Classical mechanics refers to the classical (i.e., non-relativistic, non-quantum) study of physics. Three major formulations of classical mechanics are newtonian mechanics, lagrangian mechanics, and hamiltonian mechanics. The latter two are rather useful in extensions to Classical Mechanics; ...

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Canonical transformation problem

(Apologies if HW questions are not allowed -- I couldn't really find a definite answer on this) Question Let $Q^1 = (q^1)^2, Q^2 = q^1+q^2, P_{\alpha} = P_{\alpha}\left(q,p \right), \alpha = 1,2$ ...
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175 views

Derivation of Scattering Equation 9.88 in Thornton & Marion

I am confused as to how a particular equation in Thornton & Marion's 'Classical Dynamics of Particles and Systems' was derived. It is equation 9.88, on page 354 of the fifth edition. An incoming ...
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124 views

Buoyancy Correction for a Kater Pendulum

Question: Consider a Kater pendulum: that is, a rod with two cylindrical masses at both ends, with a knife edge between the two masses and another between one ...
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79 views

How to analyze this constraint question

Let $\gamma$ be a smooth curve in the plane, and introduce curvilinear coordinates $q_1,q_2$ on a neighborhood of $\gamma$; $q_1$ is the direction of $\gamma$ and $q_2$ is distance from the curve. ...
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127 views

A discrete approach to the catenary

I'm trying to work out a model for the system above, that is, $N$ particles of unitary mass subject to the constraints: $$1=\varphi _i(\mathbf r _1,\mathbf {r}_2,...,\mathbf r _n)=|\mathbf ...
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332 views

Anharmonic oscillator solution function

I am solving a CLASSICAL an-harmonic oscillator problem with Hamiltonian given by $H= (1/2)\dot{x}^2+(1/2)x^2-(1/2)x^4$ with all the constants (k's) and mass being taken as 1 (one). I find that $x= ...
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60 views

Movement of a gyroscope with non-fixed axis

Assume one has a gyroscope rotating around an axis with both ends leaning on a dedicated semiplane as shown on the picture below. There is no friction either between the rotor and the axis or between ...
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135 views

Friction From an Object On-Top of a Sliding Object

Consider a block $A$ lying on a flat and frictionless table, and a block $B$ lying on top of block $A$. A horizontal force $F$ is applied to block $A$. If there is no friction between blocks $A$ and ...
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568 views

projectile that splits into two fragments of equal mass

I am studying for an exam, and this is part of a problem in my book. A projectile is launch from level ground and is intended to hit a target 100m away. Instead, it explodes into two fragments of ...
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198 views

At what angle do billiard balls scatter if they collide off center?

The angle defined by joining a line from the centers of the balls must be important. But do they follow this angle when viewed in the rest frame of one of the balls or in the CM frame? The spheres ...
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128 views

Single particle trajectory in a quadrupole potential

I am wondering if there are any studies of a single (classical) particle trajectory in quadrupole potential: $$ V(x,y,z)=A\sqrt[]{\frac{x^2 + y^2}{a} + \frac{z^2}{b}} $$
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78 views

What is the optimal diameter for the exhaust hole in a pressurized vessel to deliver highest acceleration ?

Imagine you have a sealed cylindrical vessel with a given radius with a compressed gas inside. Let's give some numbers, 5 cm radius and 100 atm pressure. You poke a hole in the vessel and the gas will ...
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41 views

Calculate the acceleration of the trailing muon bunch

Two separate suitably short but intense bunches of muons, "A" and "B", are both supposed to be constantly accelerating (in an otherwise sufficiently flat region) with constant proper acceleration ...
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60 views

Hamiltonian system: match transformations and constants of motion

I have a problem about the interpretation of an exercise. Given the following Hamiltonian $$H=\frac{\mathbf{p_0}^2}{2m}+\frac{\mathbf{p_1}^2}{2m}+\frac{\mathbf{p_2}^2}{2m}-2V(\mathbf{r_1}- ...
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88 views

A quicker way to verify that a function is a constant of motion?

I have three particles that we can indicate with $\alpha$ ($\alpha$=0,1,2), they are identified by the $r^i_\alpha$ coordinates and $p^\beta_j$ conjugata momenta ($\beta=0,1,2$ and $i,j=1,2,3$). I ...
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453 views

Why is the angle of impact complementary to the angle of launch in the simple equations for the range of a projectile?

I'm using the standard equation for the range of a projectile: \begin{align} d &= \frac{v\ \text{cos}\theta}{g} \left( v\ \text{sin}\theta + \sqrt{v^2\ \text{sin}^2\theta + 2gy_0}\right) ...
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45 views

Acceleration by spherical particles (micron-scale) by an external force

I am looking for an expression for the velocity of a micron sized (1 - 10 micron diameter) sized particles under accelerating forces. I have aerosols in mind. This is what I have in mind The ...
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302 views

Torque, lever and mass

The Force used in a catapult is exerted near its axis. If we double the length of the arm of the catapult, but still use the same Force at the same point as before near the same axis, does the ...
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257 views

Limitations on the choice of axis of rotation regarding rolling wheels

Consider a situation where a wheel is rolling without friction on a level surface. Call the center of the wheel $C$, the point where the wheel contacts the ground $G$, and some arbitrary other point ...
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67 views

A question related to tractrix

I'm a novice to physics, so maybe it's rather stupid. According to wiki, the tractrix could be considered a trajectory: Suppose $AB$ is a stick on a smooth plane $\pi$, and the initial position of ...
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55 views

Fading transition and rotation of and object in 2D

I'm looking for sources about I guess dynamics subject. The model I'd like to solve is reduced to a question of: How does a force applied on a certain point of an object results in both fading ...
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27 views

Dual methods in structure optimization

I am studying Structure Optimization, and the teacher took a few weeks to detail dual methods, but I don't get why it is useful. From an optimization problem with ...
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364 views

Normal modes of oscillation: how to find them

Are normal modes the eigenvectors of the matrix $(\omega ^2 T- V)$ where $T$ is the matrix of kinetic energy and $V$ is the matrix of potential energy? Is it the only way to express them? How can I ...
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342 views

What is the angle of the string with the vertical?

A heavy uniform sphere of radius $a$ has a light inextensible string attached to a point on its surface. The other end of the string is fixed to a point on a rough vertical wall. The sphere ...
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21 views

On Bolte's semiclassical law

i have seen on internet the following, for $ E >> 1 $ the Eigenvalue Staircase can be approximated by $ N(E)= \frac{1}{\pi}argZ(1/2+i \sqrt E ) $ ...
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277 views

moment of tennis racket & volley

I've been looking for the relation between moment of a tennis racket and its maneuverability. If I define maneuverability as the easiness to get in position for quick reaction strokes like volleys and ...
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112 views

Is there analysis library for stress-strain data?

I have three column data that has time-displacement-force from 1D tensile/compression test. Now I would like to get the standard mechanical properties of the material, like Young modulus, yield ...
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441 views

How equivalent are heat energy and work energy in connection with a spinning flywheel?

Let's say we have two identical spinning flywheels, that have arbitrary geometry, and are made of copper. Now we apply some heat energy at the center point of flywheel A, causing it to slow down a ...
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83 views

How can I model a polyatomic molecule as a system of coupled oscillators?

(Classical Mechanics) Let's say I have a polyatomic molecule, what is the best way for finding the equations of oscillations if they are bounded by a torsion spring?
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159 views

Conveyor scales modeling

Assume we have a conveyor scales. Which consists of scales, and motor with conveyor belt placed above, so that the boxes can be measured (weight) while moving above. What I want is to create the model ...
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255 views

Flattening rate of a parabola

If you have a piece of paper that is furled and unfurled so that it's in the shape of part of a parabola, and knowing that if you leave it, it'll flatten itself after time, would it flatten faster if ...
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146 views

How do I determine where the maximum air velocity around an object is?

I have a freestream of air directed at an object. Is there a way to find where the maximum air velcoty occurs with only having initial air speed and static pressure at the object?
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Calculating/estimating heat transfer losses for hot air balloon (lantern)

I'm trying to build a flying lantern / hot air balloon that flies as close to hovering as possible (as opposed to up-up and awaaay). To see if this is feasible I'm trying to simulate as much as I can ...
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23 views

What justification is necessary for convolutional variational principles to be considered legitimate?

I recently asked a related question and was interested in why/or why we cannot use convolutional variational principles in practice or in theory. Summarizing the points I made in the earlier post: ...
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47 views

Stability of Mathieu's equation and parameteric resonance

I am given the following equation (Mathieu's equation) in my subject of Numerical Analysis : $$ \frac{d^2 x}{dt^2}=-\omega^2(1+\epsilon\cos(t))x $$ I am supposed to find those frequencies $\omega$ ...
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29 views

Motion Integrals of a Particle in a Force Field

I am trying to wrap my head around the following problem: A point particle is moving in a field, where its potential energy is U=-α/r. Find first motion integrals. In our university we have no ...
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12 views

Non-dimensionalizing the “bead on a rotating hoop, with viscous damping” problem

This is not a homework question. Rather, this is an exercise I have taken up on myself. In particular, I am trying to find an algorithmic way to non-dimensionalize known equations, using the ...
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23 views

Rotation of Thin street sign

I am attempting to complete a home question in which a shop sign in the shape of a thin rectangle of size p x q (with q being the longer side), and mass m, that rotates about an axis that passes ...
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37 views

Derivable Concepts in Mechanics and Electromagnetism

In Classical Mechanics, one of the possible foundations is based on three concepts aka mass(equivalent to energy), length and time. This is a foundation because we can model everything ( pressure, ...
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63 views

Deriving Snell's law via Lagrangian mechanics

A particle moves with kinetic energy $K_1$ in a region where its potential energy has a constant value $U_1$. After crossing a certain plane, its potential energy changes discontinuously to a new ...
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21 views

Will the center of mass of the whole system change when object swims on curved surface?

In the example given here, the object can move on the frictionless surface of the sphere by changing its shape periodically. So will the center of mass of the whole system change after the object ...
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26 views

Force in colliding snooker balls

If a snooker ball is traveling at 2m/s and hits another ball, the first ball will stop dead and the second will accelerate instantaneously to 2m/s. F=ma, so this would seem to imply an infinite force. ...
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43 views

Time reversed Abraham-Lorentz reaction force

The Abraham-Lorentz radiation reaction force on a charged particle is given by: $$\mathbf{F_{rad}} = \frac{q^2}{6\pi\epsilon_0c^3}\mathbf{\dot{a}}$$ I understand the situation where one fires a ...
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36 views

Quantum chaos vs classical chaos

There is this popular conjecture from Bohigas, which says: When the analogeous classical system of a quantum system shows chaotic behaviour then the spacing distribution of the quantum system ...
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17 views

Parametric impulse on driven, damped oscillator

I've been thinking about driven harmonic oscillators recently. I know how to calculate their response to a sinusoidal drive, and their response to an impulse or more generally an arbitrary drive via ...
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46 views

Canonical transformation from Hamiltonian without external source to Hamiltonian with external source

Let a system with time-independent Hamiltonian, $H_0(q,p)$ be subjected to an external oscillating field $E_0\sin(wt)$, so that the Hamiltonian becomes $H=H_0(q,p)-qE_0\sin(wt)$. Find a canonical ...
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Impact strength test of a container filled with liquid

as a university project I have to project(choose material and dimensioning) a liquid container (NaClO, density 1100 g/L), we ...
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38 views

Symplectic Structure without predefined Hamiltonian

Here there is a link which has helped me understanding the relationship between symplectic geometry and classical mechanincs. In short, the symplectic form transforms the derivative of the ...
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What is the temperature effect on trajectories in phase space in molecular dynamics?

In molecular dynamics simulations or microcanonical ensemble (fixed-energy), what is the effect of temperature on the trajectories of the reacting systems (let's say two reactants react to form two ...
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27 views

Isolate an electronic drum from the ground

Please note that I know nothing about this part of physics, so sorry if I make some mistakes. Drum is an awesome instrument, yet it can easily make your neighbours very angry.The vibrations caused by ...