Linked Questions

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1answer
4k views

What is the common difference between partial time derivative and ordinary time derivative? [duplicate]

What is difference between partial and ordinary time derivative? for example: what is difference between $\frac {\partial v}{\partial t}$ and $\frac {dv}{dt}$? where the $v$ is velocity.
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3answers
1k views

Why there is added a partial time derivative in formula for time derivative of potential energy? [duplicate]

In proving the total energy in conservative field is constant we have this equation(picture) why it added partial derivative? Why? I mean where it did come from?
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1answer
317 views

Difference between $dM/dt $ and $\partial M/\partial t$ [duplicate]

$\frac{dM}{dt} = 0$ represents a constant of motion $M.$ Why not $\frac{\partial M}{\partial t}$ represent a constant of motion $M$?
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0answers
43 views

How to explain implicit time dependence to someone? [duplicate]

I am trying to explain what implicit time dependence is and how it differs from explicit time dependence, but I'm unsure how "sound" my explanation is. Here is what I said: Suppose I have a function $...
31
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2answers
3k views

Symbols of derivatives

What is the exact use of the symbols $\partial$, $\delta$ and $\mathrm{d}$ in derivatives in physics? How are they different and when are they used? It would be nice to get that settled once and for ...
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4answers
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Hamilton equations from Poisson bracket's formulation

Referring to Wikipedia we have that the equation of motion for a $f(q, p, t)$ comes from the formula \begin{equation} \frac{\mathrm{d}}{\mathrm{d}t} f(p, q, t) = \frac{\partial f}{\partial q} \frac{\...
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5answers
4k views

Which derivative with respect to time is which in the Heisenberg picture of quantum mechanics?

For an observable $A$ and a Hamiltonian $H$, Wikipedia gives the time evolution equation for $A(t) = e^{iHt/\hbar} A e^{-iHt/\hbar}$ in the Heisenberg picture as $$\frac{d}{dt} A(t) = \frac{i}{\hbar} ...
14
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4answers
4k views

Conserved quantities and total derivatives?

I am having a bit of a crisis in understanding of the physical meanings of total derivatives. When a quantity $\rho$ (be it a vector or a scalar) is said to be conserved, then (mathematically) $$\...
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4answers
3k views

Which Schrödinger equation is correct?

In the coordinate representation, in 1D, the wave function depends on space and time, $\Psi(x,t)$, accordingly the time dependent Schrödinger equation is $$H\Psi(x,t) = i\hbar\frac{\partial}{\...
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2answers
3k views

Conserved quantities commute with Hamiltonian — don't understand proof

http://en.wikipedia.org/wiki/Constant_of_motion#In_quantum_mechanics I understand the derivation to get $$\frac{d}{dt}⟨\psi|Q|\psi⟩ = -\frac{1}{i\hbar}⟨\psi|[H,Q]\psi⟩ + ⟨\psi|\frac{d}{dt}Q|\psi⟩$$ ...
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3answers
993 views

Can the kinetic energy be a function of the position vector?

,I got one confusion when reading Goldstein's Classical Mechanics (page 20, third edition). After getting the equation $$ \sum \left\{\left[\frac{\mathrm{d}}{\mathrm{d}t}{\left(\frac{\partial T}{\...
3
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2answers
695 views

Meaning of time derivative of an operator

Today when my professor was deriving this equation: $$\frac{\mathrm d\langle A\rangle}{\mathrm dt}=\frac{i}{\hbar}\langle\left[H,\,A\right]\rangle+\left\langle\frac{\partial A}{\partial t}\right\...
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1answer
1k views

Full time-derivative, Poisson brackets and Hamilton's equations (classical mechanics)

While studying Poisson brackets in classical mechanics and the derivation of $\dot{q_j}=\{q_j,H\}$ and $\dot{p_j}=\{p_j,H\}$ form of Hamilton's equations I encountered a surpsing identity, which led ...
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2answers
203 views

Lagrangian mechanics not relying on time or independent of time [closed]

If neither the potential energy nor kinetic energy depends on time, then Lagrangian is explicitly independent of time I find this statement a little bit odd because velocity is distance over time or ...
3
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2answers
341 views

Principle of least action: $\frac{d S_{cl}}{dt_b} = \frac{\partial S_{cl}}{\partial t_b} + \frac{\partial S_{cl}}{\partial x_b}\dot{x}_b$

Question I cannot see how I can obtain the yellow highlighted section on the RHS from that of the LHS. The following equation can be found in both my lecture notes(*1) (page 9, equation 2.7) and is ...

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