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Velocities - Equation 1.46 of Goldstein 3rd edition
In his derivation of the Euler-Lagrange equations from D'Alembert's principle, Goldstein
uses the parametrization (equation 1.45')
$$\displaystyle{\vec{r_i}=\vec{r_i}(q_1,q_2, ..., q_n, t)}\tag{1.45'}$...
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Cartesian coordinate velocity and generalized coordinate velocity
use $x_k$ to denote the kth component of cartesian coordinate, and $q_k$ to denote the generalized coordinate.
Taking the derivate of $x_k(q_1,q_2,q_3,t)$ w.r.t. time, we have
$$\frac{d x_k(q_1,q_2,...
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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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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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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 $$\...
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Why does cancellation of dots $\frac{\partial \dot{\mathbf{r}}_i}{\partial \dot{q}_j} = \frac{\partial \mathbf{r}_i}{\partial q_j}$ work?
Why is the following equation true?
$$\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 ...
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