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6
votes
7answers
990 views

Physical intuition for higher order derivatives

Could somebody give me an intuitive physical interpretation of higher order derivatives (from 2 and so on), that is not related to position - velocity - acceleration - jerk - etc?
1
vote
0answers
21 views

Position, velocity, acceleration, jolt, and [duplicate]

I am familiar with the fact that $\displaystyle{\frac{dx}{dt}}=v$, $\displaystyle{\frac{dv}{dt} =a}$, and $\displaystyle{\frac{da}{dt}=J}$ where $J$ denotes the 'jolt', or jerk. Are further ...
0
votes
0answers
38 views

Different subscripts for $\nabla$ operators while deriving force on system of many particles

Consider a system of 4 particles in an external conservative field. So force acting on each particle is derived from potential energy $U(x,y,z)$of the particle+field system: Total (external) force on ...
15
votes
2answers
559 views

Lagrangian Mechanics - Commutativity Rule $\frac{d}{dt}\delta q=\delta \frac{dq}{dt} $

I am reading about Lagrangian mechanics. At some point the difference between the temporal derivative of a variation and variation of the temporal derivative is discussed. The fact that the two are ...
1
vote
0answers
31 views

Derivative with respect to a difference of independent variables

I am dealing with an equation from nonlinear acoustics (Khokhlova-Zabolotskaya-Kuznetsov equation) where a strange term (for me as a mathematician) is used. The equation looks like this $$ ...
3
votes
1answer
48 views

Is this equation $\nabla_a\sqrt{-g}=0$ correct? [duplicate]

Is the equation $$\nabla_a\sqrt{-g}=0$$ correct? Here $\nabla_a$ is the Levi-Civita connection, and $g$ is the determinant of metric $g_{ab}$. Apparently, we have $\nabla_ag_{bc}=0$, but I am not sure ...
4
votes
2answers
117 views

Why is the covariant derivative of the determinant of the metric zero?

This question, metric determinant and its partial and covariant derivative, seems to indicate $$\nabla_a \sqrt{g}=0.$$ Why is this the case? I've always learned that $$\nabla_a f= \partial_a f,$$ ...
1
vote
2answers
215 views

How to find Tangential/Radial/Angular Velocity for motion in any curve?

Is the radial velocity responsible only for changing distance between objects and the component perpendicular to it only for change in direction? If so why? Please try to give a different explanation ...
1
vote
0answers
71 views

Euler-Lagrange equations in General Relativity

When obtaining the Euler Lagrange equations for a scalar field with higher order derivatives in curved space is it the same to use $$ -\partial_\nu\partial_\mu\frac{\partial \sqrt{-g} ...
1
vote
0answers
37 views

Are physical functions always differentiable [duplicate]

I know that physicist usually don't really think too much about differentiabillity of functions. Usually there are at most finite many points where functions aren't differentiable and if there are ...
0
votes
1answer
52 views

Can there be a wave function that is physically possible but is non differerentiable (maybe even non-continous)?

The definition of a wave function demands continuity and differentiability so that it can satisfy the Schrödinger Equation. My question is whether this assumption is necessary for reality. Does ...
1
vote
1answer
34 views

Quick question - infinitesimals proofs [duplicate]

In a few of my courses in mechanics certain statements/equations have been proved by assuming that two infinitesimals multiplied by each other are zero. For instance in the equation : $dx + dy + dx^2 ...
0
votes
1answer
29 views

Directional derivative of the potential energy in the direction of the displacement in three dimensions

For a conservative force $\vec{F}=-\vec{\nabla } U \implies \mathrm dW= -\vec{\nabla} U \cdot \mathrm d\vec{s} $ Where $\mathrm d\vec{s}$ is the infinitesimal displacement. For a differentiable ...
0
votes
1answer
29 views

Infinitesimal time intervals use

I've a question, that maybe will sound obvious, on the use of infinitesimal quantities. Consider the expression for the acceleration in non inertial frames. ...
-1
votes
1answer
48 views

Uncertainty in Range of Projectile [closed]

If we are given that a projectile is launched with velocity 10m/s at an angle of $45^\circ$ and uncertainty in angle is of $0.5^\circ$ . What is the uncertainty in the range of projectile. The problem ...
0
votes
2answers
59 views

Calculate divergence via partial derivative [closed]

I need to calculate the divergence and curl for a vectorfield. I've done that before so that's no problem :) Or I've done it using partial derivative, maybe there are multiple ways to solve for ...
0
votes
1answer
46 views

Help with relativistic notation (Derivative of Lagrangian)

I am trying to learn QFT, but I haven't taken a course in general relativity so the relativistic notation stuff is taking me a bit to get used to. I do not understand how to do the following. For a ...
1
vote
2answers
64 views

Free falling and bouncing back

My confusion arises with free falling body. For a free falling body the displacement ~ time graph has a kink (at the time when the body hit the ground ). at a kink point, a function is not ...
2
votes
2answers
72 views

Do integrals of position make any sense? Do they have an application? [closed]

I know that taking the derivative of position with respect to time defines what we call velocity, but I've never heard of physicist going in the opposite direction with position. Is there any ...
1
vote
1answer
55 views

Killing equation manipulation

Why does the killing equation $$K_{\mu;\nu}+ K_{\nu;\mu} = 0$$ equal $$K_{\mu,\nu}+ K_{\nu,\mu} -2\Gamma^{\rho}_{\mu\nu}K_{\rho} = 0 $$ when in general a covariant derivative $V_{\beta;\alpha} = ...
0
votes
1answer
34 views

Calculating motion of equation in tensor form

for the Lagrangian density $$\mathscr{L}=\frac{1}{2}(\partial_{\mu}A^{\mu})^2$$ how can I get this $$\frac{\partial{\mathscr{L}}}{\partial(\partial_{\mu}A_\nu)}=(\partial_\rho A^\rho)\eta^{\mu\nu}$$ ...
1
vote
0answers
50 views

Partial derivative vs Total derivative

This is essentially a follow up to my question here since I seem to have some difficulties regarding the differences between partial and total derivatives. Consider a Lagrangian density ...
1
vote
1answer
37 views

Integral of a velocity profile?

Part of my fluid mechanics homework asks me to solve: $${\partial u\over \partial x} = 0$$ Which represents how the velocity profile, u, changes in the x. I'm not sure whether you can integrate ...
1
vote
1answer
50 views

Meaning of dt/dx when deriving the law of reflection

One way to derive the law of reflection, you can use the principle of least action to minimize the action path of motion of light. They key concept while doing this is to take the derivative of the ...
0
votes
0answers
42 views

Confused about something in Landau classical mechanics

On page 7 in Landau and Lifshitz Mechanics He writes: We have $L'=L(v'^2)=L(v^2+2ve+e^2)$ now the confusing part comes (for me): He writes: Expanding this expression in powers of e and neglecting ...
1
vote
0answers
58 views

Why is this differential added instead of subtracted?

I was looking at a derivation of the Barometric formula which reads like this: Consider a flat disc of air of mass $\mathrm{d}m$ at distance $h$ above the ground of mass $\mathrm{d}m$ and ...
0
votes
0answers
29 views

What is the derivative of the length vector in the direction of the original unit vector?

What is the derivative of the length vector? I am asking this question becuase I have seen, in Prof. M. S. Sivakumar's youtube video, the result that $\dot{|r|} \hat{r} = \dot{r}$ Where, $ ...
0
votes
0answers
33 views

Advection Operator shift in scalar product

Can someone help me with advection operator shifts? I can't figure out the rule for the shift inside of a scalar product. The terms $(u,(v\cdot \nabla)\delta v)_\Omega$ and $(u,(\delta v\cdot ...
1
vote
0answers
46 views

How does the gradient operator pick up a minus sign when the reference frame is switched from one particle to another? [closed]

A potential between two particles, $i$ and $j$, is given as a function only of the separation distance, $$V_{ij} = V_{ij}(|r_i − r_j|)$$ It should follow that the force by $j$ on $i$ is equal and ...
3
votes
1answer
60 views

Partial derivatives vs total derivatives in thermodynamics

The specific heat of a system is defined as $$C_z = T \left( \frac{\partial S}{\partial T} \right)_{z=\text{const}}$$ Sometimes however, I find the same definition, but with total derivatives ...
0
votes
1answer
63 views

Divergence of inverse of metric tensor

I know that the Levi-civita connection preserves the metric tensor. Is the divergence of the inverse of metric tensor zero, too?! I'm not so familiar with the divergence of the second ranked tensor. ...
3
votes
1answer
26 views

Dimensional interpretation of inverse gradient length $\frac{d}{dx} \ln(Y)$

Preliminary definition: inverse gradient length Let me first explain what I mean by that term. The inverse gradient length of some quantity $Y$ (often thermodynamic temperature $T$) $L_Y^{-1}$ is ...
3
votes
2answers
90 views

Exact differentials and state functions

I was reading a Wiki article on the relationships between heat capacities And during the derivation I came across this formula (and others like it): This equation was used as a tool in a ...
0
votes
1answer
72 views

A question when using $E= - \nabla V$

This problem is from Problems and Solutions on Electromagnetism. A thin but very massive disc of insulator has surface charge density $\sigma$ and radius $R$. A point charge $+Q$ is on the axis of ...
1
vote
0answers
43 views

Higher order versions of momentum? Can conservation principles be established and used? [closed]

Question Can higher order derivatives of momentum be useful in creating theories of dynamics if they have conservation principles? Even if they aren't needed, could it be done in theory? For ...
1
vote
3answers
238 views

Time derivative of a function in Phase Space

Consider a function $\mathcal{H}(q_i,p_i;t)$ such that it obeys the equation: $$ \frac{d\mathcal{H}}{dt}=\frac{\partial\mathcal{H}}{\partial t}$$ What does this equation imply (read: mean), ...
1
vote
1answer
158 views

Proof for Negele and Orland equation (2.34)

The equation (2.34) of Negele and Orland has $$\mathcal H_\text{A}(\hat{\mathbf p},\hat{\mathbf x}) = \frac{1}{2m}\left(\hat {\mathbf p} - \frac e c \mathbf A(\hat{\mathbf x})\right)^2.\tag{2.34a}$$ ...
2
votes
2answers
496 views

Derivation of the Riemann tensor confusion

I'm trying to understand the derivation of the Riemann curvature tensor as given in Foster and Nightingale's A Short Course In General Relativity, p. 102. They start by giving the covariant derivative ...
2
votes
0answers
98 views

Supercovariant Derivative action

My query is with Weinberg Vol3 equation just above 26.7.22 Weinberg follows Majorana Superfield formalism. Where, covariant derivative is defined as, $$D_{R\alpha}=-\epsilon_{\alpha ...
0
votes
0answers
46 views

Mode expansions of fields

This is a very simple question but would appreciate it if someone could clarify - I've heard different things from different people so I'm a little bit confused yet the question is simple: Given the ...
1
vote
2answers
914 views

Time derivative of angular velocity in rotating reference frame

I am going through a section in a textbook regarding the Newton Euler equations for a system of rigid bodies (robotics text). There is a particular line in the derivation I don't understand, I've ...
0
votes
1answer
67 views

Variation of a tensor

Let a change of coordinates be given by $x^{\mu}\to x^{\mu '}=x^{\mu}+\varepsilon \xi^{\mu}(x)$ with epsilon a small quantity. Given a tensor $T$ we define $\delta T:=T'(x)-T(x)$. I guess this means ...
1
vote
3answers
170 views

What is the physical meaning of the Levi-Civita connection?

I'm taking a course in General Relativity and I have studied the fundamental theorem of Riemannian geometry: Let $M$ be a manifold with metric $g$. Then exists an unique torsion-free connection ...
1
vote
1answer
86 views

Derivation Of Euler-Lagrange Equation [closed]

I want the proof of this relation in details, $$ \frac{\rm d}{{\rm d}t}\left(\frac{\partial\vec{r}_v}{\partial q_\alpha}\right)=\frac{\partial\vec{\dot{r}_v}}{\partial q_\alpha} $$
30
votes
4answers
7k views

What is the physical meaning of the connection and the curvature tensor?

Regarding general relativity: What is the physical meaning of the Christoffel symbol ($\Gamma^i_{\ jk}$)? What are the (preferably physical) differences between the Riemann curvature tensor ($R^i_{\ ...
25
votes
6answers
3k views

Why are Killing fields relevant in physics?

I'm taking a course on General Relativity and the notes that I'm following define a Killing vector field $X$ as those verifying: $$\mathcal{L}_Xg~=~ 0.$$ They seem to be very important in physics ...
1
vote
3answers
132 views

Is there any physical interpretation for $\nabla\cdot(\nabla \times F)=0$?

It is well known that the divergence of the curl is always 0. Mathematically I understand why this happens ($d^2=0$ where $d$ is the exterior derivative) but today I was wondering what is the physical ...
1
vote
2answers
95 views

Two different formulas

My problem is simple : given a particle of mass $m$, charge $q$ and velocity $\bf{v}$. If $\bf{A}$ denotes the magnetic potential satisfying $\bf{B}= \nabla \times \bf{A}$. I want to etablish the ...
0
votes
0answers
40 views

Force of Leaky Bucket over time

Indie Lab Procedure: I attached a cylindrical can to a force sensor. On the bottom of the can I drilled holes of various diameters. I then added a volume of water while collecting data from the force ...
1
vote
4answers
619 views

Why is $F=-\nabla V$?

I came across this equation $$F=-\nabla V$$ where $V$ is potential energy. I do understand that $$F(r)=-\frac{dV}{dr}.$$ Hence does this mean the nabla operator in this case means derivative? Because ...