All Questions
56
questions
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Derivation of Lagrange's equation form d'Alembert's Principle
Im studying Mechanics form Goldstein. I cross this equation in "Derivation of Lagranges equation from d'Alembert's Principle",section 1.4. I have two questions from this derivation.
The ...
1
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0
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20
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Do partial derivation respect to velocity and total derivation respect to time commute? [duplicate]
Imagine we have a function of position $x^i$ and velocity $v^i$ $f(x,v)$. Position and velocity are both functions of time $t$. If the function doesn't depend explicitely on time, then we have the ...
2
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2
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161
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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 ...
1
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2
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398
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Total time derivatives and partial coordinate derivatives in classical mechanics
This may be more of a math question, but I am trying to prove that for a function $f(q,\dot{q},t)$
$$\frac{d}{dt}\frac{∂f}{∂\dot{q}}=\frac{∂}{∂\dot{q}}\frac{df}{dt}−\frac{∂f}{∂q}.\tag{1}$$
As part of ...
1
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3
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141
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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
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1
answer
123
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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:
$$\delta L= \frac{\partial L}{\partial q}\delta q + \frac{\partial L}{\partial \dot q}\...
1
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3
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306
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In Euler-Lagrange equations, why we take ${\partial T}/{\partial {x}} $ as zero (when no terms of $x$ is present)?
Basically, why we treat them as independent quantities. I know what a partial derivative is, It means if a function depends on multiple variables, the partial derivative with respect to a particular ...
4
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1
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How do total time derivatives of partial derivatives of functions work?
Say im trying to prove $\frac{\partial \dot{T}}{\partial \dot{q}^i} - 2\frac{\partial {T}}{\partial {q^i}} = - \frac{\partial {V}}{\partial {q^i}}$ from the Lagrangian equation: $L = T - V$, and the ...
0
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3
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195
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Having trouble taking derivative of a cross product when finding Lagrangian to find force equation for rotating non-inertial frame
I've been working on a problem for my classical mechanics 2 course and I am stuck on a little math problem. Basically, I am trying to prove this equation of motion with a Lagrangian:
$$m\ddot{r} = F + ...
2
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1
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225
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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'...
0
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1
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83
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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 ...
2
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2
answers
466
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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 $\mathbf{x} = (x, y, z)$. ...
2
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1
answer
100
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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 ...
0
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0
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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. $q_j$).
$q$ ...
0
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3
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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 ...