Questions tagged [classical-mechanics]
For questions on classical mechanics from a mathematical standpoint. This tag should not be the sole tag on a question.
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Proof of Identity for Lagrangians
My textbook states (in a derivation of the Euler-Lagrange equation) the following which I would like to understand:
\begin{equation*}
S[\bar{x}+\delta x] =\int_{t_a}^{t_b} L(\dot{x}+\delta \dot{x}, x+\...
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How to prove the brachistochrone to be Isochronous
When I was reading the calculus of Variations chapter in Classical Mechanics by John R. Taylor, he mentioned that cart rolling back and forth on cycloid-shaped track are exactly isochronous (period ...
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I need a solution to this double integral. [closed]
I am stuck on this problem in my thesis research. I need to solve this double integral:
$$T(x,t)=\frac{i}{\pi}\int_{0}^{t}\int_{-\infty}^{\infty}{e^{-i\xi x}\sin{(\xi t_{0})}\sum_{k=0}^{\infty} \frac{(...
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Simple Kinematics (v-t graph) [closed]
A particle moves in a straight line. Its velocity–time graph is shown.
a. After 165 seconds the particle returns to the starting point. Find the value of T.
b. At what time does the particle have ...
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Question in Gelfand & Fomin's derivation of Hamilton's equations
I'm having some difficulty interpreting the mathematical underpinnings of this derivation given in chapter 4, section 16.
We have $F(x, y_1,...,y_n, y'_1,...,y'_n), p_i=F_{y'_i}$, and the familiar ...
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Find time at which a certain distance occurs between moving objects
I have two objects moving away in space whose positions given by $r_1$ and $r_2$, their velocities $v_1$ and $v_2$ and relative vectors R=$r_2$-$r_1$ and V=$v_2$-$v_1$. How to find the time at which ...
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Rigid bodies: proof of existence of internal forces that preserve the distances
I am new to Physics and I have a pure Math background. I am currently studying mechanics and I have the following question regarding rigid bodies. I am posting here the 2D version of the question. If ...
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The "same old" dy/dx question? Separation of differentials?
I completely understand this question has been addressed one too many times. But I still simply cannot wrap my head around the concept of dy/dx.
Simply put, when can we treat dy/dx as a ratio and when ...
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Reference request for texts that treat classical mechanics in a mathematical manner
I am looking for books and papers that treat the physics theory of classical mechanics in a rigorous, mathematical manner. I would love to read such a text. Are there even such texts out there? I have ...
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Is there a rigorous textbook on step by step development for coming up with the equations of motion of classical dynamical systems?
I was trying to find some references for modelling the equations of motion of a simple dynamical system (say a pendulum on a moving mass) when I realized that the very vast majority of the material ...
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Where does this factor of $\pi$ come from in the period of small oscillations about equilibrium points?
I am working through some exercises in Arnold's Mathematical Methods of Classical Mechanics book, specifically the second problem on page 20. For context, $T(E)$ is the period of motion along a closed ...
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A basic question involving decomposition of forces
I'm working on an unassessed course problem containing this diagram:
The solution booklet uses $g\sin(\pi/6)=g/2$ for the component of gravity acting in the $x$ direction (which I'll denote $g_x$). ...
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What properties does the flow of a Hamiltonian vector field have compared to the flow of a symplectic vector field?
Let $(P,\omega)$ be some real $2n$-dim symplectic manifold. A symplectic vector field, $Z\in\mathfrak{X}_{sp}(P)$, is one for which the Lie derivative satisfies $\mathcal{L}_Z \omega =0$ or, ...
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Riemannian Metric from Hamiltonian?
Suppose I have a robot arm configuration space (c-space) $Q$ and with Lagrangian $L = T - V$ on $TQ$, and I set up the equation of motion
$$\frac{\partial}{\partial t}\left\{\frac{\partial L}{\partial ...
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Question about partial derivative in the proof of the Poisson bracket of the Hamiltonian and a function of its flow
Let $M$ be an open set in $\mathbb{R}^d \times \mathbb{R}^d$ such that the flow of the Hamiltonian maps $M$ to itself and $G_t$ be the flow map of the Hamiltonian i.e. $G_t(x_0,p_0) = (x(t),p(t))$ if $...
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Prove that the average velocity over an interval is the average of instantaneous velocities without calculus
I am re-reading my old first semester Physics textbook and came across the well known formula that for a particle in constant acceleration $a_x$ from time $0$ to time $t$ $$v_{av-x}=\frac{1}{2}\left(...
user689775
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Preserving the symplectic 2-form vs phase space volume
Say I have a Hamiltonian system of $N$ particles in 3D-3V phase space. I'm using some sort of update scheme taking the system from $t^{n-1}$ to $t^{n}$ to $t^{n+1}$. I want to know if the update ...
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The use of subscripts in L11 E04 - Classical Mechanics by Leonard Susskind
I am struggling to get my head around the soultion to exercise 4 lecture 11 in The Theoretical Minimum (Classical Mechanics).
The exercise and its solution can be found here: https://tales.mbivert.com/...
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Clarification on how a system of differential equations is defined on Arnold's Mathematical Methods of Classical Mechanics
Definition: given a function $f:\mathbb{R}^n\to\mathbb{R}^m$, the partial derivative of $f$ at $a\in\mathbb{R}^n$ in the direction $v\in\mathbb{R}^n$ is defined as
$$\lim_{t\to 0}\frac{f(a+tv)-f(a)}{t}...
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What assumption of Noether's theorem fails in this Hamiltonian system with infinitely many particles?
A consequence of Noether's theorem is that the energy of a Hamiltonian system is conserved if and only if the Hamiltonian is time-translation invariant. However, to my surprise I found some ...
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Newton's method: partial or total derivative for composed functions?
Suppose I have the following Jacobian
$J_{ij}(\underline{\Delta\sigma})=\displaystyle\frac{\partial \Delta\varepsilon_i}{\partial \Delta\sigma_j} \in \mathbb{R}^{6\times6}$
within a Newton-Raphson ...
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Pendulum with Oscillating Fulcrum with Newton's Laws?
I was given an honors project to solve for the equations of motions of a pendulum with an oscillating fulcrum. I (somewhat) understand the procedure on how to solve it with lagrangian mechanics and ...
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A question regarding partial derivative in Mechanics
I am reading Mechanics: From Newton's Laws to Deterministic Chaos by Florian Scheck Chap2 The Principles of Canonical Mechanics
Right now I have some doubts regarding the derivation process which ...
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Derivative of a curve on tangent bundle $TQ$ and second-order equations (where is the acceleration?)
This is very basic question that I should have resolved long ago but didn't and it still plagues me.
Let $TQ$ be the tangent bundle of some (configuration) manifold, $Q$, and let $(\pmb{q},\pmb{v}) = (...
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Prove that if air resistance is proportional to velocity, then impact speed will be lower than projection speed.
The full question is:
A particle of mass m is projected vertically upwards through a resistive medium with initial speed $u$. It experiences a resistive force of $mkv$. and a weight force $mg$. When ...
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unknown manipulation to deal with variation and obtain energy balance equation?
I'm studying a paper relevant to using d'Alembert's principle to describe the motion of fluid. The authors shows an interesting manipulation to obtain energy balance equation, which makes me confused. ...
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How to model a ball moving on the graph of $z=f(x,y)$ by gravity action? [closed]
I want to model the movement of a ball constrained on the graph $z = f(x, y)$ (for instance $z = \cos(x^2 + y^2)$ by the action of gravity only (no friction or drag and no air resistance) do you have ...
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What is the differential on the space of trajectories and why does it anti-commute with the regular differential on $\mathbb{R}$?
From Classical Field Theory by Deligne and Freed, page 143 (page 6 in this pdf) $x$ is a map from $\mathbb{R}$ to $X$ which is Euclidean space with standard inner product $\langle \cdot, \cdot \...
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Conditions for a vector field to be complete.
If we consider in a symplectic N-dimensional manifold $(M,\omega)$ the infinitely differentiable functions at all points of $M$ denoted by $C^\infty(M,\mathbb{R})$, and a Hamiltonian vector field ...
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Finding the equation of motion based on the diagram and its transfer function
I have a diagram that looks like below:
Description: By applying force p(t), point A is moving with displacement $y = asin𝜔t$. $k_1$ and $k_2$ stand for the coefficients of the corresponding spring, ...
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How many objects can I perceive to all be travelling at speed arbitrarily close to $\sqrt 2\cdot c$ relative to each other?
Classically, space is a $\Bbb R^3$ manifold. According to relativity, spacetime is $4$d although I understand this is Minkowski space. The thing I was curious about, is whether the two theories ...
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How to compare alignment between a normal vector and inertia tensor
I'm working with data of biological cells, and we want to measure how the shape of a cell aligns with the normal vector of its surrounding tissue (where the cell is at the boundary).
One idea was to ...
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Can we recover dynamics from coisotropic reduction?
I am trying to understand the significance of coisotropic reduction in a mechanical way.
Let $(M,\omega)$ be a symplectic manifold and $i: N \hookrightarrow M$ a coisotropic submanifold i.e. $T_qN^{\...
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Using trigonmetry to compute valid configuration(s) of a 2D four bar linkage given a specific constraint
I'm trying to find an analytical solution to an inverse kinematics problem for a software project, and after simplifying things as much as possible I've come up with the following problem statement.
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Modelling a differential equation for a practical problem in fluid mechanics
I have been working on modeling the time taken to empty a cylindrical tank with a sealed top using a moving lid.
The lid, in the form of a disk, moves down as the water level decreases. The discharge ...
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Derivation of Spatial Delaunay Elements
I am trying to calculate the Spatial Delaunay Elements with actions $(L, G, H)$ and angles $(l, g, h)$ from polar coordinate $\left( \rho, \theta, \phi\right)$ and momentum $\left( P, \Theta, \Phi\...
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What is a dispersion relation?
What exactly is a dispersion relation?
It shows up all the time in my physics classes and I never fully understood its meaning.
Currently I am working on the linear chain model in crytsallography and ...
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Integrating the velocity vector
If we integrate the velocity vector with respect to time, do we get the displacement vector or the position vector?
What I mean is that, do we integrate the velocity vector to get the displacement ...
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Exterior Derivative and Lie Derivative on infinite dimensional manifolds
Lately I have been trying to understand the chapter in Abraham and Marsden's Foundations of Mechanics on infinite-dimensional Hamiltonian systems. Now that I've finally got a feeling for the canonical ...
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Pullback of vector fields in mechanics
I am studying article On the Lagrange-Dirichlet converse in dimension three, by Burgos. I am trying to understand this Lemma 2.13, more specifically, understanding the lemma and where it came from, as ...
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Motion of a Particle Newton's Second Law
There exists a particle of mass $m$ acted on by a force $F = kt^2$. $k$ is a constant and $t$ is time. The particle starts at $x=0$ with constant velocity $u$. Find the acceleration, velocity and ...
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Determining the polar moment of an inertia for a non-uniform circular cross section
I know the integral for determining the polar moment of inertia for a given geometry is given by:
$$J=\iint R^2dA$$
For the attached geometry there is a circular cross section with distinct areas. I ...
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Find the angle of a pendulum given the time
You are given a pendulum with the distance $d$ from the anchor point to the center of the massive sphere, radius $r$ and mass $m$. Gravity is given by $g$.
Pendulum
I want to derive a formula for the ...
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Gravitational Attraction Central Force
I understand that a central force is one always pointing to the same fixed point, in the same direction (parallel) as the position vector $\mathbf r$, and whose magnitude only varies depending on the ...
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Lagrangian field theories define a canonical $\mathbb R$-torsor/$\mathbb R$-bundle
Why is it possible to define an $\mathbb R$-torsor/$\mathbb R$-bundle out of the Lagrangian and to construct a projective limit? (I'm guessing this has to do with time evolution acting on the space of ...
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Motion in a vertical circle (Mechanics / Physics) solution error?
I wonder if someone would be kind enough to check a solution that I feel contains an error?
Question
A small bead, of mass $m$, is threaded on a smooth circular wire, with centre $O$ and radius $a$, ...
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Necessary condition for a curve to provide a weak extremum(calculus of variation)
Necessary condition for a curve to provide a weak extremum.
Let $x(t)$ be the extremum curve.
Let $x=x(t,u) = x(t) + u\eta(t)$ be the curve with variation in the neighbourhood of $(\varepsilon,\...
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Why only "small" variation considered in the formulation of Euler Lagrange equations?
For the Euler Lagrange equation formulation, they only use a "small" variation $\varepsilon$ for
\begin{equation}
y(x,\varepsilon) = y(x) + \varepsilon\eta(x)
\end{equation}
where $y(x)$ is ...
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How to solve the equation to plot the deformation curve of the actuator?
Thanks for reading this.
I ask the question from the following research article (https://pubs.rsc.org/en/content/articlelanding/2019/sm/c9sm01672d#:~:text=When%20the%20magnetic%20field%20strength,...
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5 balanced equal masses on a circumference with varying distances between them
I'm looking for a way to calculate possible balanced positions of 5 equal masses on the circumference of a spinning wheel such that the distance along the circumference between the masses are all ...
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