Questions relating to Newton's Laws of Motion

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Relation between Poisson bracket and commutator.

In quantum case, we have commutators. In classical case, we have Poisson bracket. Let $T$ be a Poisson group, $a, b \in \mathbb{C}_q[T].$ It seems that we have $$ [a, b]=q\{a,b\}+o((q-1)^2). $$ Is ...
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1answer
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Expression for Hamiltonian vector field!

How does one prove that the Hamiltonian vector field has the following expression, what is the reasoning? \begin{equation} X_H=\sum ^n_{i=1}\frac{\partial H}{\partial q_i}\frac{\partial }{\partial ...
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Computing the angular momentum in spherical coordinates [migrated]

How to compute the angular momentum of a particle in spherical coordinates? It's given by: $$x_1=r\cdot\cos(\phi)\cdot\sin(\theta)$$ $$x_2=r\cdot\sin(\phi)\cdot\sin(\theta)$$ ...
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Angular momentum in Cylindrical Coordinates

How to calculate the angular momentum of a particle in a cylindrical coordinates system $$x_1 = r \cos{\theta}$$ $$x_2 = r \sin{\theta}$$ $$x_3 = z$$ Thanks.
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+50

Classical perturbation theory + KAM theory

In classical canonical perturbation theory of many degrees of freedom we encounter the problem of small divisors when attempting to find a solution for the generating function of the canonical ...
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22 views

Stopping distance of a car on inclined surface

I am making a simulator of car stopping distances in various conditions for a computing project. My math knowledge of this subject is quite limited and i was hoping for a equation or formula to use to ...
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32 views

Applying linear algebra to solve a problem in mechanical equilibrium

I came across the following problem in "Introduction to Applied Mechanics" by Gilbert Strang, and am a little confused about the solution to this problem. The following figure shows the problem. ...
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44 views

What is the formal mathematical representation of a “force”?

In mechanics, it is usual to represent a force by a 3-vector. When it is necessary to consider the turning effect of a force, the 3-vector is commonly "attached" to a point on its line of action. In ...
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30 views

Lagrangian and Hamiltonian Mechanics

I am interested in how Lagrangian and Hamiltonian mechanics and then symplectic geometry was developed starting from Newtonian mechanics. We can start by assuming that Newtonian mechanics tells us ...
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21 views

Invariants under Hamiltonian mechanics?

I am interested in certain properties of measures evolving according to Hamiltonian mechanics. Say we have a point $z$ in phase space: $z = (p,q)$ where $p$ is a generalized momentum vector and $q$ is ...
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71 views

Cambridge Mathematical Tripos Question - 1871

This is a question from the Cambridge Mathematical Tripos in 1871, Scanned copy added at the end of the post. A ship $A$ sees another ship $B$ whose course is not known. Given that they have ...
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1answer
72 views

Solving Collision Problem (Momentum conserved) Systematically in more than 2 dimensions

I know that the equations for conservation of momentum and energy $m_1v_{i1}+m_2v_{i2} = m_1v_{f1}+m_2v_{f2},\;\frac{1}{2}\epsilon(m_1v_{i1}^2+m_2v_{i2}^2) = \frac{1}{2}(m_1v_{f1}^2+m_2v_{f2}^2)$, ...
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1answer
56 views

Help with a 3-body problem

If I have three particles with masses $ m_1, m_2, m_3$ with their respective position vectors $ x_1, x _2, x_3 $ and their speeds $ v_1, v_2, v_3 $ how could I find a parametric function that would ...
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2answers
83 views

Differential Geometry and classical mechanics basics.

This is so far my understanding of classical mechanics, Interspaced are a few questions where I am still not entirely sure what is going on. Thank you for your help !! A tangent space can be thought ...
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1answer
45 views

Show that particle whose position satisfies $\frac{d \mathbf{r}}{dt}= \mathbf{c} \times \mathbf{r}$ moves in a circular path with constant speed

A particle P moves so that its position vector r at time $t$ satisfies the differential equation $$\frac{d \mathbf{r}}{dt}= \mathbf{c} \times \mathbf{r},$$ where c is a constant vector. ...
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37 views

Lax Pairs and constant eigenvalues

Can someone tell me whether the following is true, and if so a hint the proof? If we have a Lax Pair $\dot{L} = [A,L]$ then the eigenvalues of $L$ are constants of the motion. (The opposite ...
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1answer
40 views

Vector spaces and bundles in classical mechanics

My understanding so far is that if we have a manifold of coordinates, the velocity field is more generally called a tangent space $T_xQ$, further the space described by position and momentum is a ...
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1answer
30 views

Vector space and form?

What is the significance of equipping the vector space with a form? For instance a symplectic space has a symlectic two form? Why does it need it + what does having it allow/benifit us?
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1answer
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mechanics piston problem involving rotational motion.

The above figure shows a piston driving a crank OP pivoted at the end $O$. The piston slides in a straight cylinder and the crank is made to rotate with constant angular velocity $ \omega $. Find ...
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representation of Eulers's equation in biharmonic form

As we know the Euler's equation $${\rm div}{\rm div}(\frac{\nabla^2F}{\|\nabla^2F\|})=0$$ Can be written in biharmonic equation form $$\Delta^2F+ (something)=0$$ I want to know in the context of solid ...
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1answer
36 views

Simple Forces - Finding Force and Tension for self learner

Problem A particle of mass 4kg is suspended from a point A on a vertical wall by means of a light inextensible string of length 130cm. a) A horizontal force, P is applied to the particle so ...
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Balance of forces in a mechanics problem

I tried to solve a particular problem of mechanics and found some difficulties in the vector analysis part that I can't get rid of. It's probably some stupid mistake I made, but I can't see it now, ...
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3answers
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Basic problem in Mechanics

A body starts from rest and builds up to a velocity of 7.2km/hr in half a minute. A) What is the acceleration? B) What distance is travelled in the half minute? Apparently the answer is ...
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21 views

At how much horizontal distance the body will strike the ground?

A very smooth plane of length 40m is inclined to horizon at an angle 30 degree . From the foot of this plain , a body starts from rest and moves with an acceleration $10m/s^2$ . After moving a ...
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1answer
24 views

Determine coefficients so that $f(x,y)$ is a conserved quantity

Given the function $f(x,y) = ax^2y + by + ct^2 $ with $x(t)=2t$ and $y(t)=3t^2$ I shall determine the coefficients a,b,c so that $f(x,y)$ is a conserved quantity. My approach is $$\frac{df}{dt} = ...
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Why does this graph produce a straight line? [duplicate]

When we graph the sin and cos of theta against the range of a projectile, we get a straight line. When we graph range against angle, we get a hyperbola. Why does the sin and cos of theta against ...
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1answer
35 views

Conservation of momentum 1-D problem: track is not level

This problem is a standard 1-D problem. You have 2 equal carts approaching each other and collide on linear air track. BUT the person setting up the track did NOT do a good job at making the track ...
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14 views

Show that after the deformation the line element is given by the following equation

Assume throughout that $X_i,x_i$ (i=1,2,3) are respectively the material and spatial coordinates of a point referred to a common rectangular Cartesian coordinate system with origin $0$, and the the ...
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1answer
29 views

How Lagrange equations imply Newton equation

Excuse my english (it's not my first language) I'm self-studying Mechanics and i have a little problem : We can see that in Landau's book or in wikipedia that when we inject the lagrangian in Euler ...
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2answers
77 views

Rigid Body Motion with Fixed Point

Suppose that $ T : \mathbb{R} \longmapsto \mathbb SO_{3} $ is a differentiable function. This means that $ T(t) $ is differentiable component wise. Does there exist a function $ \Omega : ...
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37 views

Using Feynman's Subscript Notation

I have a homework problem that wants me to calculate the force $\vec{F} = \vec{\nabla}_{\vec{X}}U + \frac{\mathrm{d}}{\mathrm{d} t} \left(\vec{\nabla}_{\dot{X}} U\right)$ where $U(\vec{X}, \dot{X}, ...
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1answer
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Circular Motion and Conservation of Energy Question

A particle $P$ of mass m is connected to a fixed point $O$ by a light inextensible string $OP$ of length $r$ and is moving in a vertical circle, centre $O$. At its lowest point, $P$ has speed $U$. ...
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Bullet hit, calculate pressure

I wanna calculate the stress that is there when a bullet hits a metall plate. i know the formula pressure = force / area But how can I calculate the force?
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Lagrangian of particle on a horizontal, square rotating hoop.

The problem of interest is worded as follows: A horizontal square wire frame with vertices $ABCD$ and side length $2a$ rotates with constant angular frequency $\omega$ about a vertical axis through ...
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2answers
50 views

Classical Mechanics Question

A toboggan travels along the path ABC shown in the diagram. The path lies in a vertical plane, and consists of two circular arcs $AB$ and $BC$. The line ABC is horizontal and there is no friction ...
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2answers
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Period of a mechanical system

Im trying to solve the following problem. Consider $\mathbb{R}^{2}$ with coordinates $(x,y)$. Let $H$ be a smooth function on $\mathbb{R}^{2}$. Also, consider the Hamilton equations: ...
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1answer
107 views

Cauchy Momentum Equation - Stress Tensor

I've been trying to understand the derivation for the Cauchy Momentum Equation for so long now, and there is one part that every derivation glides over very quickly with practically no explanation ...
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Estimate the period of motion of a particle

a) Evaluate the period of motion of a particle in the potential in the field $U(x)$ if it's energy value lies in the vicinity of $U_m$ (see the picture below). b) Estimate at what part of of period ...
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1answer
23 views

projectile motion (dynamic) stone

2) A stone is projected downwards with a velocity of 20m/s at an angle of 30degrees below a horizontal line through the point of projection. Find the velocity of the stone after 2 sec. t=0 => ...
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1answer
40 views

complicated projectile motion about throwing thing [closed]

A stone is projected with a velocity of $20$ m/s at an elevation of $\theta$, given $\tan(\theta)=3/4$, from the top of a wall $9$m high. Find the range of the horizontal plane through the bottom ...
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1answer
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projectile motion (with height) complicated

When a child standing on a horizontal path throws a ball, it leaves her hand from a point that is 90cm vertically above the path. The ball clears a 4.5 m high wall that is 10.5 m away from ...
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1answer
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Show that the action variable is $J = \sqrt{A^2 + 2E} - A$

I have the hamiltonian: $$H = \frac{1}{2}p^2 + \frac{1}{2}A^2 \tan^2(q)$$ And I would like to show that the action variable is $J = \sqrt{A^2 + 2E} - A$, where $E$ is the energy. I'm having a ...
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1answer
15 views

Vector finding out the value of P?

Given two force 2P and P , when the first one is doubled , i.e 4P and the 8 is added to the second one i.e P+8 , the direction of the resultant remains unaltered . What is the value of P ? N.B : I ...
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1answer
63 views

How to derive the hamiltonian from a non-classical lagrangian

For the non-classical lagrangian of a hydrogen atom: $$L = -mc^2 \sqrt{1-\frac{v^2}{c^2}} + \frac{e^2}{4 \pi \epsilon r}$$ We get that two conserved quantities are: $J = \gamma mr^2 \dot{\phi}$ and ...
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1answer
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Find the smallest $t$ such that the object reaches the height of $96\text{ feet}$ at time $t$.

An object is tossing upwards with an initial speed of $64 \text{ feet/sec}$.Suppose the gravitational acceleration is $32\text{ feet/sec}^2$. Find the smallest $t$ such that the object reaches the ...
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1answer
28 views

Workout initial velocity of projectile without knowing launch angle

I am trying to work out the speed of a tennis serve. I know the following: X Distance: 18m, Time: 0.5s, g: 9.8, Launch height: 3m. Is it possible to work out the initial velocity of the serve ...
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1answer
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Problem on Finding the speed using Intertia

I did the first part (using parallel axis theorem) and showed that intertia. The problem is in the second part, I know that $C=I\frac{d^2\theta}{dt^2}$ , where C is the moment. So in this case it ...
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111 views

What are sliding vectors mathematically?

What is the mathematical definition of sliding vectors and their operations, as used in mechanics? What kind of mathematical structure do they form? Does the operation of constructing the "space" of ...
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2answers
99 views

Find the time interval between oscillations of SHM.

Parts i) and ii) I can solve. But for part iii) I can't do, as I don't know which equation describes the SHM motion? Is it $y=0.5sin(1.2t)$ or $y=0.5cos(1.2t)$ or $x=0.5sin(1.2t)+2.5$? I thought ...
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Deriving the $F_3$ type generating function in Hamiltonian formulation

I'm working on some practice questions and I am a bit confused with this one: Generating functions of the type $F_1(q,Q)$ satisfy the condition: $$pdq-PdQ = dF_1$$ Starting from this condition ...