All Questions
Tagged with differentiation classical-mechanics
126 questions
1
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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
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1
answer
141
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Reasoning behind $\delta \dot q = \frac{d}{dt} \delta q$ in deriving E-L equations [duplicate]
Consider a Lagrangian $L(q, \dot{q}, t)$ for a single particle. The variation of the Lagrangian is given by:
$$\delta L= \frac{\partial L}{\partial q}\delta q + \frac{\partial L}{\partial \dot q}\...
1
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3
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354
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In Euler-Lagrange equations, why we take ${\partial T}/{\partial {x}} $ as zero (when no terms of $x$ is present)?
Basically, why we treat them as independent quantities. I know what a partial derivative is, It means if a function depends on multiple variables, the partial derivative with respect to a particular ...
- 69
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1
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60
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Goldstein's generalized forces
In Goldstein's
$$
Q_{j}=\sum_{i} \mathbf{F}_{i} \cdot \frac{\partial \mathbf{r}_{i}}{\partial q_{j}}=-\sum_{i} \nabla_{i} V \cdot \frac{\partial \mathbf{r}_{i}}{\partial q_{j}}
$$
which is exactly ...
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1
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How do total time derivatives of partial derivatives of functions work?
Say im trying to prove $\frac{\partial \dot{T}}{\partial \dot{q}^i} - 2\frac{\partial {T}}{\partial {q^i}} = - \frac{\partial {V}}{\partial {q^i}}$ from the Lagrangian equation: $L = T - V$, and the ...
- 75
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2
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205
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Multivariable chain rule in classical mechanics; example of physical system [closed]
I'm a teaching assistant in calculus and my students who are studying mechanical engineering asked me to explain the multivariable chain rule. So I thought it could be fun if I could give an example ...
- 127
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1
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How do I show that $\dfrac{dE}{dt} = \dfrac{\partial U}{\partial t}$ where $U(\mathbf{r}_1,...,\mathbf{r}_N,t)$ is the potential energy?
I'm working through Chapter 1 of Analytical Mechanics for Relativity and Quantum Mechanics, and in Section 1.8, they derive the equality in the question. To show this, they claim that
$$\dfrac{dT}{dt} ...
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3
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232
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Having trouble taking derivative of a cross product when finding Lagrangian to find force equation for rotating non-inertial frame
I've been working on a problem for my classical mechanics 2 course and I am stuck on a little math problem. Basically, I am trying to prove this equation of motion with a Lagrangian:
$$m\ddot{r} = F + ...
- 17
2
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1
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244
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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
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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
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The use of $x_\varepsilon (t) = x(t) + \varepsilon (t)$ and $x_\varepsilon (t) = x(t) + \varepsilon \eta (t)$ in proving Hamilton's principle
The following Wikipedia page uses $x_\varepsilon (t) = x(t) + \varepsilon (t)$ in the proof.
https://en.wikipedia.org/wiki/Hamilton%27s_principle#Mathematical_formulation
But in my mechanics book (by ...
- 485
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2
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486
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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
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1
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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 ...
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316
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D'Alembert's principle derivation in Goldstein's Classical Mechanics book [duplicate]
(I could not find any answer to the following question in other related questions posted on SE, so asking it here.)
In the derivation of D'Alembert's principle in his "book", Goldstein uses the ...
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3
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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 ...
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2k
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Spring Constant and Second Derivative
According to my book the spring constant is given by $k=V''(X_o)$. Where $V(X)$ is the potential energy function. if I use the function $V(X)=X^6+X^4$ the spring constant is zero at $X_o=0$. However, ...
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3
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237
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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$$
Where $r=|{\bf r}|$ is the distance and $U(r)$ is a potential for a central force.
This is from page 13 of "...
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2
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146
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Work done as change of potential, how total derivative is converted to partial derivative
I am reading Goldsetein's Classic Mechanics 3rd edition in Chapter 1 it says,
If work done in moving form point 1 to 2 denoted by $W_{12}$, is independent of the path it should be possible to ...
- 113
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45
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About Lagrange equation [duplicate]
$$\frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial L}{\partial \dot{q}_j} \right ) = \frac {\partial L}{\partial q_j}.$$
I don't understand partial derivative by "function" (e.g. $q_j$).
$q$ ...
- 37
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3
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2k
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Time derivative of the Lagrangian
I have the time derivative of the lagrangian:
$$\frac{\mathrm d \mathcal L}{\mathrm d t}=\sum_i\left(\frac{\partial \mathcal L}{\partial q_i}\frac{\mathrm d q_i}{\mathrm d t}+\frac{\partial \mathcal ...
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55
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Math question about point transformations
I am trying to prove the classic problem to showcase Lagrangian's scalar invariant property.
Namely, that if you have $x_i = \{ x_1, ...., x_n; t \}$ , you can then represent $L(x_1,....,\dot{x_1},.....
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2
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1k
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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 ...
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2
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295
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Taylor expansion in derivation of Noether-theorem
In my classical mechanics lecture we derived the Noether-theorem for a coordinate transformation given by:
$$ q_i(t) \rightarrow q^{'}_i(t)=q_i(t) + \delta q_i(t) = q_i(t) + \lambda I_i(q,\dot q,t).$$...
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3
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140
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Mathematical identity related to d'Alembert's Principle
In Hand & Finch's book on Analytical Mechanics, I came across this mathematical identity Eq. 1.19 in Chapter 1, page 5, which is related to the description of d'Alembert's principle:
$$\dot{\vec{...
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1
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58
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Change of variable in function
Suppose I have a function $h(\theta)$ measuring the height of a piston, with $\theta = \omega t$. I would like to know the vertical acceleration of this piston as $\omega$ changes at the point $\theta ...
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1
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320
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What is the meaning of $d$? [duplicate]
What is the meaning of $d$? Is is Delta? If it is Delta, why is it then not $\Delta$? I am still confused with that. Can someone help explain it to me?
user224451
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1
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Lagrange equations in a conservative system, understanding $\nabla_i$
For a system of multiple particles with conservative forces: $\mathbf{F}_i = - \nabla_i V$, with $V \equiv V(\mathbf{r}_1,\dots,\mathbf{r}_N)$ the potential in function of the position of the $N$ ...
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145
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Constraints and time derivative
Consider a system of $N$ particles. There are $C$ holonomic time independent constraints, $$ \begin{aligned} f_1(\mathbf{r}_1,\dots,\mathbf{r}_N) & =0 \\ f_2(\mathbf{r}_1,\dots,\mathbf{r}_N) & ...
- 265
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1
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852
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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 ...
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686
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Derivative of Lagrangian with respect to a vector
Sometimes to find an equation of motion, the Lagrangian is derivated with respect to the (position) vector. How can this be possible?
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1
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359
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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
$$\frac{\mathrm d}{\mathrm dt}\left(\frac{\partial {r_i}}{\partial {q_j}}\right) = \frac{\partial {\dot r_i}}{\partial {q_j}}=\sum_k \...
- 595
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7
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Why isn't the Euler-Lagrange equation trivial?
The Euler-Lagrange equation gives the equations of motion of a system with Lagrangian $L$. Let $q^\alpha$ represent the generalized coordinates of a configuration manifold, $t$ represent time. The ...
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4
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The definition of the hamiltonian in lagrangian mechanics
So going through the "Analytical Mechanics by Hand and Finch". In section 1.10 of the book, the Hamiltonian $H$ is defined as: $$H = \sum_k{\dot{q_k}\frac{\partial L}{\partial \dot{q_k}} -L}.\tag{1.65}...
user63248
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2
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Derivation of generalized velocities in Lagrangian mechanics
So I know that: $$r_i = r_i(q_1, q_2,q_3,...., q_n, t)$$
Where $r_i$ represent the position of the $i$th part of a dynamical system and the $q_n$ represent the dynamical variables of the system ($n$ =...
user63248
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2
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Derivation of Rotational Motion Equations using Calculus
How are the equations for rotational motion derived using calculus and the following general equations ?
$$\mathbf{v}(t) = \mathbf{v}_0+\int_{t_0}^t \mathbf{a}(t')dt'$$
$$\mathbf{r}(t) = \mathbf{r}...
1
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1
answer
412
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What does lowercase-delta mean in Noether's first theorem?
Most expressions of Noether's Theorem I have come across do not use lowercase delta, but a couple sites do. I am confused....... Check out page 21 of the June 23 issue of 'Science News' ...
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Squaring the momentum operator in QM becomes a second derivative. How?
$\frac{p^2}{2m}$ is the Kinetic energy in classical mechanics. However, the same $p^2$ becomes the second derivative $\frac{\partial ^2}{\partial x^2}$ in the Kinetic Energy operator in QM. I mean it ...
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2
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Order of derivatives in Euler-Lagrange equations
The Euler-Lagrange equations are
$$\frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial L}{\partial \dot{q}_i} = \frac{\partial L}{\partial q_i}.\tag{1}$$
Is it equivalent to switch the derivatives on the ...
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0
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Functional derivative of a symmetrized field
I'm confused whether a symmetrisation/antisymmetrization of a function with respect to its arguments, i.e., $$F(x_1,x_2,...,x_n)=\frac{1}{\sqrt{N!}}\sum_{\pi}\textrm{sgn}(\pi)~f(x_{\pi(1)},...,x_{\pi(...
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Why $\sum\limits_{i} \frac{\partial L}{\partial \dot{q_i}} \dot{q_i} = \sum\limits_{i} \frac{\partial T}{\partial \dot{q_i}} \dot{q_i} = 2T$? [closed]
From Landau and Lifschitz's "Mechanics"; section 6.
I understand up to this point
$$E \equiv \sum\limits_{i} \dot{q_i}\frac{\partial L}{\partial \dot{q_i}} - L $$
Then the author states:
Using ...
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2
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123
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Physics of small values and differentials
In some formulas in physics having a ratio, for example $ pressure={F \over\ A}$, the denominator is chosen to be a small quantity ($\Delta A$) and is written like,
$$P= {\Delta F\over \Delta A}.$$
...
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2k
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Velocity in generalized coordinates
Consider the expression of velocity in generalized coordinates, $\mathbf v = \frac {d \mathbf x}{dt}$, where $\mathbf x = \mathbf x (\mathbf q(t), t)$.
We end up with a total derivative, i.e $$\...
- 647
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2
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207
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Take derivative to a cross product of two vectors with respect to the position vector [closed]
I'm doing classical mechanics about Lagrange formulation and confused about something about vector differentiation.The Lagrangian is given:
$$\mathcal{L}=\frac{m}{2}(\dot{\vec{R}}+\vec{\Omega} \times \...
- 103
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2
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627
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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
1
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2
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1k
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Formulation of acceleration in general
I know that in fluid dynamics, we use Lagrangian description of acceleration. That is, a material derivative $$\frac{dv}{dt}=\frac{\partial v}{\partial t}+(v\cdot\nabla )v .$$ My question is can we ...
- 1,246
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1
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2k
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What is the curl of $k\hat{r}/r^n$?
I'm trying to find the curl of ${\bf F}(r) = k \hat{r}/r^n$. I think that this converts to:
$$
k\left(\frac{\hat{x}}{r} + \frac{\hat{y}}{r} + \frac{\hat{z}}{r}\right)\frac{1}{(x^2 + y^2 + z^2)^{n/2}}
...
- 13
1
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0
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260
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Partial derivative of $v$ w.r.t. $x$ in Lagrangian dynamics [duplicate]
In Lagrangian dynamics, when using the Lagrangian thus:
$$
\frac{d}{dt}(\frac{\partial \mathcal{L} }{\partial \dot{q_j}})-
\frac{\partial \mathcal{L} }{\partial q_j} = 0
$$
often we get terms such ...
- 11
1
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3
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123
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Lagrange classical relation
I have been studying theoretical mechanics and just now I came cross a formula
called "Lagrange classical relation", that is, if we let $q_1$, $q_2$,$\cdot $$ \cdot $$\cdot $, $q _ m$, $t$ be the $...
- 115
1
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1
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671
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Determining the change in radius of water flowing from a faucet - more general question on differentiation
I've been outside of the academic world for several years now, and I'm forcing myself to go back through old textbooks and resources and work through the information in there. I can tell I'm losing ...
- 11
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1
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483
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Least action principle : is $ \delta \frac{dx}{dt} = \frac{d \delta x}{dt} $ always true?
(Just some recalls)
We have an action on which we want to apply Least action principle.
$$ S=\int_{t_i}^{t_f} L(q,\dot{q},t)dt$$
We assume that $t \mapsto q(t)$ is the function that will extremise ...
- 1,560