39
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 ...
- 4,679
19
votes
Accepted
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 ...
- 51.1k
17
votes
Accepted
What exactly is a virtual displacement in classical mechanics?
Let $Q$ denote the set of all possible configurations of the system (the configuration manifold). Consider a point $q_0\in Q$. For the sake of conceptual clarity, and to make contact with physics ...
- 54.4k
17
votes
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Compute the Legendre transform for a singular Lagrangian
As a quick note, the equations of motion that come from that Lagrangian are
$$\frac{d}{dt}\left(\dot q_1 + \dot q_2\right) = -2kq_1^3$$
$$\frac{d}{dt}\left(\dot q_2 + \dot q_1\right) = -2kq_2^3$$
...
- 51.8k
16
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 ...
- 95.3k
12
votes
Accepted
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 ...
- 170k
12
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 ...
- 23.6k
11
votes
How to find the rank of the matrix $\frac{\partial ^2\mathcal{L}}{\partial \dot{X^\mu} \partial \dot{X^\nu} }$ for the Nambu-Goto string Lagrangian?
I) In this answer we will consider the standard Nambu-Goto (NG) string and show that the Hessian has co-rank 2. The target space (TS) metric $G_{\mu\nu}(X)$ has sign convention $(-,+,\ldots,+)$, and $...
- 170k
11
votes
Reduction of Nambu-Goto action to true degrees of freedom
Here is an outline of the reduction from the Nambu-Goto (NG) action to the light-cone (LC) formulation from a Hamiltonian perspective:
The starting point is the Hamiltonian formulation of the NG ...
- 170k
11
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
$$ S = \int \left(\dot{q}^i p_i - u^\alpha \chi_\alpha - H\...
- 107k
11
votes
Accepted
How many degrees of freedom in a massless $2$-form field?
It is natural to generalize to an Abelian $p$-form gauge field
$$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 ...
- 170k
10
votes
Accepted
Hamiltonian from a Lagrangian with constraints?
Comments to the question (v2):
To go from the Lagrangian to the Hamiltonian formalism, one should perform a (possible singular) Legendre transformation. Traditionally this is done via the Dirac-...
- 170k
10
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 ...
- 170k
9
votes
Accepted
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 ...
- 107k
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, ...
- 6,517
9
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 ...
- 47.1k
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 ...
- 170k
8
votes
Accepted
Why are D'Alembert's Principle and the Principle of Least Action Related?
The principle of Least (Stationary) Action (aka Hamilton's Principle) is derived from Newton's axioms plus D'Alembert's principle of virtual displacements.
Because D'Alembert's principle allows to ...
- 4,866
8
votes
Accepted
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_{\...
- 170k
8
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 ...
- 22k
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 $(y^1, \ldots, ...
- 170k
8
votes
What are holonomic and non-holonomic constraints?
For completeness: There is also a notion of semi-holonomic constraints.
Recall that a holonomic constraint$^1$
$$f(q,t)~=~0\tag{H}$$
only depends on the generalized coordinates$^2$ $q^j$ and time $t$,...
- 170k
7
votes
Do primary first class constraints change the electric field in the Hamiltonian form of Maxwell's theory?
The problem lies in what we learn about good old constrained dynamics from traditional Dirac approach is not complete and is somehow inconsistent, and the above is one example of this. This was the ...
- 81
7
votes
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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}
\...
- 13.4k
7
votes
How to find the rank of the matrix $\frac{\partial ^2\mathcal{L}}{\partial \dot{X^\mu} \partial \dot{X^\nu} }$ for the Nambu-Goto string Lagrangian?
I) In this alternative answer we resolve the singular Hessian $H_{\mu\nu}$ of the Nambu-Goto string action by introducing two auxiliary variables from the onset, thereby indirectly showing that the ...
- 170k
7
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 ...
- 77.9k
7
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 ...
- 7,581
7
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 ...
- 124k
7
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 ...
- 38.3k
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 ...
- 1,387
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