43 votes
Accepted

What are holonomic and non-holonomic constraints?

If you have a mechanical system with $N$ particles, you'd technically need $n = 3N$ coordinates to describe it completely. But often it is possible to express one coordinate in terms of others: for ...
DK2AX's user avatar
  • 4,759
25 votes
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Hamiltonian for relativistic free particle is zero

...what I would like to know is why we get a zero Hamiltonian. I suspect that this is due to the reparametrization invariance... Will this always happen? Why? Yes, it is due to reparameterization ...
Chiral Anomaly's user avatar
19 votes

What are holonomic and non-holonomic constraints?

The question has been well-answered several times. I'll just add some geometrical context. In geometry, the holonomy group of a connection is the set of transformations an object can experience when ...
knzhou's user avatar
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17 votes

Confusion with Virtual Displacement

Here on SE, you may already find many answers to your question. Even if most of them are correct, I feel that a plain and correct answer is still missing. Where plain does not mean non-rigorous. But ...
GiorgioP-DoomsdayClockIsAt-90's user avatar
17 votes
Accepted

Compute the Legendre transform for a singular Lagrangian

As a quick note, the equations of motion that come from that Lagrangian are $$\frac{d}{dt}\left(\dot q_1 + \dot q_2\right) = -2kq_1^3$$ $$\frac{d}{dt}\left(\dot q_2 + \dot q_1\right) = -2kq_2^3$$ ...
J. Murray's user avatar
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13 votes

Why is the Hamiltonian zero in relativity?

The correct argument for how reparametrization invariance implies vanishing Hamiltonian is as follows: Given a generic Hamiltonian action $$ S = \int \left(\dot{q}^i p_i - u^\alpha \chi_\alpha - H\...
ACuriousMind's user avatar
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13 votes
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Why are Hamiltonian Mechanics well-defined?

This is a good but quite broad question. Let us suppress position dependence $q^i$, $i\in\{1, \ldots, n\}$, and explicit time dependence $t$ in the following to keep the notation simple. Given a ...
Qmechanic's user avatar
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12 votes
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What is the precise relationship between a non-invertible Hessian matrix for the Lagrangian and the presence of a gauge symmetry?

These conditions are not equivalent, only under several assumptions. A good reference are chapters 1 and 3 of Henneaux' and Teitelboim's "Quantization of gauge systems". The "proper" definition of a ...
ACuriousMind's user avatar
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12 votes
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How many degrees of freedom in a massless $2$-form field?

It is natural to generalize to an Abelian $p$-form gauge field $$A~=~\frac{1}{p!} A_{\mu_1\mu_2\ldots\mu_p} \mathrm{d}x^{\mu_1}\wedge\ldots\wedge \mathrm{d}x^{\mu_p}\tag{1}$$ with $\begin{pmatrix} D ...
Qmechanic's user avatar
  • 200k
11 votes

How to find Hamiltonian from this simple Lagrangian? (tricky)

Legendre transformation. OP's question contains some subtle aspects that we would like to illuminate. Note that a Legendre transformation (singular or not) should be an involutive operation. (It would ...
Qmechanic's user avatar
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11 votes
Accepted

Is the magnetic Lorentz force $\vec{F} = q(\vec{v}\times\vec{B})$ a force of constraint?

No, it isn't. When you talk about a constraint force, you're talking about a force which constrains the motion of a particle to a particular spatial region, such as a curve or a surface. The Lorentz ...
J. Murray's user avatar
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10 votes

Hamilton's principle with nonholonomic constraints in Goldstein

TL;DR: Note that the treatment of Lagrange equations for non-holonomic constraints in Refs. 1 & 2 is inconsistent with Newton's laws, and has been retracted on the errata homepage for Ref. 2. See ...
Qmechanic's user avatar
  • 200k
9 votes

Hamiltonian for relativistic free particle is zero

Here's another way: Suppose that your lagrangian has the following property, for any $\theta$ (it could be a function of time $t$): \begin{equation}\tag{1} L(q, \, \theta \, \dot{q}) = \theta \, L(q, ...
Cham's user avatar
  • 7,183
9 votes

Are constraint forces infinite?

OP already seems to have thought long and hard about this and makes good points. In this answer we will review the argument for why constraint forces could be infinite. We will assume that OP talks ...
Qmechanic's user avatar
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8 votes
Accepted

Free body diagram of block on accelerating wedge

Rather than answer your individual questions I will give you an overview and then discuss some of the points that you have raised. There are many ways of tackling such problems but drawing a few FBDs ...
Farcher's user avatar
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8 votes

How to find Hamiltonian from this simple Lagrangian? (tricky)

The Hamiltonian is undefined. Converting a Lagrangian to a Hamiltonian requires: Finding $p$ Writing $H=p\dot q-L$ Expressing $H$ in terms of $p$ and $q$, eliminating all dependence on $\dot q$. For ...
Jahan Claes's user avatar
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8 votes
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What is wrong with my argument to derive the Hamiltonian in relativity?

The problem is that the Legendre transformation from 4-velocity to 4-momentum is singular: The 4 components of the 4-momentum $p_{\mu}$ are constrained to live on the mass-shell $$p_{\mu}g^{\mu\nu}p_{\...
Qmechanic's user avatar
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8 votes
Accepted

Finding generalized coordinates when the implicit function theorem fails

Generally speaking, given a set of coordinates $x_1,\ldots,x_N$ under a set of $h=N-n$ holonomic constraints of the form $F_j(x_1,\ldots,x_N)=0$, you won't be able to find a subset $x_1,\ldots,x_n$ of ...
Emilio Pisanty's user avatar
8 votes
Accepted

Matrix derivative of a matrix with constraints

Setup. Let there be given an $m$-dimensional manifold $M$ with coordinates $(x^1, \ldots, x^m)$. Let there be given an $n$-dimensional physical submanifold $N$ with physical coordinates $(y^1, \ldots, ...
Qmechanic's user avatar
  • 200k
8 votes

What are holonomic and non-holonomic constraints?

A holonomic constraint is a constraint that places a definite relationship between the coordinates you're using. For example, consider a cylinder of radius $R$ rolling along a table in 1-D. The ...
Michael Seifert's user avatar
8 votes

What are holonomic and non-holonomic constraints?

For completeness: There is also a notion of semi-holonomic constraints. Recall that a holonomic constraint$^1$ $$f(q,t)~=~0\tag{H}$$ only depends on the generalized coordinates$^2$ $q^j$ and time $t$,...
Qmechanic's user avatar
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8 votes

Connection between different kinds of "Lagrangian"

The first two definitions of the Lagrangian are not equivalent but very closely related in the sense that one is a generalization of the other. The second one may be a non-definition, but that can be ...
Bence Racskó's user avatar
8 votes
Accepted

Why are first class constraints harder to quantize than second class constraints?

First-class constraints generate gauge transformations (assuming the Dirac conjecture), i.e. map physically equivalent states onto each other. Even if you do not assume the Dirac conjecture, then ...
ACuriousMind's user avatar
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8 votes
Accepted

Hamiltonian systems without a corresponding Lagrangian system

The Lagrangian can be constructed directly by performing a Dirac-Bergmann constraint analysis of OP's Hamiltonian (1). In eq. (3) OP has already correctly identified the primary constraint$^1$ $$\dot{...
Qmechanic's user avatar
  • 200k
8 votes

Are constraint forces infinite?

I think there is no single answer to this question; it depends on circumstances. If you have a three-dimensional region of space, with a sphere sitting in it, for example, then you might have a ...
Andrew Steane's user avatar
7 votes
Accepted

What is the position as a function of time for a mass falling down a cycloid curve?

1. Brachistochrone \begin{equation} \boxed{\: \begin{matrix} x\left(\theta\right) = R\left(\theta-\sin \theta\right)\\ y\left(\theta\right) = R\left( 1-\cos \theta\right) \end{matrix}\:} \tag{b-01} \...
Frobenius's user avatar
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7 votes
Accepted

Inconsistency in Lagrangian vs Hamiltonian formalism?

The problem here is that, because there exist constraints of the form $f(q,\,p)=0$, the phase space coordinates of the usual Hamiltonian formulation aren't independent. I'm not sure how you ...
J.G.'s user avatar
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7 votes
Accepted

(Anti)commutation of ghosts and fermions

It is not completely clear what OP is looking for, but here are some hopefully helpful comments: Classically (meaning when Planck constant $\hbar\to 0$), two fields $A$ and $B$ are super-commutative ...
Qmechanic's user avatar
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7 votes
Accepted

What does "degrees of freedom " mean in classical mechanics?

Degrees of freedom can be defined as the number of independent ways in which the space configuration of a mechanical system may change. Suppose I place an ant on a table with the restriction that the ...
drvrm's user avatar
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7 votes
Accepted

Confusion with Virtual Displacement

Let there be given a manifold $3N$-dimensional position manifold $M$ with coordinates $({\bf r}_1, \ldots, {\bf r}_N)$. Let the time axis $\mathbb{R}$ have coordinate $t$. Let there be given $m\leq 3N$...
Qmechanic's user avatar
  • 200k

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