Linked Questions
73 questions linked to/from Calculus of variations -- how does it make sense to vary the position and the velocity independently?
6
votes
1answer
1k views
Why does the partial derivative with respect to $x$ of a function depending only on $\dot{x}$ vanish? [duplicate]
In Classical Mechanics by Goldstein it says:
$$ \sum \left\{ \left[ \frac{d}{dt} \left( \frac{\partial T}{\partial \dot q_j} \right) - \frac{\partial T}{\partial q_j} \right] - Q_j \right\} \delta ...
2
votes
5answers
1k views
Why do we automatically assume that the velocity vector $\vec{v}$ and location vector $\vec{r}$ are independent? [duplicate]
I'm not sure if it's relevant, but I'm talking about a situation where a particle is moving in an electro-magnetic field.
As I understand, if we see the term $\nabla \cdot \vec{v}$ or $\nabla \times \...
0
votes
1answer
3k views
Independence of position and velocity in Lagrangian from the point of view of physics? [duplicate]
I would like to continue discussion from my previous post on time dependence of lagrangian Time dependence of the Lagrangian of a free particle?. I have also read this old post Why does calculus of ...
1
vote
2answers
1k views
Independent Variables of a Lagrangian [duplicate]
Let us consider a particle in one spatial dimension $x$ and one temporal dimension $t$. Its Lagrangian $L$ is given by
\begin{eqnarray*}
L &=& T- V \\
&=& \frac{1}{2} m\dot{x}^2 - ...
3
votes
2answers
1k views
Again, why is kinetic energy and velocity independent of position coordinates in Cartesian coordinates [duplicate]
This might be a very simple question. I read one previous post
Can the kinetic energy be a function of the position vector?
I know that in Cartesian coordinates, the kinetic energy $T=\frac{1}{2}mv^...
1
vote
1answer
239 views
How come $\frac{d}{dt}\left(\frac{\partial {r_i}}{\partial {q_j}}\right) = \frac{\partial {\dot r_i}}{\partial {q_j}}$ in Lagrangian mechanics? [duplicate]
It is written in the Goldstein's Classical Mechanics text that
$$\frac{\mathrm d}{\mathrm dt}\left(\frac{\partial {r_i}}{\partial {q_j}}\right) = \frac{\partial {\dot r_i}}{\partial {q_j}}=\sum_k \...
2
votes
2answers
257 views
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 ...
1
vote
0answers
221 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 ...
1
vote
0answers
76 views
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 ...
1
vote
0answers
71 views
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 ...
0
votes
1answer
42 views
Partial derivative of position in respect to velocity [duplicate]
Out of interest What is the derivative of $\frac{\partial x(t)}{ \partial v(t)}$
0
votes
0answers
44 views
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
votes
0answers
41 views
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 ...
0
votes
0answers
39 views
Question on basic tensorial calculus on field theory [duplicate]
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 ...
1
vote
0answers
38 views
How to explain Newton's second law using Lagrange's equation? [duplicate]
I know there are many related questions already posted on this topic. And there are several answers too. But I am still confused with obtaining Newton's second law from Lagrange's equation. (I don't ...