All Questions
Tagged with differentiation classical-mechanics
126 questions
2
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When can one omit a total time derivative in the Lagrangian formulation?
I am studying Lagrangian and Hamiltonian mechanics and i am using Landau & Lifshitz and Goldstein books. Both of them state that a modified lagrangian $$L'=L+\frac{df}{dt}$$ gives the same ...
- 45
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1
answer
389
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Liouville's Theorem. True or False?
In my quantum theory course, there is a question ask for checking whether the expectations in quantum and classical Liouville theory are identical.
Here is the original version:
"Assume the system ...
- 431
2
votes
3
answers
420
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Potential energy with constraints moving body
I know that for conservative forces $\vec{F}=-\nabla{U}$. Let's consider the case of gravitational potential energy, I know that $U=mgy$. Just to check: $\vec{F}=-\nabla{U}=(0,-mg)$: perfect!
Now, let'...
4
votes
2
answers
993
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Why do we consider potential energy function $U(x)$ differentiable?
Recently when skimming through my physics-text I encountered an interesting definition of Force
$$F(x) = -\frac{\mathrm dU(x)}{\mathrm dx}$$
We were taught that some functions are continuous but not ...
- 373
0
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0
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359
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Different subscripts for $\nabla$ operators while deriving force on system of many particles
Consider a system of 4 particles in an external conservative field. So force acting on each particle is derived from potential energy $U(x,y,z)$of the particle+field system:
Total (external) force on ...
- 1,034
1
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0
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80
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How does the gradient operator pick up a minus sign when the reference frame is switched from one particle to another? [closed]
A potential between two particles, $i$ and $j$, is given as a function only of the separation distance,
$$V_{ij} = V_{ij}(|r_i − r_j|)$$
It should follow that the force by $j$ on $i$ is equal and ...
2
votes
0
answers
207
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Higher order versions of momentum? Can conservation principles be established and used? [closed]
Question
Can higher order derivatives of momentum be useful in creating theories of dynamics if they have conservation principles? Even if they aren't needed, could it be done in theory? For instance,...
- 1,127
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3
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Time derivative of a function in Phase Space
Consider a function $\mathcal{H}(q_i,p_i;t)$ such that it obeys the equation:
$$ \frac{d\mathcal{H}}{dt}=\frac{\partial\mathcal{H}}{\partial t}$$
What does this equation imply (read: mean), physically?...
- 165
1
vote
1
answer
223
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Derivation Of Euler-Lagrange Equation [closed]
I want the proof of this relation in details,
$$
\frac{\rm d}{{\rm d}t}\left(\frac{\partial\vec{r}_v}{\partial q_\alpha}\right)=\frac{\partial\vec{\dot{r}_v}}{\partial q_\alpha}
$$
- 27
2
votes
1
answer
3k
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Full time-derivative, Poisson brackets and Hamilton's equations (classical mechanics)
While studying Poisson brackets in classical mechanics and the derivation of $\dot{q_j}=\{q_j,H\}$ and $\dot{p_j}=\{p_j,H\}$ form of Hamilton's equations I encountered a surpsing identity, which led ...
- 390
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1
answer
91
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In central-force mechanics, how do we substitute $ξ=\frac{1}{r}$?
I have taken a look at central-force mechanics in the past, but I still cannot understand how $ξ=\frac{1}{r}$ is substituted to find $\frac{d^2r}{dt^2}$ in terms of ξ.
So I know from $F=ma$ that:
$$(...
- 21
0
votes
1
answer
126
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Clarification about some steps in the derivation of the Lie derivative (mechanics)
First of all, this question may seem to be undefined, because I'm not sure how to connect this (to me) newly introduced concept with the abstract notion of the Lie derivative. I'm not even sure if I ...
- 912
1
vote
1
answer
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Trouble understanding Landau & Lifshitz writing about Lagrangians and Galilean Relativity [duplicate]
We have two inertial coordinate systems, $K'$ and $K$. $K$ is moving with infinitesimal velocity ${\epsilon}$ relative to $K'$. Using Galilean relativity we can transform this into $v'=v+{\epsilon}$. ...
- 387
1
vote
1
answer
211
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Lagrangian formalism (demonstration)
My question is about the multiplicity of the Lagrangian to a Physics system.
I pretend to demonstrate the following proposition:
For a system with $n$ degrees of freedom, written by the Lagrangian ...
- 1,092
2
votes
1
answer
463
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What is the function type of the generalized momentum?
Let
$$L:{\mathbb R}^n\times {\mathbb R}^n\times {\mathbb R}\to {\mathbb R}$$
denote the Lagrangian (it should be differentiable) of a classical system with $n$ spatial coordinates. In the action
$...
- 8,703
0
votes
1
answer
72
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Show $\frac{\partial T}{\partial \dot q_j} = m_i \dot r_i^T\frac{\dot r_i }{\partial \dot q_j} $ [closed]
This is a basic result in lagrangian mecanics. Let $T$ be the kinetic energy, $r_i$ be the position of the $i^{th}$ particle in the system I need to show $$\frac{\partial T}{\partial \dot q_j} = \frac{...
- 1,754
4
votes
2
answers
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Trouble with Landau & Lifshitz's expansion of the Lagrangian with respect to $\epsilon$ and $v$ [duplicate]
Hello I have a quick question on what I have been reading in Landau & Lifshitz's book on classical mechanics. I am in the very beginning of the book and I am having trouble with his derivation on ...
- 43
7
votes
2
answers
1k
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A confusion about notation in Goldstein
On treating systems of particles, Goldstein starts with the consideration that whenever there are $k$ particles on a system, the $i$-th one obeys the relation
$$\dfrac{d}{dt}{\bf p}_i = {\bf F}_i^{(e)...
- 37.4k
0
votes
2
answers
460
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Taking time derivative of two dependant variables
I'm not entirely sure if this is correct. I have to take the time derivative of the following:
$$\frac{d}{dt}mr^{2}\dot{\phi}$$
Now, both $r$ and $\dot{\phi}$ depends on the time $t$, so I have to ...
- 2,577
25
votes
2
answers
2k
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Lagrangian Mechanics - Commutativity Rule $\frac{d}{dt}\delta q=\delta \frac{dq}{dt} $
I am reading about Lagrangian mechanics.
At some point the difference between the temporal derivative of a variation and variation of the temporal derivative is discussed.
The fact that the two are ...
- 281
5
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2
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When does the total time derivative of the Hamiltonian equal its partial time derivative?
When does the total time derivative of the Hamiltonian equal the partial time derivative of the Hamiltonian? In symbols, when does $\frac{dH}{dt} = \frac{\partial H}{\partial t}$ hold?
In Thornton &...
- 935
8
votes
1
answer
712
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When motion begins, do objects go through an infinite number of position derivatives?
This might be a very vague and unclear question, but let me explain. When an object at rest moves, or moves from point $A$ to point $B$, we know the object must have had some velocity (1st derivative ...
- 83
2
votes
1
answer
598
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Clarification on a Goldstein formula steps (classical mechanics)
At page 20 of Classical Mechanics' Goldstein (Third edition), there are these two steps given between eqs. (1.51) and (1.52):
$$\sum_i m_i \ddot {\bf r}_i \cdot \frac{\partial {\bf r_i}}{ \partial ...
- 1,143
3
votes
2
answers
718
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What does $\textbf{f} = -\boldsymbol{\nabla} u$ mean in practice and how is it computed?
In classical computer simulations such as molecular dynamics (MD) simulations, one integrates Newton's equations of motion to determine particle trajectories. If we think of Newton's Second Law as ...
- 1,163
15
votes
2
answers
4k
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Why does cancellation of dots $\frac{\partial \dot{\mathbf{r}}_i}{\partial \dot{q}_j} = \frac{\partial \mathbf{r}_i}{\partial q_j}$ work?
Why is the following equation true?
$$\frac{\partial \mathbf{v}_i}{\partial \dot{q}_j} = \frac{\partial \mathbf{r}_i}{\partial q_j}$$
where $\mathbf{v}_i$ is velocity, $\mathbf{r}_i$ is the ...
- 1,483
154
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9
answers
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Calculus of variations -- how does it make sense to vary the position and the velocity independently?
In the calculus of variations, particularly Lagrangian mechanics, people often say we vary the position and the velocity independently. But velocity is the derivative of position, so how can you treat ...
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