# Tag Info

### Constraint equation for an elastic pendulum

If you need to evaluate the reaction at the constraint with a Lagrangian approach, you need to add the constraint in the computation of the stationary point of the action. Let's call $P$ the mass ...

### Constraint equation for an elastic pendulum

The elastic pendulum have two generalized coordinate $~x,y~$ from here the constraint equation $$\tan(\theta)=\frac xy\quad,\omega\,t=\arctan(x/y)$$ and with $~x=r\,\sin(\theta),y=r\,\cos(\theta)$ you ...

### "Natural frequency" seems to be a poorly defined concept

The Wikipedia definition of natural frequency is a poor one, since (as you point out) it implies that a system can only oscillate at its natural frequency if there is no driving force. This is ...
1 vote

### "Natural frequency" seems to be a poorly defined concept

The problem with an eigen-frequency is that it is exact, and the phase space to hit an exact frequency is 0, but if you do hit it, the response diverges...classically. In quantum mechanics, the eigen ...

Accepted

### Solving forced harmonic oscillator differential equation using fourier transform

It is often helpful to add a regularization factor $e^{-\epsilon \left| t \right|}$ to the integrands in the Fourier transformation to ensure convergence. You then take the limit $\epsilon \to 0$ ...
1 vote

### Solving forced harmonic oscillator differential equation using fourier transform

Without any claiming to be a complete answer with all the details you need, you could start using the (unilateral) Laplace transform \begin{equation} F(s) := \int_{t=0}^{\infty} f(t) e^{-st} dt \ . \...

Accepted

### Some books write $V(\vec{r})$ instead of $V(r)$ as a notation for the electric potential, so which one is right?

In general, electric potential is a function of the vector position. Sometimes the vector notation is reduced to a scalar due to spherical symmetry.
Accepted

### Confusion of variable vs path in Euler-Lagrange equation, Hamiltonian mechanics, and Lagrangian mechanics

The Lagrangian is a function, not a functional. The action is a functional, and is defined as $$S[q; t_0,t_1] = \int_{t_0}^{t_1} L\big(q(t), \dot q(t), t\big) \mathrm dt$$ The partial derivatives ...

### Does work depend on a point of reference?

I am so fit that my mass is 0.5kg, so when a force of 1N in the direction of my movement acts on me for 1s, I accelerate to 1002m/s and in that time I travel 1001m (all relative to Earth). All this ...
1 vote

### The Hamilton's equations of a charged particle in electromagnetic field

Here: let's do it a little bit differently that will help make things more clear. For a charged body, with charge $e$, the potential energy $eφ$ has the same relation to the body's energy $E$, as $e𝐀$...

### Why does a higher frequency mechanical wave have more energy?

I’m not really familiar with the mathematics of this, so I’ll propose a simple answer: First we need to look at the definition of a mechanical wave. A mechanical wave is a vertical transfer of energy ...

### Why does a higher frequency mechanical wave have more energy?

Wave equation in classical mechanics. Sometimes it's possible to recast the governing equation of an elastic medium as a wave equation, $\dfrac{1}{c^2} \partial_{tt} u - \nabla^2 u = f$, being $u$ the ...
In general, the lagrangian of a simple pendulum of longitude $l$ that moves in radial and angular directions is: \mathcal{L}(r, \theta) = \frac{1}{2}m(\dot{r}^2 + r\dot{\theta}^2) - mgr(1-\cos(\...