Tagged Questions

The Hamiltonian formalism is a formalism in Classical Mechanics. Besides Lagrangian Mechanics, it is an effective way of reformulating classical mechanics in a simple way. Very useful in Quantum Mechanics, specifically the Heisenberg and Schrodinger formulations. Unlike Lagrangian ...

Is there an analytical mechanics with SR? Of course you can write down the Lagrangian and Hamiltonian of a free particle. What about non-free? Are there any problems? To be specific: what would the ...

It seems that when trying to identify the physical degrees of freedom for the string some authors$^1$ use:
$$ q^-=\frac{1}{\ell}\int_0^{\ell} X^-(\tau,\sigma)d\sigma$$
Then, the commutation relation ...

I am looking at Hamiltonians for specific physical situations. I have taken a given Hamiltonian $\vec{H}(\vec{p}, \vec{x})$ and have found the following Hamiltonian equations:
$$\frac{d\vec{x}}{dt} = ...

My understanding of the Jacobi energy function $h$ as defined in Goldstein is that it is the total energy $T+V$ expressed as,
\begin{equation}
h(q,\dot q,t)=\sum \frac{\partial L}{\partial \dot q}\dot ...

I am looking for a good book doing classical mechanics and statistical mechanics in terms of the Liouville operator. I have not found a lot on this subject and even books like Mathematical Methods of ...

How do we find the phase space density from the Hamiltonian?
For example: Consider a classical gas made of N identical non-interacting particles in 1d. Each molecule is characterised by centre mass ...

I am working through a problem that has caused me difficulties in the past. I have the Hamiltonian
$$\mathcal{H}=\frac{p_1^2}{2m_1}+\frac{p_2^2}{2m_2}+\frac{k}{2}(q_1-q_2)^2$$
I want to express the ...

Consider a system of $N$ identical harmonic oscillators in 1d. The Hamiltonian will be given by
$$\mathcal{H}_N=\sum_{i=1}^N \frac{p_i^2}{2m}+\frac{m\omega^2}{2}q_i^2$$
Now supposedly the Hamiltonian ...

A Lorentz covariant equation is one that takes the same form even when a Lorentz transformation is applied to each variable. Lorentz covariance is generally made manifest by writing the equation with ...

I have read different types of constraints like primary, secondary, 1st class and 2nd class. I have a little idea but not enough. wikipedia couldn't help here. It will be so nice if some one explains ...

Imagine we have a particle described by $x \in M$, where $M$ is some manifold, then it is very intuitive I think that a velocity is an element of the tangent space at $x$, so $x' \in T_{x}M.$ Thus, by ...

Apologies if this is a really basic question, but what is the physical interpretation of the Poisson bracket in classical mechanics? In particular, how should one interpret the relation between the ...

Take the ordinary Hamiltonian from non-relativistic quantum mechanics expressed in terms of the fermi fields $\psi(\mathbf{x})$ and $\psi^\dagger(\mathbf{x})$ (as derived, for example, by A. L. Fetter ...

In this question, it is discussed why, in Lagrangians we usually stick to first derivatives and quadratic terms we never see higher derivatives.
The selected answer shows that, if a Lagrangian $L(q, ...