# Time derivatives in variational calculus

• I
• feynman1

#### feynman1

Taking the variation w.r.t f(x) of the integral over some x domain of F[f(x), f'(x), df(x)/dt], why doesn't df(x)/dt need to be taken a variational derivative and is treated as if it were constant?

Can you provide some more context about what problem you are working on or what class this came from?

These kinds of details help us decide on the level of response.

Can you provide some more context about what problem you are working on or what class this came from?

These kinds of details help us decide on the level of response.
Physics problem. It's one of taking the variation of an energy of an object with energy density F in the OP varying in space. f(x) can be a displacement, x can be a coordinate. df(x)/dt can be a speed. Clear?

Taking the variation w.r.t f(x) of the integral over some x domain of F[f(x), f'(x), df(x)/dt], why doesn't df(x)/dt need to be taken a variational derivative and is treated as if it were constant?
From your notation, ##f## appears to be a function of only one variable; namely, x. In that case ##\frac{d(f(x))}{dt} = 0##.
For ##f## to have nonzero (partial) derivatives wrt x and t, it should use this notation: ##f(x, t)##. Then ##\frac{\partial f}{\partial x} = f_x((x, t)## and ##\frac{\partial f}{\partial t} = f_t((x, t)## would make sense.

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From your notation, ##f## appears to be a function of only one variable; namely, x. In that case ##\frac{d(f(x))}{dt} = 0##.
For ##f## to have nonzero (partial) derivatives wrt x and t, it should use this notation: ##f(x, t)##. Then ##\frac{\partial f}{\partial x} = f_x((x, t)## and ##\frac{\partial f}{\partial t} = f_t((x, t)## would make sense.
right because I couldn't put a dot on top of f(x).

##f## appears to be a function of only one variable; namely, x. In that case ##\frac{d(f(x))}{dt} = 0##.
Well, not if ##x## itself depends on ##t##.

To @feynman1: could you point us to a suitable reference? I'm not yet sure what you are asking.

Well, not if ##x## itself depends on ##t##.

To @feynman1: could you point us to a suitable reference? I'm not yet sure what you are asking.
Sorry there's no direct reference as it's a work in progress. Physics problem. It's one of taking the variation of an energy of an object with energy density F in the OP varying in space. f(x,t) can be a displacement, x is a fixed coordinate not dependent on t. df(x,t)/dt can be a speed. Clear?

Not clear at all, sorry. I can't help until there's a clear problem statement.

I’m well aware of the calculus of variations! What I mean is that you have not adequately stated your problem.

• • russ_watters, Hamiltonian and Mark44
Well, not if x itself depends on t.
Which would show up in a clearly posed question. If someone writes "f(x)" without anything further, the reasonable assumption is that f depends only on x, with no relationship to t.

• ergospherical and feynman1
Which would show up in a clearly posed question. If someone writes "f(x)" without anything further, the reasonable assumption is that f depends only on x, with no relationship to t.
sorry unable to edit the OP any more

I’m well aware of the calculus of variations! What I mean is that you have not adequately stated your problem.
only difference from trivial cases: the integrand has time dependent terms but the integral isn't over time, which isn't covered by variational calculus. Clear?

[...] because I couldn't put a dot on top of f(x). [...] only difference from trivial cases: the integrand has time dependent terms but the integral isn't over time, which isn't covered by variational calculus. Clear?
No, not clear. You said "Physics Problem" more than once, but with no further detail, as if we're supposed to know clairvoyantly which physics scenario you're talking about. If the integral isn't over time (but assuming ##t## and ##x## are independent variables), then you treat ##g(x,t) := \dot f(x,t)## as simply another field in the integrand.

[Aside: You need to spell out your specific situation/problem explicitly. If you won't put more effort into composing your questions, why should other people put more effort into helping you? ]

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• russ_watters and Hamiltonian
[Aside: You need to spell out your specific situation/problem explicitly. If you won't put more effort into composing your questions, why should other people put more effort into helping you? ]
This...

• russ_watters
v(x,t) is velocity, x is position, t is time. Kinetic energy of a body~ a volume integral of v(x,t)*v(x,t) over a domain (x). The time rate of the kinetic energy thus~ a volume integral of ##v(x,t)v_t(x, t)##. How to take the variational derivative of this kinetic energy rate w.r.t v(x,t)?