Lagrangian mechanics Definition and 77 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.
Let E be a fixed immutable quantity. E can be freely exchanged between T and V, as long as $$T + V = E$$
1. What does the quantity $$\int_x T - V $$ signify? What is the importance of this quantity?
--------------------
Let E now be the budget of a factory. E can either be spent on T or V in...
I had used the same constraint as the solution manual says.
So my two Lagrangian would be
##L_1=\frac{1}{2}m_A\dot{x_A}^2+\frac{1}{2}m_B\dot{x_B}^2+\frac{1}{2}m_C\dot{x_C}^2+m_Cgx_C+T(x_A+x_B+2x_C-c)##
whereas c is just a constant.
Of course, I have to write my Lagrangian using constraints...
I am having trouble to find the moment of inertia of the second rod!
Is it related to the first rod??
At the beginning I thought It's not!
But when took those as constant,the equation had become way much simpler and there is nothing about chaos!
My approach is given below
This is from Taylor's classical mechanichs, 11.4, example of finding the Lagrangian of the double pendulum
Relevant figure attached below
Angle between the two velocities of second mass is
$$\phi_2-\phi_1$$
Potential energy
$$U_1=m_1gL_1$$
$$U_2=m_2g[L_1\cos(1-\phi_1)+L_2(1-\phi_2)]$$...
In Lagrangian mechanics we learn about generalized forces. However, I haven't seen these explicitly mentioned in books on QFT. Can the Lagrangians of QED or QCD be expressed in terms of generalized forces or is there some connection there, in particular to the Nielsen form.
Principle of stationary action allows us to find equations of motion if we plug appropriate lagrangian into Euler - Lagrange equation. In classical mechanics, this is the difference in kinetic and potential energy of the system.
However, how did Lagrange came to the idea that matter behaves...
Hello all, so I’ve been reading Jennifer Coopersmith’s The Lazy Universe: An Introduction to the Principle of Least Action, and on page 72 it says:
If I understand it right, she’s saying that in our Euler-Lagrange equation ## \frac {\partial L} {\partial q} - \frac {d} {dt} \frac {\partial L}...
Lagrangian mechanics is built upon calculus of variation. This means that we want to find out function which is a stationary point of particular function (functional) which in Lagrangian mechanics is called the action.
To know what this function is, action needs to be defined first. Action is...
I have sometimes seen the claim that one advantage of Lagrangian mechanics is that it works in any frame of reference, instead of like Newtonian mechanics which will hold only in the inertial frame of reference. However isn't this gain only at the sacrifice that the Lagrangian will need to take...
My understanding of the system from the image (which was given in book)
I could see there's 3 tension in 2 body. Even I had seen 2 tension in a body. It was little bit confusing to me. I could find tension in Lagrangian from right side. But left side was confusing to me...
I'm trying to solve the Goldstein classical mechanics exercises 1.7. The problem is to prove:
$$\frac{\partial \dot T}{\partial \dot q} - 2\frac{\partial T}{\partial q} = Q$$
Below is my progress, and I got stuck at one of the step.
Now since we have langrange equation:
$$\frac{d}{dt}...
The given lagrangian doesn't seem to correspond to any of the basic systems (like simple/ coupled harmonic oscillators, etc). So I calculated the momentum ##p## which is the partial derivative of ##L## with respect to generalized velocity ##\dot{q}##. Doing so I obtain
$$p =...
Hello,
It might sound silly, but when I try to calculate the kinetic energy of a rotating rod to form the Langrangian (and in general), why it has both translational and rotational kinetic energy?
Is it because when I consider the moment of Inertia about the centre I need to include the...
I am new to Lagrangian mechanics and I have gone through basic examples of solving the Euler Lagrange equation for simple pendulums or projectiles and things like that. But I am unable to understand what we are exactly solving the equation for or what is the significance of the differential...
I am new to Lagrangian mechanics and I am unable to comprehend why the Euler Lagrange equation works, and also what really is the significance of the lagrangian.
Here is the picture on the system.
I have to find the period (T). The masses, R and dX is given. The systam at first is at rest, then at t = 0 we pull the plank to dX distance from its originial position.
In the thread...
Hi, I am an undergraduate student in the 3rd sem, we have Lagrangian Mechanics in our course but I am unable to follow it properly. Can you please suggest me a book that will introduce me to Lagrangian and Hamiltonian Mechanics and slowly teach me how to do problems. I am beginner, so please...
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:
$$\delta J = \frac{\partial J}{\partial...
In his book, Landau mentioned varying the relativistic lagrangian
However, I do not understand how he got from varying the integral of ds to varying only the contravariant components.
Would the general procedure not be varying
$$\delta S = -mc\delta\int_a^b\frac{dx_idx^i}{\sqrt{ds}}$$ and...
Homework Statement
Homework Equations
L = T-V
For constant frequency tangential velocity is (radius)*(w)
The Attempt at a Solution
I need to find r(t) using the Langrangian L = T-V
I just was not sure whether I am on the right track for calculating the total kinetic energy for the above...
In my physics education, I shied away from heavily theoretical stuff like General Relativity. I took the required sequence in Quantum Mechanics but having never used it on the job, much of that knowledge has faded too. I started a course in Quantum Field Theory but dropped it. I had friends...
In Chapter 7: Hamilton's Principle, in the Classical Dynamics of Particles and Systems book by Thornton and Marion, Fifth Edition, page 258-259, we have the following equations:
1. Upon squaring Equation (7.117), why did the authors in the first term of Equation (7.118) are summing over two...
Homework Statement
A yoyo falls straight down unwinding as it goes, assume has inner radius a, outer radius b and Inertia I. What is the generalised coordinates and the lagrangian equation of motion?
Homework Equations
L=T-U where T is kinetic energy and U is potential
The Attempt at a...
In Chapter 8: Central-Force Motion, in the Classical Dynamics of Particles and Systems book by Thornton and Marion, Fifth Edition, page 323, Problem 8-5, we are asked to show that the two particles will collide after a time ##\tau/4√2##.
I don't have any problems with the derivations and with...
Homework Statement
Consider a particle moving over the curve ##z=a-bx^2## under the force of gravity. If the particle starts from rest at point ##(0,0)## (I'm guessing it means point ##(0,a)##), tell if the particle ever separates from the curve; if yes, find the point at which it does...
How is it that when using "natural" units we drop the units themselves. I understand that you can arbitrarily change the magnitude of a parameter by choosing a new unit. For example Oliver R. Smoot is exactly 1 smoot tall.
However, in natural units with [c]=[h/(2π)]=1 the "smoot" part is...
<<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...
Lagrangian Mechanics uses generalized coordinates and generalized velocities in configuration space.
Hamiltonian Mechanics uses coordinates and corresponding momenta in phase space.
Could anyone please explain the difference between configuration space and phase space.
Thank you in advance for...
A very simple question. How do we represent a vector with Newton's notation (writing the arrow symbol with the overdot notation)? Can we write them both above each other. First, the overdot notation and then the arrow symbol?
Thank you a lot for your help...
Homework Statement
Find the acceleration of a uniform solid sphere (of mass ##m## and radius ##R##) rolling without slipping down an incline at angle ##\alpha## using the Lagrangian method.
Homework Equations
Euler-Lagrange equation which says, $$\frac{\partial\mathcal{L}}{\partial...
(note: I'm going to represent the lagrangian as simply L because I don't know how to do script L in latex.)
Homework Statement
Two particles of equal masses m are confined to move along the x-axis and are connected by a spring with potential energy ##U = \frac{1}/{2}kx^2## (here x is the...
Homework Statement
The carbon dioxide molecule can be considered a linear molecule with a central carbon atom, bound
to two oxygen atoms with a pair of identical springs in opposing directions. Study the longitudinal
motion of the molecule. If three coordinates are used, one of the normal...
Homework Statement
Take the x-axis to be pointing perpendicularly upwards.
Mass ##m_1## slides freely along the x-axis. Mass ##m_2## slides freely along the y-axis. The masses are connected by a spring, with spring constant ##k## and relaxed length ##l_0##. The whole system rotates with...
Homework Statement
The Attempt at a Solution
So I first tried by saying consider a time t in which mass m is directly above the origin O. I.e., mass m at the Cartesian coordinate (0, 4l/3). I wrote a = a(t) as the extension function of the spring, which has 0 natural length. So, I...
Homework Statement
Hello!
I have some problems with constructing Lagrangian for the given system:
(Attached files)
Homework Equations
The answer should be given in the following form: L=T-U=...
The Attempt at a Solution
I tried to construct the Lagrangian, but I'm not sure if I did it...
Hello. I solve this problem:
1. Homework Statement
The particles of mass m moves without friction on the inner wall of the axially symmetric vessel with the equation of the rotational paraboloid:
where b>0.
a) The particle moves along the circular trajectory at a height of z = z(0)...
Can Lagrangian densities be constructed from the physics and then derive equations of motion from them? As it seems now, from my reading and a course I took, that the equations of motion are known (i.e. the Klein-Gordon and Dirac Equation) and then from them the Lagrangian density can be...
In the paper http://physics.unipune.ernet.in/~phyed/26.2/File5.pdf, the author solves the LC-circuit using Euler-Lagrange equation. She assumes that the Lagrangian function for the circuit is $$L=T-V$$ where
$$T=L\dot q^2/2$$ is the kinetic energy part $$V=q^2 / 2C$$ is the potential energy.She...
I can't for the life of me figure out what virtual work or D'alemberts principle mean and what the intuition behind them is. As far as I'm concerned D'alemberts principle is just a restatement of Newton's second law but considering the work instead of just the forces. What am I missing? I'm...
Hello everyone,
Reading Landau and Lifshitz Course of Theoretical Physics Volume 1: Mechanics (page 3) I got suck in the following step (and I cite in italics):
The change in S when q is replaced by q+δq is
\int_{t_1}^{t_2} L(q+δq, \dot q +δ\dot q, t)dt - \int_{t_1}^{t_2} L(q, \dot q, t)dt...
Consider a sphere constrained to roll on a rough FLAT HORIZONTAL surface. A book on classical mechanics says it requires 5 generalized co-ordinates to specify sphere's configuration: 2 for its centre of mass and 3 for its orientation.
I did not understand why 3 for orientation. I guess only 2...
Homework Statement
From the homework:
In General Relativity it is found that the radial equation of an object orbiting a non-rotating black hole has the form $$\dot r^2 + (1 - 2 \frac {V_o} {r} ) (\frac {l^2} {r^2} + 1) = E^2$$ where ##r## is the radial coordinate, ##l## is the angular...
In lagrangian variation we are trying to minimize the action
S = ∫t2t1 L dt.
Consider a simple case of free particle.
Imagine In a world that everyone one only knows how to solve ODE, Using euler lagrange equation, one has
d2x/dt2 = 0 , give that we know the initial position of particle in the...
Homework Statement
String is wrapped around two identical disks of mass m and radius R. One disk is fixed to the ceiling but is free to rotate. The other is free to fall, unwinding the string as it falls. Find the acceleration of the falling disk by finding the lagrangian and lagrange's...
In quantum mechanics, position ##\textbf{r}## and momentum ##\textbf{p}## are conjugate variables given their relationship via the Fourier transform. In transforming via the Legendre transform between Lagrangian and Hamiltonian mechanics, where ##f^*(\textbf{x}^*)=\sup[\langle \textbf{x}...
Homework Statement
Find the Lagrangian for the double pendulum system given below, where the length of the massless, frictionless and non-extendable wire attaching m_1 is l. m_2 is attached to m_1 through a massless spring of constant k and length r. The spring may only stretch in the m_1-m_2...
Homework Statement
A uniform cylindrical drum of mass M and radius a is free to rotate about its axis, which i is horizontal. An elastic cable of negligible mass and length l is wrapped around the drum and carries on its free end a mass m. The cable has elastic potential energy \tfrac12...
Hello,
I'm a second year physics student. We are going to use "hand and finch analytical mechanics", however the reviews I saw about this book are bad.
I've already taken calculus for mathematicians, linear algebra, classical mechanics, special relativity, and electromagnetism.
The topics it...
I am trying to establish a Rationalist approach to Physics as a side project, and have taken Hamilton's Principle as one of the few postulates in my work. I've developed the concept enough to arrive at the usual stuff, like the Euler-Lagrange equations, Newton's First Law and Nöther's Theorem...