All Questions
Tagged with differentiation classical-mechanics
126 questions
154
votes
9
answers
19k
views
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 ...
- 2,235
57
votes
7
answers
10k
views
Why isn't the Euler-Lagrange equation trivial?
The Euler-Lagrange equation gives the equations of motion of a system with Lagrangian $L$. Let $q^\alpha$ represent the generalized coordinates of a configuration manifold, $t$ represent time. The ...
- 1,883
25
votes
3
answers
3k
views
Why don't we see the covariant derivative in classical mechanics?
I am wondering why I have seen the covariant derivative for the first time in general relativity.
Starting from the point that the covariant derivative generalise the concept of derivative in curved ...
- 873
25
votes
2
answers
2k
views
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
15
votes
2
answers
4k
views
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
8
votes
1
answer
712
views
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 ...
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7
votes
3
answers
1k
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In equation (3) from lecture 7 in Leonard Susskind’s ‘Classical Mechanics’, should the derivatives be partial?
Here are the equations. ($V$ represents a potential function and $p$ represents momentum.)
$$V(q_1,q_2) = V(aq_1 - bq_2)$$
$$\dot{p}_1 = -aV'(aq_1 - bq_2)$$
$$\dot{p}_2 = +bV'(aq_1 - bq_2)$$
Should ...
7
votes
2
answers
1k
views
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
$$\dfrac{d}{dt}{\bf p}_i = {\bf F}_i^{(e)...
- 37.4k
6
votes
7
answers
255
views
Is every $dm$ piece unequal when using integration of a non-uniformly dense object?
When we want to find the total charge of an object or total mass, usually we start off with a setup such as:
$$ m = \int dm \:\;\:\text{or} \:\;\:q = \int dq$$
in which we then use (and to keep it ...
- 63
5
votes
2
answers
10k
views
When does the total time derivative of the Hamiltonian equal its partial time derivative?
When does the total time derivative of the Hamiltonian equal the partial time derivative of the Hamiltonian? In symbols, when does $\frac{dH}{dt} = \frac{\partial H}{\partial t}$ hold?
In Thornton &...
- 935
5
votes
4
answers
1k
views
Partial derivative in Newtons Second law
Newton's second law states Force is the time derivative of momentum. But is it a total derivative or partial derivative? What is the reason behind it?
- 53
4
votes
2
answers
993
views
Why do we consider potential energy function $U(x)$ differentiable?
Recently when skimming through my physics-text I encountered an interesting definition of Force
$$F(x) = -\frac{\mathrm dU(x)}{\mathrm dx}$$
We were taught that some functions are continuous but not ...
- 373
4
votes
4
answers
440
views
Variation of a function
I'm studying calculus of variations and Lagrangian mechanics and i don't understand something about the variational operator
Let's say for example that i got a Lagrangian $L [x(t), \dot{x}(t), t] $ ...
- 321
4
votes
2
answers
1k
views
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
4
votes
1
answer
2k
views
How do total time derivatives of partial derivatives of functions work?
Say im trying to prove $\frac{\partial \dot{T}}{\partial \dot{q}^i} - 2\frac{\partial {T}}{\partial {q^i}} = - \frac{\partial {V}}{\partial {q^i}}$ from the Lagrangian equation: $L = T - V$, and the ...
- 75
4
votes
1
answer
389
views
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
3
votes
2
answers
155
views
How to prove that $ \delta \frac{dq_i}{dt} = \frac{d \delta q_i}{dt} $? [duplicate]
During the proof of least action principle my prof used the equation $ \delta \frac{dx}{dt} = \frac{d \delta x}{dt} $. We were not proved this equality. I was curious to know why this is true so I ...
3
votes
2
answers
718
views
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
2
votes
6
answers
274
views
Lagrangian - How can we differentiate with respect to time if $v$ not a function of time?
In the Lagrangian itself, we know that $v$ and $q$ don't depend on $t$ (i.e - they are not functions of $t$ - i.e., $L(q,v,t)$ is a state function.)
Imagine $L = \frac{1}{2}mv^2 - mgq$
Euler-Lagrange ...
- 535
2
votes
1
answer
742
views
Proof that the Euler-Lagrange equations hold in any set of coordinates if they hold in one
This is a question about a specific proof presented in the book Introduction to Classical Mechanics by David Morin. I have highlighted the relevant portion in the picture below.
In the remark, he ...
anon
2
votes
2
answers
207
views
Take derivative to a cross product of two vectors with respect to the position vector [closed]
I'm doing classical mechanics about Lagrange formulation and confused about something about vector differentiation.The Lagrangian is given:
$$\mathcal{L}=\frac{m}{2}(\dot{\vec{R}}+\vec{\Omega} \times \...
- 103
2
votes
1
answer
1k
views
What does a Umlaut (double dot) above an angle mean?
I'm reading a paper on double pendulums and there is an equation of motion that contains a double dot (Umlaut) above an angle. What does this mean / is this a standard notation in equations of motion?...
- 287
2
votes
1
answer
244
views
Is Goldstein's matrix formalism to Hamiltonian mechanics necessary? [closed]
I am trying to see whether the matrix formalism of the Hamiltonian formalism (used in Goldstein's textbook) is truly necessary to solve problem in this framework.
It appears so based on the problem I'...
- 1,081
2
votes
1
answer
106
views
Confusion regarding the time derivative term in Lagrange's equation
I am solving a pendulum attached to a cart problem. Without going into unnecessary details, the generalised coordinates are chosen to be $x$ and $\theta$. The kinetic energy of the system contains a ...
- 192
2
votes
2
answers
1k
views
Confusing Total Derivative and Partial Derivative in Classical Field Theory - Noether Theorem
I'm really confused about total derivatives and partial derivatives.
My multivariable calculus book (Guidorizzi vol 2 Um Curso de Calculo) says that if I have a function like $f(a(u,v),b(u,v))$ then ...
2
votes
1
answer
3k
views
Lagrange equations in a conservative system, understanding $\nabla_i$
For a system of multiple particles with conservative forces: $\mathbf{F}_i = - \nabla_i V$, with $V \equiv V(\mathbf{r}_1,\dots,\mathbf{r}_N)$ the potential in function of the position of the $N$ ...
- 265
2
votes
1
answer
852
views
Hamiltonian time-independent, partial derivative always zero?
For conceptual simplicity, let's restrict the discussion to systems with a two-dimensional phase space $\mathcal P$ with generalized coordinates $(q,p)$.
Hamiltonian is a function that maps a pair ...
- 51
2
votes
4
answers
1k
views
The definition of the hamiltonian in lagrangian mechanics
So going through the "Analytical Mechanics by Hand and Finch". In section 1.10 of the book, the Hamiltonian $H$ is defined as: $$H = \sum_k{\dot{q_k}\frac{\partial L}{\partial \dot{q_k}} -L}.\tag{1.65}...
user63248
2
votes
1
answer
383
views
Having trouble deriving the exact form of the Kinematic Transport Theorem
The Kinematic transport theorem is a very basic theorem relating time derivatives of vectors between a non rotating frame and another one that's rotating with respect to it with a uniform angular ...
- 3,358
2
votes
2
answers
486
views
Derivative of Lagrangian with respect to velocity
My question revolves around this lecture notes on page $109$ equation $(5.1.10)$.
Let's stick to $\mathbb{R}^3$ and consider a particle in $3$-space with position vector $\mathbf{x} = (x, y, z)$. ...
- 121
2
votes
2
answers
627
views
What conditions are required for the derivative of kinetic energy to be F.v?
In Ch. 1 Derivation 1 of Goldstein's mechanics, we have
Show that for a single particle with constant mass the equation of motion implies
$$
\frac{dT}{dt} = \vec{F}\cdot\vec{v}
$$
The first step ...
- 647
2
votes
1
answer
3k
views
Full time-derivative, Poisson brackets and Hamilton's equations (classical mechanics)
While studying Poisson brackets in classical mechanics and the derivation of $\dot{q_j}=\{q_j,H\}$ and $\dot{p_j}=\{p_j,H\}$ form of Hamilton's equations I encountered a surpsing identity, which led ...
- 390
2
votes
1
answer
463
views
What is the function type of the generalized momentum?
Let
$$L:{\mathbb R}^n\times {\mathbb R}^n\times {\mathbb R}\to {\mathbb R}$$
denote the Lagrangian (it should be differentiable) of a classical system with $n$ spatial coordinates. In the action
$...
- 8,703
2
votes
1
answer
598
views
Clarification on a Goldstein formula steps (classical mechanics)
At page 20 of Classical Mechanics' Goldstein (Third edition), there are these two steps given between eqs. (1.51) and (1.52):
$$\sum_i m_i \ddot {\bf r}_i \cdot \frac{\partial {\bf r_i}}{ \partial ...
- 1,143
2
votes
1
answer
103
views
Time derivative of a "general" vector $\vec A$ in an accelerating frame: what about e.g. velocity $\vec v$?
According to Morin "Classical Mechanics" (Section 10.1, page 459), the derivative of a general vector $\vec A$ in an accelerating frame may be given as
$$\frac{d\vec A}{dt}=\frac{\delta \vec ...
- 23
2
votes
5
answers
348
views
Why does $\vec{r}\cdot\dot{\vec{r}}=r\dot{r}$?
Why is $$\vec{r}\cdot\dot{\vec{r}}=r\dot {r}$$ true? Before saying anything, I have seen the proofs using spherical coordinates for $$\dot{\vec {r}}= \dot{r}\vec{u_r}+r\dot{\theta}\vec{u_\theta}+r\sin\...
- 69
2
votes
2
answers
172
views
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 ...
- 41
2
votes
1
answer
145
views
Constraints and time derivative
Consider a system of $N$ particles. There are $C$ holonomic time independent constraints, $$ \begin{aligned} f_1(\mathbf{r}_1,\dots,\mathbf{r}_N) & =0 \\ f_2(\mathbf{r}_1,\dots,\mathbf{r}_N) & ...
- 265
2
votes
2
answers
1k
views
When can one omit a total time derivative in the Lagrangian formulation?
I am studying Lagrangian and Hamiltonian mechanics and i am using Landau & Lifshitz and Goldstein books. Both of them state that a modified lagrangian $$L'=L+\frac{df}{dt}$$ gives the same ...
- 45
2
votes
1
answer
64
views
Implications of Galilei-Invariance on a time-independent potential
I'm trying to compute a result shown in my classical mechanics lecture on my own. Namely, consider that a system composed of $n$ particles follows a law of force
$m_k\ddot{\vec{x_k}} = \vec{F_k}(\vec{...
- 324
2
votes
0
answers
207
views
Higher order versions of momentum? Can conservation principles be established and used? [closed]
Question
Can higher order derivatives of momentum be useful in creating theories of dynamics if they have conservation principles? Even if they aren't needed, could it be done in theory? For instance,...
- 1,127
2
votes
3
answers
420
views
Potential energy with constraints moving body
I know that for conservative forces $\vec{F}=-\nabla{U}$. Let's consider the case of gravitational potential energy, I know that $U=mgy$. Just to check: $\vec{F}=-\nabla{U}=(0,-mg)$: perfect!
Now, let'...
1
vote
3
answers
2k
views
Time derivative of a function in Phase Space
Consider a function $\mathcal{H}(q_i,p_i;t)$ such that it obeys the equation:
$$ \frac{d\mathcal{H}}{dt}=\frac{\partial\mathcal{H}}{\partial t}$$
What does this equation imply (read: mean), physically?...
- 165
1
vote
2
answers
268
views
Why the $\Delta$ in the definition of pressure? (fluid mechanics)
I'm an engineering student (first year) studying Physics 1 (now an introduction to fluid mechanics).
Q1
In my physics textbook, the "medium pressure" is defined as:
$$p_m = \frac{\Delta F_{\...
- 141
1
vote
3
answers
353
views
In Euler-Lagrange equations, why we take ${\partial T}/{\partial {x}} $ as zero (when no terms of $x$ is present)?
Basically, why we treat them as independent quantities. I know what a partial derivative is, It means if a function depends on multiple variables, the partial derivative with respect to a particular ...
- 69
1
vote
3
answers
237
views
Why is $\nabla U(r) = \frac{dU(r)}{dr} \nabla r$?
Does anyone have a proof for the equation:
$$\nabla U(r) = \frac{dU(r)}{dr} \nabla r$$
Where $r=|{\bf r}|$ is the distance and $U(r)$ is a potential for a central force.
This is from page 13 of "...
- 435
1
vote
3
answers
142
views
Proof of Lagrangian equations [closed]
Context: Trying to proof Lagrangian equations without an explicit usage of the concept of virtual displacement.
(disclaimer for happy close-vote triggers: I'm not related to any academic institution ...
- 1,158
1
vote
3
answers
3k
views
Second derivative of energy as frequency of oscillations [closed]
Is there a way to algebraically see why when I take the second derivative of a potential energy in a point where it is minimal (force is zero), I generally get the frequency (squared) of the ...
- 1,048
1
vote
1
answer
359
views
How come $\frac{d}{dt}\left(\frac{\partial {r_i}}{\partial {q_j}}\right) = \frac{\partial {\dot r_i}}{\partial {q_j}}$ in Lagrangian mechanics? [duplicate]
It is written in the Goldstein's Classical Mechanics text that
$$\frac{\mathrm d}{\mathrm dt}\left(\frac{\partial {r_i}}{\partial {q_j}}\right) = \frac{\partial {\dot r_i}}{\partial {q_j}}=\sum_k \...
- 595
1
vote
2
answers
274
views
Order of derivatives in Euler-Lagrange equations
The Euler-Lagrange equations are
$$\frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial L}{\partial \dot{q}_i} = \frac{\partial L}{\partial q_i}.\tag{1}$$
Is it equivalent to switch the derivatives on the ...
- 319