Linked Questions

3
votes
1answer
746 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 ...
1
vote
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
2k 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
656 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
827 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
188 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
240 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
169 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
69 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 ...
0
votes
0answers
50 views

Definition of integral functional [duplicate]

I'm reading the section of Marion and Thornton devoted to basics on the Calculus of Variations, and came across this definition for the functional: $$J = \int f(y(x), y'(x);x) dx$$ implying that $f$ ...
0
votes
0answers
38 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 ...
0
votes
0answers
38 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 ...
1
vote
0answers
35 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
0answers
21 views

Derivation of Equation of motion from Euler-Lagrange Equation [duplicate]

Hello I am new to Lagrange's dynamics and have some doubts regarding derivation of equation of motion given in a text : Deriving Equation of motion for a free body(no-external forces) ,in one ...
27
votes
10answers
4k views

Will a ball slide down a lumpy hill over the same path it rolls down the hill?

Suppose I have a lumpy hill. In a first experiment, the hill is frictionless and I let a ball slide down, starting from rest. I watch the path it takes (the time-independent trail it follows). ...

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