# Tag Info

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### 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 ...
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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 ...
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### 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 ...
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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 $$\frac{d}{dt}\left(\dot q_1 + \dot q_2\right) = -2kq_1^3$$ $$\frac{d}{dt}\left(\dot q_2 + \dot q_1\right) = -2kq_2^3$$ ...

### 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 ...
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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 ...

### 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 ...

### 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 $t$,...

### 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 ...
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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} \...

### 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 ...
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### 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 ...

### 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 ...
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### 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 ...
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 ...