Questions tagged [classical-mechanics]
Classical mechanics discusses the behaviour of macroscopic bodies under the influence of forces (without necessarily specifying the origin of these forces). If it's possible, USE MORE SPECIFIC TAGS like [newtonian-mechanics], [lagrangian-formalism], and [hamiltonian-formalism].
8,522
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Resource on quantum to classical
I am looking for a book/paper which derives classical mechanics starting from quantum mechanics, to better understand the transition. Expected level of mathematical rigour is equivalent to graduate ...
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Relation between old and new Hamiltonian for Canonical Transformations
The question I am asking was asked here but was never given a satisfactory answer and so I will rephrase it and add more detail.
In chapter 9 of Goldstein,when talking about a canonical transformatio $...
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Mass rotating on a vertical circular path [closed]
I am considering the following exercise:
Let's consider a mass moving on a vertical rigid hoop, subjected only to the force of gravity. What are the equations of motion?
I know that there is a simple ...
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What exactly is a linear-elastic material?
Imagine a large, soft, 3D linear-elastic medium containing a small, hard object in the middle (e.g., a marble). If the marble is displaced from equilibrium and then released, it will oscillate back ...
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Using the principle of inertia to motivate the principle of least action?
Can we motivate the principle of least action with the principle of inertia that causes a mass particle to resist changes in its momentum? After all, the principle of inertia is the starting point and ...
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Equivalence of various definitions of reversibility in classical mechanics
I was reading Classical Mechanics: The Theoretical Minimum by Leonard Susskind, and the definition of reversibility in that was:
Given a state of a system, then we know exactly what state it came ...
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Derivation of Hamilton-Jacobi (HJ) Equation
In the Derivation of Hamilton Jacobi Equation, I didn't understand the bold parts:
we can write (1) formally as,
$$
\frac{\partial F\left(q_i, Q_i, t\right)}{\partial t}=-H\left(p_i, q_i, t\right)=-H\...
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Existence Of Phase Flow Provided Potential Energy is Positive
I am reading through Arnold's "Mathematical Methods Of Classical Mechanics". In the section 4D on p. 21 concerning Phase Flow there is a question that reads as follows:
Show that if ...
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On the definition(s) of flux/current density
There are three common definitions of flux. Citing Wikipedia:
In transport phenomena (heat transfer, mass transfer and fluid dynamics), flux is defined as the rate of flow of a property per unit area....
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Range of angular speeds for which bead at the bottom of rotating hoop is stationary
A bead lies on a frictionless hoop of radius $R$ that rotates around a vertical diameter with constant angular speed $\omega$, as shown in the figure below.
As the title suggests, I am particularly ...
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What would be a good particle point-of-view model of a self-gravitating molecular cloud: with orbit or not?
Excuse my poor English, I have a lot to learn.
Should we imagine a particle falling directly to the center without an orbit like an apple falling from a tree to Earth, converting its potential energy ...
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When a car curves, if friction points towards the centre which force makes the wheel rotate?
The translational motion of the wheel is due to the friction right. If friction is towards the centre during a turn, which provides the translational motion to the wheels? Or is it the component of ...
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A doubt regarding $L=T-V$ and explicit time dependence
Edit: After having some clarity, I chose to write an answer instead of editing the question itself. Scroll down to read it after reading the problem that follows.
Let's say $\vec{r}=\vec{r}(q_1,q_2 ......
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What is the vector field associated with potential energy?
The mere concept of a line integral is defined for a vector field, and I thus thought the following was a rigorous and general definition of potential energy:
Definition: Given a conservative force ...
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Why is it useful to learn about Hamiltonian Mechanics in the framework of Symplectic Geometry?
...Other than providing a deeper insight into the mathematical background of dynamical systems.
Does casting certain classes of problems in terms of symplectic geometry make solving them easier/...
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Why do we construct Lagrangian submanifolds after symplectic reductions
I am learning about Hamilton-Jacobi actions, symplectic reductions and Lagrangian submanifolds and I am trying to understand the relation between these concepts.
I have read that Lagrangian ...
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How to solve the Helmholtz equation in damped oscillator BCs?
Given the surface of the vibrating object $\partial \Omega$, I am trying to simulate its outer sound pressure field $p(x)$ by the equivalent source method.[1]
For ...
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Lagrangian Mechanics - Is the Given Answer Incorrect? [closed]
A heavy symmetric top rotating about a fixed point has Lagrangian
$$L=\frac{I_1}{2}(\dot \theta ^2 + \dot \phi^2 \sin ^2 \theta)+\frac{I_3}{2}(\dot \psi + \dot \phi \cos \theta)^2-mgl\cos \theta$$
...
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Car moving on a ball in space
Consider a car of mass $m$ moving on the surface of a ball (think of it as earth) with moment of inertia $I$, floating in a vacuum.
Let the car slowly (adiabatically) drive around the circle of ...
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2
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Solving for the radius of a sphere as a function of time
I have tried to realistically model the famous game Agar.io, which can described as the following: A sphere of initial mass $m_0$ expels part of its mass at a given rate ($\frac{dm_l}{dt}$) for thrust ...
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Rotated square during relativity
I'm working with relativity and I can't seem to understand the following explanation for why a square with side length $L$ is perceived as rotated when travelling at a velocity $v$ (relativistic ...
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Intuition for the negative sign of gravitational potential energy
The gravitational potential energy is introduced to us as $U = mgy$. We usually set $U = 0$ on ground level and hence, for an object at height $y$, we have a potential energy equal to $U = mgy$.
I ...
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A Question about Diffeomorphism Invariant Action
I remember that the canonical Hamiltonian of a diffeomorphism-invariant theory, in general, is zero. For example, the geodesic equation is derived from the action of arc length $$S[g(\tau)]=\int_{a}^{...
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What's the meaning of a complex momentum in classical mechanics?
I'm looking at a section of Griffiths and Schroeter's Introduction to Quantum Mechanics, pp. 355. It states a straightforward set of equations that got me thinking about the exact way in which complex ...
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System of 6 harmonic oscillators [closed]
I have seen this question in given image many times in past few years. And everytime i see this question, i instinctively conclude that its has atleast three possible answers $π/3$ , $2π/3$ and $π$ .
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Why does friction tend to slow the objects?
I was biking on the road when suddenly realized a question, that why does friction tend to slow the objects(when the "engine" wasn't doing the "external" work, of course)?
A naive ...
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When discussing non-steady current, why is Newton's 3rd Law valid only if momentum carried by the Electromagnetic field taken into account? [duplicate]
I can't understand what the underlying statement implies -
'When currents vary with time, Newton's third law is valid only if momentum carried by the electromagnetic field is taken into account'.
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Dispersion of finite 2D lattice
Following problem: I have the coupling matrix for an $N$-by-$N$ finite lattice of coupled masses (only nearest-neighbour coupling, periodic until terminated). I would like to numerically calculate its ...
2
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Conflict between a system lowering its potential energy and kinetic energy
Every system in the universe has a natural tendency to attain a state of minimum energy. The state of minimum energy corresponds to a state of maximum stability.
These are from a renowned book. I am ...
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Is there a classical limit that connects classical volume/phase space with the scalar product of momentum and position eigenstates?
The book Understanding Molecular Simulation: From Algorithms to Applications, 2002, Daan Frenkel and Berend Smit, states the following
$$
\langle r|k\rangle \langle k|r\rangle = 1/V^N
$$
where $|r\...
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Is cofficient of restitution equation valid for collisions involving more than 2 bodies?
I couldn't solve a question and the only equation I was missing was of cofficient of restitution i.e. $e=\frac{v_{final2} - v_{final1}}{u_{intial1} - u_{initial2}}$ .
But the collision was among 3 ...
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Are there non-orthogonal "normal" modes for non-identical coupled oscillators?
The question is broad, I will specify an example to elaborate what I'm asking.
Suppose I have two different LC circuits inductively coupled (or capacitively, but the question I have will be relevant ...
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Confusion about friction during pure rolling
Friction comes into play with the relative motion between the surface and the points of the body in contact with it. In a perfectly rolling sphere, the instantaneous velocity of the bottommost point ...
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Why is the gravitational potential energy lost not subtracted from the required work done in the given problem? [closed]
An elastic string of natural length $l \;\text{m}$ is suspended from a fixed point $O$. When a mass of $M \;\text{kg}$ is attached to the other end of the string, its extension is $\frac {l}{10} \;\...
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Does one loose constraints with the total Hamiltonian?
Let me recall some of the discussion in M. Henneaux and C. Teitelboim, Quantization of Gauge Systems. Princeton University Press, 1992. for context. Assume we have a constrained Hamiltonian system ...
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Rigid Body Simulation with Joints
I have read the basics of how force based physics simulation works and how you can create joints by enforcing certain constraints on the forces between two rigid bodies. What is unclear to me is what ...
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Independence of position and momentum in action
Why are position and momentum independent with respect to the Hamiltonian Action $S_H$ given by
$$
S_H = \int_{t_1}^{t_2} (p \dot q - H) dt \ \ \ ? \tag{1}
$$
While deriving Hamilton's equations from ...
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2
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How does angular momentum contributes to increase the normal force?
I found the following in an exercise of gyroscopes:
In a grain mill, grain is ground by a massive wheel that rolls without slipping in a circle on a flat horizontal surface driven by a vertical shaft....
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Trouble showing from first principles the equation describing the motion of a water drop through a cloud
If we had a water drop, initially of mass $m_0$ and speed $v_0$, falling through a cloud for a time
T and during this time its mass increases at a constant rate $\alpha m_0$, through accretion of
...
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3
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Oscillating disc on flat table
Suppose we have a uniform disc of mass $2m$ and radius $R$, and we place a small piece of putty of mass $m$ onto the disc at a position very close to the rim.
Now let's say the disc with putty is now ...
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Physics of bursting balloons?
This is a reference request for a theory of bursting balloons:
Let's say I have a balloon and this balloon has an impurity along a small strip. Let's assume for example the balloon had a hole there at ...
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3
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How is buoyant force still upwards in a mixture of fluids
For objects 1 and 2, how is the buoyant force by oil still upwards? Shouldn't the pressure caused due to the oil push it down?
2
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Conjugate momentum notation
I was reading Peter Mann's Lagrangian & Hamiltonian Dynamics, and I found this equation (page 115):
$$p_i := \frac{\partial L}{\partial \dot{q}^i}$$
where L is the Lagrangian. I understand this is ...
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2
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Pressure in an accelerated container
Let us take a closed container completely filled with some liquid.Let us assume that the container moves with some acceleration in the horizontal direction.
Usually when we have such situations,we ...
2
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1
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How is the relationship between the old and the new canonical variables justified?
In Classical Hamiltonian Mechanics, a canonical transformation of the phase-space coordinates $(p,q,t) \to (P,Q,t)$ is such that the general form of Hamilton's equations is followed and Hamilton's ...
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1
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Semiholonomic Constraints in Rolling Without Slipping
I was reading Goldstein's Classical Mechanics and it is mentioned there, on page 50 (third edition), that rolling without slipping is a semiholonomic constraint, which is defined on page 49 as a ...
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1
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Trust in wave-particle duality [closed]
I am doing an essay in school to consider how much trust is needed to accept knowledge within the natural sciences. For one of my points I decided to look into W-P duality. I know the photoelectric ...
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2
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Calculating coordinates of extra balancing mass [closed]
(https://i.stack.imgur.com/AjdKz.jpg)
Hello, I have unbalanced mechanism which center of rotation is not in line with center of its mass and I'm trying to calculate coordinates $x_1$ and $y_1$ of mass ...
2
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Can liquids exert/be affected by normal force?
I am doing a problem in which I am drawing free-body diagrams for different objects in a system, and two of the objects which are included in my scenario are Earth and a body of water which is resting ...
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$\partial F_2/dt$ part of a time dependent canonical transformation
Suppose we have a time-dependent canonical transformation - say generated by a function of the type $F_2(q,P,t)$. The resulting Kamiltonian picks up an extra partial $\partial F_2/\partial t$:
\...