For questions involving the Lagrangian formulation of a dynamical system. Namely, the application of an action principle to a suitably chosen Lagrangian or Lagrangian Density in order to obtain the equations of motion of the system.

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Why are there only derivatives to the first order in the Lagrangian?

Why is the Lagrangian a function of the position and velocity (possibly also of time) and why are dependences on higher order derivatives (acceleration, jerk,...) excluded? Is there a good reason for ...
31
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6answers
3k views

Why does calculus of variations work?

How does it make sense to vary the position and the velocity independently? Edit: Velocity is the derivative of position, so how can you treat them as independent variables? Doesn't every physics ...
26
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5answers
3k views

Hamilton's 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 ...
28
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7answers
6k views

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 ...
28
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13answers
2k views

Lagrangians 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 ...
14
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2answers
3k 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 ...
9
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4answers
891 views

Is there a proof from the first principle that the Lagrangian L = T - V?

Is there a proof from the first principle that for the Lagrangian $L$, $$L = T\text{(kinetic energy)} - V\text{(potential energy)}$$ in classical mechanics? Assume that Cartesian coordinates are ...
16
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3answers
1k views

How general is the Lagrangian quantization approach to field theory?

It is an usual practice that any quantum field theory starts with a suitable Lagrangian density. It has been proved enormously successful. I understand, it automatically ensures valuable symmetries of ...
5
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1answer
974 views

Invariance of Lagrangian in Noether's theorem

Often in textbooks Noether's theorem is stated with the assumption that the Lagrangian needs to be invariant $\delta L=0$. However, given a lagrangian $L$, we know that the Lagrangians $\alpha L$ ...
16
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3answers
2k views
10
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4answers
2k views

Deriving Newton's Third Law from homogeneity of Space

I am following the first volume of the course of theoretical physics by Landau. So, whatever I say below mainly talks regarding the first 2 chapters of Landau and the approach of deriving Newton's ...
6
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4answers
702 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
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2answers
416 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 ...
12
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9answers
5k views

Book about classical mechanics

I am looking for a book about "advanced" classical mechanics. By advanced I mean a book considering directly Lagrangian and Hamiltonian formulation, and also providing a firm basis in the geometrical ...
21
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5answers
3k views

Why not using Lagrangian, instead of Hamiltonian, in non relativistic QM?

When we studied classical mechanics on the undergraduate level, on the level of Taylor, we covered Hamiltonian as well as Lagrangian mechanics. Now when we studied QM, on the level of Griffiths, we ...
3
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1answer
781 views

Deriving D'Alembert's Principle

The wiki article states that D'Alembert's Principle cannot derived from Newton's Laws alone and must stated as a postulate. Can someone explain why this is? It seems to me a rather obvious principle.
12
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2answers
2k views

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 ...
3
votes
1answer
365 views

What variables does the action $S$ depend on?

Action is defined as, $$S ~=~ \int L(q, q', t) dt,$$ but my question is what variables does $S$ depend on? Is $S = S(q, t)$ or $S = S(q, q', t)$ where $q' := \frac{dq}{dt}$? In ...
15
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3answers
3k views

Physical meaning of Legendre transformation

I would like to know the physical meaning of the Legendre transformation, if there is any? I've used it in thermodynamics and classical mechanics and it seemed only a change of coordinates?
22
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6answers
3k views

Can Noether's theorem be understood intuitively?

Noether's theorem is one of those surprisingly clear results of mathematical calculations, for which I am inclined to think that some kind of intuitive understanding should or must be possible. ...
20
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7answers
12k views

What is the difference between Newtonian and Lagrangian mechanics in a nutshell?

What is Lagrangian mechanics, and what's the difference compared to Newtonian mechanics? I'm a mathematician/computer scientist, not a physicist, so I'm kind of looking for something like the ...
14
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6answers
3k views

What is the physical meaning of the action in Lagrangian mechanics?

The action is defined as $S = \int_{t_1}^{t_2}L \, dt$ where $L$ is Lagrangian. I know that using Euler-Lagrange equation, all sorts of formula can be derived, but I remain unsure of the physical ...
8
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4answers
2k 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 ...
1
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2answers
320 views

What is the relativistic action of a massive particle?

all Lorentz observers watching a particle move will compute the same value for the quantity $$ds^2 = -(c \, dt)^2 + dx^2 + dy^2 + dz^2,$$ $$ds^2 = g_{\mu\nu}dx^{\mu}dx^{\nu},$$ and ''ds/c'' is then ...
19
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1answer
603 views

Why does charge conservation due to gauge symmetry only hold on-shell?

While deriving Noether's theorem or the generator(and hence conserved current) for a continuous symmetry, we work modulo the assumption that the field equations hold. Considering the case of gauge ...
2
votes
3answers
414 views

Does a four-divergence extra term in a Lagrangian density matter to the field equations?

Greiner in his book "Field Quantization" page 173, eq.(7.11) did this calculation: ${\mathcal L}^\prime=-\frac{1}{2}\partial_\mu A_\nu\partial^\mu A^\nu+\frac{1}{2}\partial_\mu A_\nu\partial^\nu ...
2
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3answers
1k 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 ...
3
votes
3answers
507 views

Type of stationary point in Hamilton's principle

In this question it is discussed why by Hamilton's principle the action integral must be stationary. Most examples deal with the case that the action integral is minimal: this makes sense - we all ...
40
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8answers
4k views

What's the point of Hamiltonian mechanics?

I've just finished a Classical Mechanics course, and looking back on it some things are not quite clear. In the first half we covered the Lagrangian formalism, which I thought was pretty cool. I ...
13
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4answers
519 views

Equivalence between Hamiltonian and Lagrangian Mechanics

I'm reading a proof about Langrangian => Hamiltonian and one part of it just doesn't make sense to me. The Lagrangian is written $L(q, \dot q, t)$, and is convex in $\dot q$, and then the Hamiltonian ...
13
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4answers
4k views

The meaning of action

The action $$S=\int L \;\mathrm{d}t$$ is an important physical quantity. But can it be understood more intuitively? The Hamiltonian corresponds to the energy, whereas the action has dimension of ...
11
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1answer
719 views

Do an action and its Euler-Lagrange equations have the same symmetries?

Assume a certain action $S$ with certain symmetries, from which according to the Lagrangian formalism, the equations of motion (EOM) of the system are the corresponding Euler-Lagrange equations. Can ...
9
votes
2answers
1k 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 ...
11
votes
3answers
2k views

Galilean invariance of Lagrangian for non-relativistic free point particle?

In QFT, the Lagrangian density is explicitly constructed to be Lorentz-invariant from the beginning. However the Lagrangian $$L = \frac{1}{2} mv^2$$ for a non-relativistic free point particle is ...
18
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4answers
924 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 ...
9
votes
4answers
2k views

Lagrangian to Hamiltonian in Quantum Field Theory

While deriving Hamiltonian from Lagrangian density, we use the formula $$\mathcal{H} ~=~ \pi \dot{\phi} - \mathcal{L}.$$ But since we are considering space and time as parameters, why the formula ...
9
votes
4answers
443 views

Is the Lagrangian of a quantum field really a 'functional'?

Weinberg says, page 299, The quantum theory of fields, Vol 1, that The Lagrangian is, in general, a functional $L[\Psi(t),\dot{\Psi}(t)$], of a set of generic fields $\Psi[x,t]$ and their time ...
22
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8answers
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Why are L4 and L5 lagrangian points stable?

This diagram from wikipedia shows the gravitational potential energy of the sun-earth two body system, and demonstrates clearly the semi-stability of the L1, L2, and L3 lagrangian points. The blue ...
1
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2answers
362 views

Why lagrangian is negative number?

In the special relativistic action for a massive point particle, $$\int_{t_i}^{t_f}\mathcal {L}dt,$$ why is the Lagrangian $$\mathcal {L}=-E_o\gamma^{-1}$$ a negative number?
10
votes
2answers
395 views

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 ...
16
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6answers
1k views

Why does a system try to minimize potential energy?

In mechanics problems, especially one-dimensional ones, we talk about how a particle goes in a direction to minimize potential energy. This is easy to see when we use cartesian coordinates: For ...
10
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5answers
588 views

Form of the Classical EM Lagrangian

So I know that for an electromagnetic field in a vacuum the Lagrangian is $\mathcal L=-\frac 1 4 F^{\mu\nu} F_{\mu\nu}$, the standard model tells me this. What I want to know is if there is an ...
10
votes
3answers
4k views

What exactly are Hamiltonian Mechanics (and Lagrangian mechanics)

What exactly are Hamiltonian Mechanics (and Lagrangian mechanics)? I want to self-study QM, and I've heard from most people that Hamiltonian mechanics is a prereq. So I wikipedia'd it and the entry ...
6
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5answers
927 views

What is the Lagrangian for a relativistic charge that includes the self-force?

The usual Lagrangian for a relativistically moving charge, as found in most text books, doesn't take into account the self force from it radiating EM energy. So what is the Lagrangian for a ...
5
votes
1answer
265 views

On a trick to derive Noether current

Suppose, in whatever dimension and theory, the action $S$ is invariant for a global symmetry with a continuous parameter $\epsilon$. The trick to get the Noether current consists in making the ...
5
votes
2answers
930 views

How the Lagrangian of classical system can be derived from basic assumptions?

It is well known that the Lagrangian of a classical free particle equal to kinetic energy. This statement can be derived from some basic assumptions about the symmetries of the space-time. Is there ...
3
votes
4answers
500 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 ...
4
votes
2answers
587 views

Conversion of the Nambo-Goto action into the Polyakov action?

I`ve read that the Nambo-Goto action containing the induced metric $\gamma_{\alpha\beta}$ $$\tag{1} S_{NG} ~=~ -T\int_{\tau_i}^{\tau_f} d\tau \int_0^{\ell} d\sigma \sqrt{-\gamma}$$ can be converted ...
4
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3answers
342 views

Writing $\dot{q}$ in terms of $p$ in the Hamiltonian formulation

In the Hamiltonian formulation, we make a Legendre transformation of the Lagrangian and it should be written in terms of the coordinates $q$ and momentum $p$. Can we always write $dq/dt$ in terms of ...
4
votes
2answers
142 views

Landau's argument for dependence of Lagrangian on magnitude of velocity

In chapter 1, of Landau-Lifshitz Mechanics' book, Landau through isotropy and homogeneity of space and homogeneity of time proves that the Lagrangian must depend of magnitude of velocity of the ...