
{\displaystyle V}
are the kinetic and potential energy of the system, respectively. Here
q
∈
M
,
{\displaystyle q\in M,}
and
v
{\displaystyle v}
is the velocity vector at
q
{\displaystyle q}
(
v
{\displaystyle (v}
is tangential to
M
)
.
{\displaystyle M).}
(For those familiar with tangent bundles,
L
:
T
M
×
R
t
→
R
,
{\displaystyle L:TM\times \mathbb {R} _{t}\to \mathbb {R} ,}
and
v
∈
T
q
M
)
.
{\displaystyle v\in T_{q}M).}
Given the time instants
t
1
{\displaystyle t_{1}}
and
t
2
,
{\displaystyle t_{2},}
Lagrangian mechanics postulates that a smooth path
x
0
:
[
t
1
,
t
2
]
→
M
{\displaystyle x_{0}:[t_{1},t_{2}]\to M}
describes the time evolution of the given system if and only if
x
0
{\displaystyle x_{0}}
is a stationary point of the action functional
S
[
x
]
=
def
∫
t
1
t
2
L
(
x
(
t
)
,
x
˙
(
t
)
,
t
)
d
t
.
{\displaystyle {\cal {S}}[x]\,{\stackrel {\text{def}}{=}}\,\int _{t_{1}}^{t_{2}}L(x(t),{\dot {x}}(t),t)\,dt.}
If
M
{\displaystyle M}
is an open subset of
R
n
{\displaystyle \mathbb {R} ^{n}}
and
t
1
,
{\displaystyle t_{1},}
t
2
{\displaystyle t_{2}}
are finite, then the smooth path
x
0
{\displaystyle x_{0}}
is a stationary point of
S
{\displaystyle {\cal {S}}}
if all its directional derivatives at
x
0
{\displaystyle x_{0}}
vanish, i.e., for every smooth
{\displaystyle \delta {\cal {S}}\ {\stackrel {\text{def}}{=}}\ {\frac {d}{d\varepsilon }}{\Biggl |}_{\varepsilon =0}{\cal {S}}\left[x_{0}+\varepsilon \delta \right]=0.}
The function
δ
(
t
)
{\displaystyle \delta (t)}
on the right-hand side is called perturbation or virtual displacement. The directional derivative
δ
S
{\displaystyle \delta {\cal {S}}}
on the left is known as variation in physics and Gateaux derivative in Mathematics.
Lagrangian mechanics has been extended to allow for non-conservative forces.
We know that all actions are invariant under their gauge transformations. Are the equations of motion also invariant under the gauge transformations?
If yes, can you show a mathematical proof (instead of just saying in words)?
I'm stuck in a problem of a spring mass system with a pendulum attached to it as showed in the figure below:
My goal is to find the movement equation for the mass, using Lagrangian dynamics.
If the spring moves, the wire will move the same amount. Therefore, we can write the x and y position...
Here is the problem :
A pendulum is composed of a mass m attached to a string of length l, which is suspended
from a fixed point. When hanging at equilibrium, the pendulum is hit with a horizontal
impulse that results in an initial angular velocity ω0. Show that if ω20 < 2g/l, the string
will...
The given lagrangian doesn't seem to correspond to any of the basic systems (like simple/ coupled harmonic oscillators, etc). So I calculated the momentum ##p## which is the partial derivative of ##L## with respect to generalized velocity ##\dot{q}##. Doing so I obtain
$$p =...
I tried to solve the problem in 2 ways, first using lagrangian mechanics and second by putting a rotating reference frame on the initial take-off point.
However I cannot be sure if the equations of motion for the two solutions came out the same.
A-) Equations of motion from Lagrangian...
Consider a free particle with rest mass ##m## moving along a geodesic in some curved spacetime with metric ##g_{\mu\nu}##:
$$S=-m\int d\tau=-m\int\Big(\frac{d\tau}{d\lambda}\Big)d\lambda=\int L\ d\lambda$$...
Summary: In QFT, if we add a gauge breaking term to the Lagrangian, do we still need to introduce Faddeev-Popov ghost particles?
Ghosts seems to be introduced to maintain gauge invariance. But suppose we have eliminated the gauge invariance, from the start, by explicitly introducing a gauge...
On page 224 of the 5th edition of Classical Dynamics of Particles and Systems by Stephen T. Thornton and Jerry B. Marion, the authors introduced the ##δ## notation (in section 6.7). This notation is given by Equations (6.88) which are as follows:
$$\delta J = \frac{\partial J}{\partial...
Homework Statement
A rigid cylinder of radius ##R## and mass ##\mu## has a moment of inertia ##I## around an axis going through the center of mass and parallel to the central axis of the cylinder. The cylinder is homogeneous along its central axis, but not in the radial and angular directions...
<<Moderator's note: Moved from a technical forum, no template.>>
Description of the system:
The masses m1 and m2 lie on a smooth surface. The masses are attached with a spring of non stretched length l0 and spring constant k. A constant force F is being applied to m2.
My coordinates:
Left of...
Homework Statement
A circular hoop of radius R rotates with angular frequency ω about a vertical axis coincident with its diameter. A bead of mass m slides frictionlessly under gravity on the hoop. Let θ be the bead’s angular position relative to the vertical (so that θ = 0 corresponds to the...
In Lagrangian mechanics, both q(t) and dq/dt are treated as independent parameters. Similarly, in Hamiltonian mechanics q and p are treated as independent. How is this justified, considering you can derive the generalized velocity from the q(t) by just taking a time derivative. Does it have...
Homework Statement
A bead of mass m slides in a frictionless hollow open-ended tube of length L which is held at an angle of β to the vertical and rotated by a motor at an angular velocity ω. The apparatus is in a vertical gravitational field.
a) Find the bead's equations of motion
b) Find...
Homework Statement
A particle of mass m is on top of a frictionless hemisphere centered at the origin with radius a"
Set up the lagrange equatinos determine the constraint force and the point at which the particle detaches from the hemisphere
Homework Equations
L=T-U
The Attempt at a...
In field theory a typical Lagrangian (density) for a "free (scalar) field" ##\phi(x)## is of the form $$\mathcal{L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi -\frac{1}{2}m^{2}\phi^{2}$$ where ##m## is a parameter that we identify with the mass of the field ##\phi(x)##.
My question is...
Homework Statement
Homework Equations
The last part of this question is an example of this result:
The Attempt at a Solution
Here is the solution
I think L' is missing a term: If we take the Earth as your frame of reference.(i.e. You are stationary, watching the movement of the railway...
In 3rd edition of Goldstein's "Classical Mechanics" book, page 335, section 8.1, it is mentioned that :
In Hamiltonian formulation, there can be no constraint equations among the co-ordinates.
Why is this necessary ? Any simple example which will elaborate this fact ?
But in Lagrangian...
Homework Statement
Two blocks of equal mass, m, are connected by a light string that passes over a massless pulley. One block hangs below the pulley, while the other sits on a frictionless horizontal table and is attached to a spring of constant k. Let x=0 be the equilibrium position of the...
Hi all,
I've recently been asked for an explanation as to why the Lagrangian is a function of the positions and velocities of the particles constituting a physical system. What follows is my attempt to answer this question. I would be grateful if you could offer your thoughts on whether this is...
Hi,
I have a conceptual question regarding Lagrangian dynamics. It has to do with the potential energy formulation. My instructor today mentioned something in class that does not make much sense to me.
Here is he most basic example that illustrates my confusion:
Take a simple 1dof...
I'm going to run through a derivation I've seen and ask a few questions about some parts that I'm unsure about.
Firstly the theorem: For every symmetry of the Lagrangian there is a conserved quantity.
Assume we have a Lagrangian L invariant under the coordinate transformation qi→qi+εKi(q)...
Homework Statement
Carry out the integration ψ = ∫[M(dr/r2)] / √(2m(E-U(r)) - (M2/r2))
E = energy, U = potential, M = angular momentum
using the substitution: u = 1/r for U = -α/r
Homework Equations
The Attempt at a Solution
This is as far as I've gotten: -∫ (Mdu) /...
Homework Statement
The pendulum of a grandfather clock consists of a thin rod of length L (and negligible mass) attached at its upper end to a fixed point, and attached at its lower end to a point on the edge of a uniform disk of radius R, mass M, and negligible thickness. The disk is free...
A point of mass m, affected by gravity, is obliged to be in a vertical plan on a parabola with equation z = a.r^2
a is a constant and r is the distance between the point of mass m and the OZ vertical axis. Write the Lagrange equations in the cases that the plan of the parabola is :
a) is...
Homework Statement
Hi all, I need to derive differantial equations of system with lagrange multiplier method, a disk is rolling and a bar is fixed onto the point of a disk
http://img130.imageshack.us/img130/1669/adsziss.jpg
By deniz120 at 2010-05-31
Homework Equations
The...
I am currently a grad student. Part of my PhD work will be to formulate a mathematical model of a manufacturing process using Lagrangian dynamics. I am just beginning to delve into the world of variational mechanics, having never had a formal course in the subject. The process involves a...
I'm very happy that I found this forum, hello everyone.
I'm studying Lagrangian Dynamics and I can't figure out how to find the generalized forces in a setup like this:
____ ____
| | ___ | |
| b1 |===|___|===| b2 |
|____| |____|
--->x...
My question pertains to Example 1.2 of Schaum's Outline of Lagrangian Dynamics by Dare A. Wells, chapter 1, page 4.
You can view the diagram and the example (1.2) by going to the following link on Amazon.com and clicking on Excerpt, and then going to page 4...
Lagrangian Dynamics problem -- need help with setup
Here's the problem:
A simple pendulum of length b and bob with mass m is attached to a massless support moving horizontally with constant acceleration a. Determine the equations of motion.
For the pendulum, x = b sin theta and y = b cos...