1
vote
1answer
45 views

How can I derive the Hamiltonian of simple harmonic oscillator from this Lagrangian?

I'm working through Leonard Susskind's Theoretical Minimum: Classical Mechanics and I can't seem to understand how the Hamiltonian of a simple harmonic oscillator is derived from the following ...
5
votes
2answers
137 views

Questions about the degree of freedom in General Relatity

I'm confused about the number of degrees of freedom in General Relatity. There are two ways to count it. However, they are contradictory. For simplicity, we consider vacuum solution. First, ...
0
votes
0answers
27 views

Conditions for hyperregularity

(Definitions from Marsden & Ratiu, ``Introduction to Mechanics and Symmetry''): A Lagrangian is regular if the Hessian $\partial^2 L/(\partial \dot{q}^i \partial \dot{q}^j)$ is weakly non ...
3
votes
1answer
64 views

Invariance of canonical Hamiltonian equation when adding the total time derivative of a function of $q_i$ and $t$ to the Lagrangian

The following is exercise 8.2 in 3rd edition (and exercise 8.19 in 2nd edition) of Goldstein's Classical Mechanics. Adding the total time derivative of a function of $q_i$ and t to the Lagrangian ...
6
votes
3answers
294 views

Eigenvalues of the Lagrangian?

It is often stated that the Lagrangian formalism and the Hamiltonian formalism are equivalent. We often hear people talk about eigenvalues of Hamiltonians but I have never ever heard a word about ...
1
vote
1answer
49 views

Where can some worked problems in classical mechanics (and more specifically the Lagrangian and Hamiltonian formalisms) be found? [duplicate]

I've been looking for a textbook in classical mechanics that's readily available (like can be found in the library of James Cook University of Townsville, Australia) and full of fully-answered ...
2
votes
1answer
81 views

Schrödinger evolution for a Klein-Gordon equation

I have a problem with the transition from quantum relativistic wave equations (specifically Klein-Gordon equation) to QFT, since a lot of assumptions seem implicit. For example I have a problem with ...
1
vote
2answers
104 views

How can I derive this Hamiltonian?

I have a Lagrangian $L$, a momentum $p$ and a Hamiltonian $H$: $$L=\frac m 2(\dot z + A\omega\cos\omega t)^2 - \frac k 2 z^2$$ $$p=m\dot z + mA\omega\cos\omega t$$ $$H=p\dot z - L=\frac m 2 \dot ...
1
vote
0answers
32 views

Hamiltonian flow is volume preserving

I was reading about advantage of Hamiltonian over Lagrangian. One of the advantage is "Hamiltonian flow is volume preserving". Can anyone help me to understand this? Means what is advantage of ...
2
votes
0answers
142 views

Can you give example of some problems with solutions in each of Newtonian, Lagrangian and Hamiltonian method? [closed]

I am a student from information system and just want to know about classical mechanics. I know Newtonian mechanics from high school and I have read about Lagrangian and Hamiltonian mechanics in ...
1
vote
0answers
48 views

path integral quantization of EM field derived from canonical quantization?

In Peskin's QFT book page 294, he formally addressed the quantization of EM field, $$propagotor_{EM}=\frac{-ig_{\mu\nu}}{k^2+i\epsilon}$$ Now that we have the functional integral quantization ...
3
votes
1answer
144 views

Finding the creation/annihilation operators

Using Minkowski signature $(+,-,-,-)$, for the Lagrangian density $${\cal L}=\partial_{\mu}\phi\partial^{\mu}\phi^{\dagger}-m^2\phi \phi^{\dagger}$$ of the complex scalar field, we have the field ...
4
votes
1answer
59 views

Why is the Legendre transformation an application of the duality relationship between points and lines?

When I read the Wiki about Legendre transformation, there is a statement The Legendre transformation is an application of the duality relationship between points and lines. What's the meaning of ...
14
votes
4answers
431 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 ...
2
votes
1answer
97 views

Lagrangian to Hamiltonian

I'm having some problems with an assignment where I have to state the Hamiltonian from the kinetic energy $T$ and potential energy $U$. These are as follows: ...
9
votes
1answer
181 views

Are the Hamiltonian and Lagrangian always convex functions?

The Hamiltonian and Lagrangian are related by a Legendre transform: $$ H(\mathbf{q}, \mathbf{p}, t) = \sum_i \dot q_i p_i - \mathcal{L}(\mathbf{q}, \mathbf{\dot q}, t). $$ For this to be a Legendre ...
0
votes
2answers
215 views

Time dependence of Lagrangian and Hamiltonian?

I am reading a online tutorial about Lagrangian mechanics. In one section, it states that if the kinetic term in Lagrangian has no explicit time dependence, the Hamiltonian does not explicitly depends ...
3
votes
1answer
177 views

Missing terms in Hamiltonian after Legendre transformation of Lagrangian

Short question Given any Lagrangian density of fields one could possibly conceive, is it the case that after one has performed a Legendre transformation, if the Hamiltonian is then expressed in terms ...
3
votes
1answer
65 views

Classical Mechanics & Coordinates [closed]

What is the meaning generalised coordinates in Classical Mechanics? How is Lagrangian formalism different from Hamiltonian formalism? How are they related to Hamilton's Principle? How are they ...
1
vote
1answer
179 views

Calculate Hamiltonian from Lagrangian for electromagnetic field

I am unable to derive the Hamiltonian for the electromagnetic field, starting out with the Lagrangian $$ \mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}-\frac{1}{2}\partial_\nu A^\nu \partial_\mu A^\mu ...
1
vote
1answer
82 views

Transforming a lagrangian to hamiltonian and vice versa

I am not refering to Legendre transform, but to something more simple. In analytical mechanics, the Lagrangian can be described as $L=T-V$, and the Hamiltonian is if the Lagrangian doesn't explicitly ...
1
vote
1answer
123 views

Hamiltonian conservation

Lagrangian formalism does not involve forces that doesn't come from a potential and Hamiltonian formalism says that even though energy is not conserved due to a force like this, the Hamiltonian is ...
4
votes
1answer
223 views

Constraints of massive relativistic point particle in hamiltonian mechanics

I try to understand constructing of Hamiltonian mechanics with constraints. I decided to start with the simple case: free relativistic particle. I've constructed hamiltonian with constraint: ...
36
votes
8answers
3k 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 ...
10
votes
1answer
321 views

What's the physical intuition for symplectic structures?

I always thought about symplectic forms as elements of areas in little subspaces because of the Darboux theorem, however I cannot get the physical intuition for it and for the hamiltonian vector ...
1
vote
1answer
87 views

Something between Lagrangian and Hamiltonian called Routhian

So, in my mechanics class, the teacher mentioned there is a special function which is kind of a midpoint between the Lagrangian and the Hamiltonian, called the Routhian. Now, I wanted to give it a ...
1
vote
1answer
71 views

Why does a particle fall in a straight line?

In Lagrangian Mechanics we choose the path of least action. Given a uniform gravitational field, and a particle of finite mass; and fixing two points the start & end-point we consider all paths ...
3
votes
0answers
84 views

Lagrangian with vanishing conjugate momentum, independent variables

Given a Lagrangian density $\mathcal L(\phi_r,\partial_\mu\phi_r,\phi_n,\partial_\mu\phi_n)$, for which we find out that for some $\phi_n$ its conjugate momentum vanishes: ...
7
votes
3answers
295 views

Why not formulate Quantum Mechanics using Lagrangians? [duplicate]

As the title implies, why is it that the most common formalisms we use in quantum mechanics prefer to describe systems in the terms of a Hamiltionian instead of a Lagrangian? Is there some ...
5
votes
0answers
415 views

Is Feynman talking about the Zeroth Law of Thermodynamics?

In Volume 1 Chapter 39 of the Feynman Lectures on Physics, Feynman derives the ideal gas law from Newton's laws of motion. But then on page 41-1, he puts a caveat to the derivation he has just ...
2
votes
1answer
142 views

Does a Lagrangian imply a well-defined quantum Hamiltonianian with a Hilbert space?

The question is about: (1) whether giving a Lagrangian is sufficient enough to (uniquely) well-define a Hamiltonianian quantum theory with a Hilbert space? The answer should be Yes, or No. If ...
6
votes
1answer
304 views

In Path Integrals, lagrangian or hamiltonian are fundamental?

When studying path-integrals one question arose to my mind... Which presentation is more fundamental to calculate the propagator? The one based on the Hamiltonian (phase space)? $$K(B|A) = \int ...
1
vote
0answers
75 views

How to introduce generating function in Hamiltonian formalism for field theories?

Let's have hamiltonian $$ H(\psi , P ,\partial_{i}\psi ) = P\partial_{0}\psi - L(\psi , \partial_{\mu}\psi ), \quad P = \frac{\partial L}{\partial (\partial_{0}\psi)}. \qquad (.0) $$ I tried to ...
0
votes
1answer
216 views

Working with a Routhian for a specific system

I asked a more general question earlier about the Routhian, but I'm still having trouble working with it. Here's my specific case. Given the following Lagrangian: ...
2
votes
1answer
309 views

How do you derive Lagrange's equation of motion from a Routhian?

Given a Routhian $R(r,\dot{r},\phi,p_{\phi})$, how do you derive Lagrange's equation for $r$? Do you just solve the following for $r$? $$\frac{d}{dt}\frac{∂R}{∂\dot{\phi}}-\frac{∂R}{∂\phi}=0$$ And ...
1
vote
1answer
100 views

Landau Lifshitz energy for uniform rotation

Landau Lifshitz claim in their Mechanics book (39.11) that for a uniform rotation we have $ E = \frac{mv^2}{2} - \frac{m}{2} (\omega \times r)^2 + U,$ where the rotation is given by $v' = v + \omega ...
4
votes
1answer
414 views

Why does Quantum Field Theory use Lagrangians rather than Hamiltonains? [duplicate]

Why does Quantum Field Theory use usually Lagrangians rather than Hamiltonains? I heard many reasons, but I'm not sure which is true. Some say it's just a matter of beauty, so Lagrangians are more ...
1
vote
1answer
183 views

Hamiltonian from Euclidean lagrangian?

Can somebody help me in deriving the Hamiltonian of system starting from Euclidean Lagrangian? Say we are given the Minkowski Lagrangian $$L_m = \frac{\dot{\phi}^2}{2} - V(\phi).$$ The Hamiltonian ...
0
votes
0answers
56 views

A posteriori solution to the Hamilton Jacobi equation

I was wondering about the following: For many simple systems it is far too cumbersome to solve the Hamilton Jacobi equation compared with the Hamilton or Lagrange formalism. Now I was wondering, ...
8
votes
6answers
607 views

What is Quantization?

In classical mechanics you construct an action (involving a Lagrangian in arbitrary generalized coordinates, a Hamiltonian in canonical coordinates [to make your EOM more "convenient & ...
1
vote
0answers
112 views

Single particle trajectory in a quadrupole potential

I am wondering if there are any studies of a single (classical) particle trajectory in quadrupole potential: $$ V(x,y,z)=A\sqrt[]{\frac{x^2 + y^2}{a} + \frac{z^2}{b}} $$
2
votes
1answer
183 views

Motivating the Legendre Transform Mathematically

If I begin with a functional of the form $$J[y] = \int_a^b f(x,y,y')dx$$ and find its Euler-Lagrange equations $$\frac{\partial f}{\partial y} - \frac{d}{dx}\frac{\partial f}{\partial y'} = 0 = ...
5
votes
1answer
342 views

What is Maupertuis' principle good for?

The strength of Hamilton's principle is obvious to me and I see the advantage. Now, for conservative systems we also have Maupertuis' principle that says: $$ \delta \int p dq =0$$ and I am not sure ...
5
votes
4answers
2k views

What is canonical momentum?

What does the canonical momentum $\textbf{p}=m\textbf{v}+e\textbf{A}$ mean? Is it just momentum accounting for electromagnetic effects?
2
votes
4answers
366 views

Are Lagrangians and Hamiltonians used by Engineers?

Analytical Mechanics (Lagrangian and Hamiltonian) are useful in Physics (e.g. in Quantum Mechanics) but are they also used in application, by engineers? For example, are they used in designing bridges ...
2
votes
2answers
174 views

A kind of Noether's theorem for the Hamiltonian

How can I (conveniently?) show that an invariance of the Lagrangian and Hamiltonian (i.e. the kinetic as well as the potential energy are independently invariant) will lead to a conservation law using ...
7
votes
5answers
301 views

Why can't we obtain a Hamiltonian by substituting?

This question may sound a bit dumb. Why can't we obtain the Hamiltonian of a system simply by finding $\dot{q}$ in terms of $p$ and then evaluating the Lagrangian with $\dot{q} = \dot{q}(p)$? Wouldn't ...
3
votes
2answers
182 views

Bertrand's theorem

I found in Goldstein's Classical Mechanics that the condition for closed orbits is given by $\frac{d^2 V_{eff}}{dr^2}>0$.(bertrand's theorem). Can somebody explain to me, how this inequality is ...
1
vote
2answers
182 views

Is this a valid derivation of the Legendre transformation from the Euler-Lagrange condition

E-L condition: $$\frac{d p}{dt}=\frac{\partial L}{\partial q}$$ Where $p=\frac{\partial L}{\partial \dot{q}}$ Are the following steps valid: $$\frac{\partial q}{dt} dp=\partial L$$ $$\dot{q} \: ...
1
vote
3answers
1k views

How is a Hamiltonian constructed from a Lagrangian with a Legendre transform

many textbooks tell me that Hamiltonians are constructed from Lagrangians like $$L=L(q,\dot{q})$$ with a Legendre transformation to obtain the Hamiltonian as $$H=\dot{q}\frac{\partial L}{\partial ...