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].

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Probability of observing harmonic oscillator at a particular position

Consider a classical harmonic oscillator whose Hamiltonian is $$H=\frac{p^2}{2m} +\frac{1}{2}mw^2x^2$$ where $w$ is the oscillating frequency. I wish to find the probability of observing the harmonic ...
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How do I approximate an angular acceleration vector from a history of orientations?

A rigid body is simulated to move in six degrees of freedom. At every past timestep, I know its displacement vectors $x^{GLOB}$ in the global coordinate frame, as well as 3x3 matrices $R$ that ...
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Question about constraint in lagrangian

We know that the lagrangian can be written using constraints. Suppose I have constraint functions $$f_1=\cos(x) - (x+t)/R=0,\qquad f_2=\sin(x)-y/R=0 .$$ But I know that $1=\cos^2+\sin^2$, so do we use ...
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Can a frame still be an inertial frame if its center varies with time relative to a “true” inertial frame?

Picture a seat on a Ferris wheel. Neglecting any rocking, is the seat of a Ferris wheel an inertial frame? My guess is that yes it is right? The frame itself isn’t rotating or accelerating relative to ...
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Does passing an electric current along a strip of metal submerged in saltwater cause anything?

If saltwater corrodes metal. Can we effectively stop this by passing some electric current? Has this been tried before?
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Is there a way to understand which variable is more influential in the dynamics of a system?

Is there any known way to identify which variable has the most impact in the dynamics of a system given its lagrangian or hamiltonian formulation? Let's say i have a system with 3 variables, two ...
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Finding a new hamiltonian from a given canonical transformation

Let us suppose we have a given Hamiltonian $$H = \frac{P_1^2}{4}+\frac{P_2^2}{2}+V(Q_2)$$ and $q_1 = Q_1 + Q_2/2$ with $q_2 = Q_1 - Q_2/2$. I need to find the new hamiltonian by using these canonical ...
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Hello, I am wondering How I can set Equation which describing the motion of an object attracted to magnet

Umm, I imagined the sphere magnet (it can rolling) which has friction. then How I can do make formula about this? I know $$ E = \int \frac{B^2}{2\mu} \,dV $$ and that is same as $1/2mv^2$ ? and $I = 2/...
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Liouville's theorem on the tangent bundle [duplicate]

One interpretation of Liouville's theorem is the determinism and reversibility of classical mechanics, i.e. the mechanical states can't converge or diverge. The theorem is often formulated on the ...
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Doubt in derivation of bending of beam, It's about derivatives and intergration

Radius of curvature of the beam in above picture is given as: $$ \frac{1}{R} = \frac{d^2 y}{dx^2}$$ Please help me two points used as steps of a derivation in my book: How was the radius of ...
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How much time after will two oppositely charged particles collide for both gravitational force and electrostatic force?

Suppose two point objects charged with opposite charges $q_1$ and $q_2$ at a distance $r$ in a vaccum. So, the net electrostatic force on both objects $= F_c = \frac {q_1q_2}{4π\epsilon_0r²}$ [$\...
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Calculating work done when the lower bound of integral is greater than the upper bound

In this video, Dr. Peter Dourmashkin explained friction as an example of a force by which the work done is not path independent. In $2$$:$$50$ min of the video, when we're coming back, he said, $d\...
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Recommend a classic physics book to prep for uni course [closed]

Biology/premed major here, always been pretty poop at physics despite being good at math, but I have a physics course next year concerning classical mechanics and was wondering if you if someone could ...
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What is the degrees of freedom (Lagrange equation) of two connected spool rolling down two inclines?

I'm quite confused as to how to use the Lagrange equation [second type] in a system which features a spool rolling down an incline. I think this particular example is quite representative of what is ...
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How does the direction of a force applied to a pulley influence the pulley's motion?

Suppose friction is negligible, we want to determine the force F that has to be applied to a pulley (whose radius is $R$) such that the body m has a constant acceleration a, which is given. What I did ...
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What does the position $x(t)$ looks like in an overdamped system?

I know that for the position $x$ as a function of time in an underdamped system (such as a mass on a spring) you can use the function: $$x(t)=Ae^{\gamma t}cos(\omega t-\phi),$$ where $$ \begin{split} ...
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Time taken by the pendulum to complete one oscillation with initial angle $ \theta = 90° $

The question that I have is pretty much the question in the image below. The question was given to my class (11th grade) by our physics teacher as a challenge problem. I think this is important to ...
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Minimum seperation of moving objects doubt

Let there be $2$ objects $P_1$(initial velocity $u$ $ms^{-1}$ & acceleration $a$ $ms^{-2}$) & $P_2$ (initial velocity $U$ $ms^{-1}$ & acceleration $A$ $ms^{-2}$) initially separated by ...
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Equation of motion of a particle in a sinusoidal well

Do you have solutions for the (classical or not) equations of motion of a particle in a sinusoidal well or just a quartic well, classicaly I would write the equations like so: $$\frac{d^2x}{dt^2}\...
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Why do shot arrows seem to naturally point tangent to their arc?

Several years ago, I posted a question about Why do archery arrows tilt downwards in their descent?. An answer was given that a torque arises from the difference in location of net force of gravity (...
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How does a spring's maximum compression ratio change with decreasing it's free length for a coiled helical spring(compression)?

I'm just looking for a simple expression formula for how maximum spring compression changes with free length of a coiled helical spring, say a Compression spring. A way of understanding what I mean by ...
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What curve does a rod form when bent to intersect 3 or more points?

Suppose that we have a sufficiently thin, flexible cylindrical rod of length $L$ made from a homogeneous, isotropic material, and that initially [at rest?] the central axis of the rod is a straight ...
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What exactly did Lagrange do, historically? [migrated]

I'm tying to understand, historically, what lead to Lagrangian mechanics (LM). What did Lagrange actually do? In the time (year 1788), when Lagrange published his work (that we nowadays call "...
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What is the answer to this spring, pulley and mass system problem? [closed]

For the first point $x_{eq}=\frac{mg}{k}$ put regarding the second point, we need to get the differential equation, after using $F=ma$ we get $mg-kx=ma$, I considered that $x$ is equal to $x+x_{eq}$ ...
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Symmetry of the second partial derivatives of the Hamiltonian

A dynamical system with generalised particle position $q$ and generalised momentum $p$, described by: $$\dot{q}=F_1(q,p)\quad\text{and}\quad\dot{p}=F_2(q,p)\tag{1}$$ is a Hamiltonian system if: $$\...
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Lagrangian Dynamics of an inverted Spherical Cart Pendulum

Introduction I have to come up with a PD-controller for an inverted Spherical Cart Pendulum, therefore I tried to compute the Dynamics of such a Pendulum. The Spherical Cart Pendulum is a hybrid ...
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Force of brake pads on a wheel

I have a hollow-cylinder wheel model, braked with brake pads located at a distance d of the wheel's center axis. The brake pads have a contact area S. They are also forced towards the wheel with a ...
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Magnitude of the variations $\delta q_i$ in the principle of stationary action

To determine the equation of motion using the principle of stationary action, one has to consider the variation of the action due to variations $\delta q_i$ in all the generalized coordinates $q_i$. ...
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Proof of principle of stationary action when the Lagrangian is not $L=T-V$

The principle of stationary action claims that the action $S$ takes a stationary value in a real system, where: $$S = \int_{t_1}^{t_2} L dt\tag{1}$$ and $L$ is the Lagrangian of the system. It can be ...
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Expected energy in micro-canonical and canonical distribution

Which relation $E(β)$ is required to ensure that he micro-canonical distribution and the canonical distribution have the same expected energy?
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Velocity- and time-dependent forces can not be conservative [duplicate]

I have already seen many posts about this specific question but in only a few of those mathematically rigorous answers were given. Unfortunately none of them are accessible at my level (first year ...
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Derivation of Hamiltonian $H=T+V$ from Lagrangian $L=T-V$

I understand that the Hamiltonian is the Legendre transform of the Lagrangian: $$ \begin{split}H(q,p,t) &= \frac{\partial L}{\partial \dot{q}}\dot{q} - L(q,\dot{q},t) \\ \implies H&=p\dot{q} -...
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Is there a limit for passing of classical waves behind a slit?

For the classical waves is there a limit for them to pass through a slit when the dimentions of the slit are much smaller then the wavelength? I think that the Hyugence principle must be valid and ...
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Can we tell anything about the velocity at which a spherical stone falls into a lake based on the sound it makes?

Suppose we have a spherical cow... I mean stone, and we throw it to a lake. Can we tell anything about the velocity (speed and angle) at which it falls based on the sound it makes? And based on a ...
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Time derivative of unit velocity vector?

Let's say I have some parametric curve describing the evolution of a particle $\mathbf{r}(t)$. The velocity is $\mathbf{v}(t) = d\mathbf{r}/dt$ of course. I am trying to understand what the expression ...
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Does existence of an analytic solution to an equation of motion given by Newton's second law depend on coordinates?

Newton's second law is a coordinate agnostic statement, we can use it to calculate the forces in a coordinate system, and hence, the motion of the body in that coordinate system. However, depending on ...
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Angular velocity of point on rigid body [duplicate]

a while back I asked this question and I still did not fully understand. Suppose we have a rigid object rotating about some central point with a given linear and angular velocity, how do we then ...
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Would it be more insightful to teach students Lagrangian/Hamiltonian mechanics before Newtonian mechanics? [closed]

What benefits would it bring to teach analytical mechanics before Newtonian (vector) mechanics? I began thinking in this after I saw a 2019 article in the magazine Physics Today that advocates ...
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Physical interpretation of mathematical operations

I have difficulty interpreting mathematical operations in a physical way. For example, I see many people reading Newton's second law as "Resultant force equals mass times acceleration", ...
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1 answer
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Confusion about the action variable definition

Suppose we have an integrable system consisting of a $2n$-dimensional phase space $M$ together with $n$ independent functions $f_{1\leq j \leq n }$ in involution. Suppose the level set $$M_f = \{ (p,q)...
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Particle moving around the inside of a semicone - how to model its position up incline?

An inverted hollow cone (cut in half) is set up with its vertex at the origin $O$ and an angle $\alpha$ between the horizontal ($x$ and $y$-axes) and the cone (so $\alpha$ close to $0$ would be a flat ...
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Conservation and potential with non-cartesian forces

I understand how to determine if a force is conservative from \begin{equation} \nabla\times \mathbf{F}=0 \implies \mathbf{F}\text{ is conservative} \end{equation} When $F$ is in cartesian coordinates. ...
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Minkowski's equation of motion

I'm trying to prove $f^{\mu}U_{\mu}=0$ for four-force $f^{\mu}=c\frac{dP^{\mu}}{ds}$ and four-velocity $U_{\mu}$. I start by using the chain rule, $f^{\mu}=c\frac{dP^{\mu}}{dt}\frac{dt}{ds}=\gamma\...
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For virtual displacement in the Lagrangian, why is $\delta \dot{x_i} = \delta \frac{dx_i}{dt} = \frac{d}{dt}\delta x_i \equiv 0$?

I am having trouble understanding why $$\delta \dot{x_i} = \delta \frac{dx_i}{dt} = \frac{d}{dt}\delta x_i \equiv 0.\tag{7.132}$$ you can see my explanation leading up to it below. I would greatly ...
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Fake Perpetual Motion Device using an Electromagnet

I was watching a video of one of those fake perpetual motion machines where a ball falls down a hole and then flies off a ramp back onto the starting platform. As suspected, the large base is hiding ...
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Find the equation for the angle $\theta$ in which the particle leaves the semicircle. No Friction [closed]

I think I missed something in this mechanics problem. We're given a polished (no friction) and homogeneous hemicircle which has mass $M$ and a particle of mass $m$ laying on the top of it. There is ...
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About the equation $\frac {d^2} {dt^2}\vec x(t) = \nabla \times \vec F(x(t))$. Motion in a curl vector field

I was wondering if there is a physical interpretation of ODEs of the form $$\frac d{dt}\vec x(t)=\vec y(t)$$ $$ \frac d{dt} \vec y(t) = \nabla \times \vec F(x(t))$$ (or equivalently $\frac {d^2} {dt^2}...
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Help with understanding virtual displacement in Lagrangian

I know that these screen shots are not nice but I have a simple question buried in a lot of information My question Why can't we just repeat what they did with equation (7.132) to equation (7.140) ...
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Where does the stored potential energy in a bar go, when I reduce the load acting on it?

Consider a deformable bar, fixed at one end and acted upon by a load P (gradually increasing), as shown, through a rigid plate attached to its end. At the end of loading, potential energy is stored in ...
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Doubt in Equations of motion of rigid body, can anyone tell me how the second step occurred from first one [closed]

Doubt in Equations of motion of rigid body, can anyone tell me how the second step occurred from first one.
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