Articles of classical mechanics

Is it possible to write the curl in terms of the infinitesimal rotation tensor?

Is it possible to write the curl in terms of the infinitesimal rotation tensor? Basically, we can write the curl as a matrix operator $$ curl=\begin{bmatrix} 0 & -\partial z & \partial y\\\partial z & 0 & -\partial x\\-\partial y & \partial x & 0\end{bmatrix} $$ and we can write the infinitesimal rotation tensor as […]

Why don't we differentiate velocity wrt position in the Lagrangian?

In Analytic Mechanics, the Lagrangian is taken to be a function of $x$ and $\dot{x}$, where $x$ stands for position and is a function of time and $\dot{x}$ is its derivative wrt time. To set my question, lets consider motion of a particle along a line: $$x: \mathbb{R} \to \mathbb{R} ~~as~~ t \mapsto x(t)$$ and […]

Given a diffeomorphism between two surfaces, is there an expression for the pullback of the covariant derivative of a vector field?

Let $A$ and $B$ be two surfaces (smooth enough) in an affine space $M$ with metric $g$. Let $g^A$, $g^B$ be the metric tensors on the two surfaces induced by $g$, and $\nabla^A$, $\nabla^B$ the Levi-Civita connections on the two surfaces. Let $f:A\rightarrow B$ be a diffeomorphism. I’m wondering if, given a vector field $X^B$ […]

How to Show Cotangent Bundles Are Not Compact Manifolds?

Hamiltonian mechanics occurs in a sympletic manifold called phase space. Lagrangian mechanics take place in the tangent bundle of the configuration manifold. Using Legendre transform makes possible to pass to Hamiltonian formulation, because this transform allows the construction of the cotangent bundle which is a sympletic manifold. Lagrangian formulation is a sub-set of Hamiltonian formulation, […]

How to introduce stress tensor on manifolds?

I want to understand the type of stress tensor $\mathbf{P}$ in classical physics. Usually in physics it is said that the force $\text d \boldsymbol F$ (vector) acting on an infinitesimal area $\text d \boldsymbol s$ (vector) equals $\text d \boldsymbol F = \mathbf{P} \cdot \text d \boldsymbol s$ where $\cdot$ is a “scalar product”. […]

Euler-Lagrange equations of the Lagrangian related to Maxwell's equations

Clarification on Lagrangian mechanics would be much appreciated: Suppose $$L(\phi,\,\,\phi_{,i},\,\,A_i, \dot A_i)=|\dot A+\nabla\phi|^2-|\nabla \times A|^2-c\phi+d\cdot A$$ Are the corresponding Euler-Lagrange equations then: $$c=0$$ by considering $\phi$, and $$2(\dot\phi_{,i}+\ddot A_i)+d_i=0$$ by considering $A_i$? I am confused by the dependent variables in this Lagrangian — they are differentiated wrt to different variables, namely $\phi$ wrt spatial elements, […]

Envelope of Projectile Trajectories

For a given launch velocity $v$ and launch angle $\theta$, the trajectory of a projectile may be described by the standard formula $$y=x\tan\theta-\frac {gx^2}{2v^2}\sec^2\theta$$ For different values of $\theta$ what is the envelope of the different trajectories? Is it a parabola itself? The standard solution to this “envelope of safety” problem is to state the […]

Tensors in the context of engineering mechanics: can they be explained in an intuitive way?

I’ve spent a few weeks scouring the internet for a an explanation of tensors in the context of engineering mechanics. You know, the ones every engineering student know and love (stress, strain, etc.). But I cannot find any explanations of tensors without running into abstract formalisms like “homomorphisms” and “inner product spaces”. I’m not looking […]

Car movement – differential geometry interpretation

The problem presented below is from my differential geometry course. The initial reference is Nelson, Tensor Analysis 1967. The car is modelled as follows: Denote by $C(x,y)$ the center of the back wheel line, $\theta$ the angle of the direction of the car with the horizontal direction, $\phi$ the angle made by the front wheels […]

Why isn't the 3 body problem solvable?

I’m new to this “integrable system” stuff, but from what I’ve read, if there are as many linearly independent constants of motion that are compatible with respect to the poisson brackets as degrees of freedom, then the system is solvable in terms of elementary functions. Is this correct? I get that for each linearly independent […]