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### Lagrange's equation implying Newton's 2nd law? [duplicate]

The typical first application of Lagrange's equation is showing that it implies Newton's law for a particle whose Lagrangian is $L=\frac{1}{2}mv^2-V(x)$. Plugging this Lagrangian into Lagrange's ...
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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 ...
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### Partial derivatives in Lagrangian formalism [duplicate]

Suppose I have a function $f = xy$. A partial derivative of $f$ with respect to $x$ implies holding $y$ constant: $$\frac{\partial f}{\partial x} = y$$ Does this mean that in order to evaluate ...
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### Independence of position and velocity vector [duplicate]

Hi I am a mathematics student with an interest in Physics. In our Physics elective our prof. said if $\vec r$ denotes the position vector then the velocity vector $\vec v = \vec {\dot r}$ is ...
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### Partial derivative of position in respect to velocity [duplicate]

Out of interest What is the derivative of $\frac{\partial x(t)}{ \partial v(t)}$
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$$\frac{\mathrm{d}}{\mathrm{d}t} \left ( \frac {\partial L}{\partial \dot{q}_j} \right ) = \frac {\partial L}{\partial q_j}.$$ I don't understand partial derivative by "function" (e.g. $q_j$). $q$ ...
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### Partial Differentiation without chain rule in Euler Lagrange Equations [duplicate]

The Euler-Lagrange equations for a bob attached to a spring are $$\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial v} = \frac{\partial L}{\partial x}$$ But $v$ is a function of $x$. Is it ...
Working on the Maxwell field as a gauge theory, at some point the following derivative comes up: $\frac{\partial(\partial_iA_0)}{\partial A_0}=0$ which must be, accordingly to the theory, zero. My ...