All Questions
56
questions
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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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votes
2
answers
285
views
Taylor expansion in derivation of Noether-theorem
In my classical mechanics lecture we derived the Noether-theorem for a coordinate transformation given by:
$$ q_i(t) \rightarrow q^{'}_i(t)=q_i(t) + \delta q_i(t) = q_i(t) + \lambda I_i(q,\dot q,t).$$...
- 13
0
votes
3
answers
136
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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:
$$\dot{\vec{...
- 113
2
votes
1
answer
3k
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Lagrange equations in a conservative system, understanding $\nabla_i$
For a system of multiple particles with conservative forces: $\mathbf{F}_i = - \nabla_i V$, with $V \equiv V(\mathbf{r}_1,\dots,\mathbf{r}_N)$ the potential in function of the position of the $N$ ...
- 265
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votes
1
answer
656
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Derivative of Lagrangian with respect to a vector
Sometimes to find an equation of motion, the Lagrangian is derivated with respect to the (position) vector. How can this be possible?
1
vote
1
answer
341
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How come $\frac{d}{dt}\left(\frac{\partial {r_i}}{\partial {q_j}}\right) = \frac{\partial {\dot r_i}}{\partial {q_j}}$ in Lagrangian mechanics? [duplicate]
It is written in the Goldstein's Classical Mechanics text that
$$\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
57
votes
7
answers
9k
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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 ...
- 1,826
2
votes
4
answers
1k
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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: $$H = \sum_k{\dot{q_k}\frac{\partial L}{\partial \dot{q_k}} -L}.\tag{1.65}...
user63248
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votes
2
answers
2k
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Derivation of generalized velocities in Lagrangian mechanics
So I know that: $$r_i = r_i(q_1, q_2,q_3,...., q_n, t)$$
Where $r_i$ represent the position of the $i$th part of a dynamical system and the $q_n$ represent the dynamical variables of the system ($n$ =...
user63248
1
vote
2
answers
260
views
Order of derivatives in Euler-Lagrange equations
The Euler-Lagrange equations are
$$\frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial L}{\partial \dot{q}_i} = \frac{\partial L}{\partial q_i}.\tag{1}$$
Is it equivalent to switch the derivatives on the ...
- 319
1
vote
2
answers
160
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Why $\sum\limits_{i} \frac{\partial L}{\partial \dot{q_i}} \dot{q_i} = \sum\limits_{i} \frac{\partial T}{\partial \dot{q_i}} \dot{q_i} = 2T$? [closed]
From Landau and Lifschitz's "Mechanics"; section 6.
I understand up to this point
$$E \equiv \sum\limits_{i} \dot{q_i}\frac{\partial L}{\partial \dot{q_i}} - L $$
Then the author states:
Using ...
- 33
0
votes
2
answers
2k
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Velocity in generalized coordinates
Consider the expression of velocity in generalized coordinates, $\mathbf v = \frac {d \mathbf x}{dt}$, where $\mathbf x = \mathbf x (\mathbf q(t), t)$.
We end up with a total derivative, i.e $$\...
- 627
2
votes
2
answers
188
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:
$$\mathcal{L}=\frac{m}{2}(\dot{\vec{R}}+\vec{\Omega} \times \...
- 103
1
vote
0
answers
258
views
Partial derivative of $v$ w.r.t. $x$ in Lagrangian dynamics [duplicate]
In Lagrangian dynamics, when using the Lagrangian thus:
$$
\frac{d}{dt}(\frac{\partial \mathcal{L} }{\partial \dot{q_j}})-
\frac{\partial \mathcal{L} }{\partial q_j} = 0
$$
often we get terms such ...
- 11
1
vote
3
answers
120
views
Lagrange classical relation
I have been studying theoretical mechanics and just now I came cross a formula
called "Lagrange classical relation", that is, if we let $q_1$, $q_2$,$\cdot $$ \cdot $$\cdot $, $q _ m$, $t$ be the $...
- 115