Linked Questions
62 questions linked to/from Is there a proof from the first principle that the Lagrangian $L = T - V$?
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How do you determine the Lagrangian? [duplicate]
I have always been puzzled by how do you arrive at Lagrangians?
That is, how do you know that the functional you need to get Newton's equations is
$$L = T-V(x)~?$$
Do you derive the Lagrangian ...
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0
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Derivation of Euler-Lagrange equations from Newton's second law [duplicate]
I was thinking about this for some time and I wanted to clarify my question. Can the Euler-Lagrange equation be somehow derived from Newton's second law? Here's a possible way to do it:
We start with ...
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2
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1
answer
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Why is the Lagrangian defined as $L=T - V$? [duplicate]
Please try to provide a sufficient answer, and when it is just „because it satisfies Newton‘s equations“, please try to give an example or explain it. If you know it, I would be very happy if you ...
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Lagrangian intuition [duplicate]
I am new to lagrangian mechanics and it just baffles me the idea of subtracting potential energy from kinetic energy. Why don't we use kinetic energy alone and the least action path (between two ...
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0
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The principle of least action [duplicate]
I have read about the principle of least action. This principle suggests that nature would allow a particle to travel in a path along which the integral of the difference between kinetic energy and ...
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0
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Reason behind $L = T - V$ (Lagrangian formalism) [duplicate]
I've been learning about the Lagrangian formulation recently, and while I'm with the process, I am still struggling somewhat with the theory behind it. As I (rather poorly) understand it, the ...
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Lagrangian in Classical Mechanics [duplicate]
Where can I find the ab-initio derivation of the Lagrangian in classical mechanics? How did Lagrange arrive at $L=T-V$ ?
Respected Experts kindly explain to me. What is the need to construct $L=T-V$ ...
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1
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How can the action can describe a movement? What is the argument behind? [duplicate]
We define the action of a system as $$S(q)=\int_{t_1}^{t_2}L(t,q(t),q'(t))dt,$$
where $q(t)$ is the evolution of the system and $L$ is the Lagrangien. How can a stationary point of $S$ can describe ...
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Lagrangian function $L = T- U$ & Lagrange's book "Mécanique analytique" [duplicate]
Lagrangian is function of generalized co-ordinates, generalized velocities and time:
$$L=L(q,\dot{q},t)$$
Why the specific form $L=T-U$ is used as a definition of Lagrangian function? Here as usual $T,...
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0
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Action in Lagrangian Mechanics [duplicate]
I editted this question since it was closed because it is a duplicate. However, answers in the referenced question didn't solve my question, so I am writing it again.
Lagrangian mechanics is built ...
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0
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Origin of Hamilton's variational principle [duplicate]
My question is what is the theoretical origin of Hamilton's principle. I mean is there any rigorous mathematical proof of this principle from some more basic principles?
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Question regarding the motivation behind finding Lagrangian [duplicate]
The Lagrangian method uses the Kinetic Energy and Potential Energy to give us the equation of motion, that's well and good.
But it seems too random to define a quantity called $L$ (action).
The ...
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0
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Why identify the Lagrangian as kinetic energy minus potential energy? [duplicate]
When deriving the Euler-Lagrange equations from the Principle of Stationary Action we start by insisting that the actual path of a particle minimizes the action $S$, defined as
$$S=\int_{t_1}^{t_2}\...
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How Lagrangian equations of motions can be obtained from Newton's laws? [duplicate]
Let's consider a system of $N$ point particles. Let's also assume that acceleration of each particle is a function of positions of all the particles.
I assume that for such a system we can prove that ...
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Is there a way to derive Principle of Least Action from Newton's laws instead of other ways around? [duplicate]
If we start from
$ m\ddot{x} = -\frac{dV}{dx} $
How could one derive/construct principle of least action? i.e. find out that this quantity
$ S = \int_{t_0}^{t_1}dt\left(\frac{m\dot{x}^2}{2} - V(x) \...
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0
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Why does the Lagrangian equal $T-V$? [duplicate]
When defining $L=T-V$, and using Euler-Lagrange equations ($\partial_x L = \frac{d}{dt} \partial_{\dot{x}} L$), we get back $m \ddot{x} = - \frac{dV}{dx}$ ONLY WHEN ASSUMING that $V(x, \dot{x}) = V(x)$...
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Derive $L = T - U$, in newtonian mechanics, assuming the principle of least action [duplicate]
I am trying to get a better understanding of the Lagrangian. From what I know, we say that each trajectory in physics must be a path that is at a minimum, which means that is satisfies the Euler-...
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What are examples of Lagrangians that 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 ...
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Why the Principle of Least Action?
I'll be generous and say it might be reasonable to assume that nature would tend to minimize, or maybe even maximize, the integral over time of $T-V$. Okay, fine. You write down the action ...
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Why should an action integral be stationary? On what basis did Hamilton state this principle?
Hamilton's principle states that a dynamic system always follows a path such that its action integral is stationary (that is, maximum or minimum).
Why should the action integral be stationary? On ...
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Motivation for form $L = T - V$ of Lagrangian
This question (in Lagrangian mechanics) might be silly, but why is the Lagrangian $L$ defined as: $L = T - V$?
I understand that the total mechanical energy of an isolated system is conserved, and ...
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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.
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3
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Are there examples in classical mechanics where D'Alembert's principle fails?
D'Alembert's principle suggests that the work done by the internal forces for a virtual displacement of a mechanical system in harmony with the constraints is zero.
This is obviously true for the ...
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24
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2
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Lagrangian Mechanics - Commutativity Rule $\frac{d}{dt}\delta q=\delta \frac{dq}{dt} $
I am reading about Lagrangian mechanics.
At some point the difference between the temporal derivative of a variation and variation of the temporal derivative is discussed.
The fact that the two are ...
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What does a Lagrangian of the form $L=m^2\dot x^4 +U(x)\dot x^2 -W(x)$ represent?
I saw this Lagrangian in notes I have printed:
$$
L\left(x,\frac{dx}{dt}\right) = \frac{m^2}{12}\left(\frac{dx}{dt}\right)^4 + m\left(\frac{dx}{dt}\right)^2\times V(x) -V^2(x).
$$
(It appears in the ...
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4
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Connection between different kinds of "Lagrangian"
Being a physic student I first heard the term: "Lagrangian" during a course about Lagrangian mechanics; at that time this term was defined to me in the following way:
For a classic, non ...
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6
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4
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Why Lagrangian is negative number for a relativistic massive point particle?
In the special relativistic action for a massive point particle,
$$S=\int_{t_i}^{t_f}\mathcal {L}dt,$$
why is the Lagrangian
$$\mathcal {L}=-E_o\gamma^{-1}$$
a negative number?
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2
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When can we add a total time derivative of $f(q, \dot{q}, t)$ to a Lagrangian?
The other day, I was listening to this lecture on the Lagrangian for a charged particle in an electromagnetic field, and at one point in the video, the lecturer mentions that we can add any total time ...
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Does Newtonian $F=ma$ imply the least action principle in mechanics?
I've learned that Newtonian mechanics and Lagrangian mechanics are equivalent, and Newtonian mechanics can be deduced from the least action principle.
Could the least action principle $\min\int L(t,...
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3
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Can Lagrangian mechanics be justified without referring to Newtonian mechanics?
Are there any ways of justifying Lagrangian mechanics as a foundation of classical physics, without referring to Newtonian mechanics? In other words, what is the deeper reason or intuition why
$$\...
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