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Results tagged with calculus-of-variations
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user 16183
Questions on the calculus of variations, which deals with the optimization of functionals mostly defined on infinite dimensional spaces.
17
votes
2
answers
973
views
Example of ODE not equivalent to Euler-Lagrange equation
I am looking for an explicit (preferably simple) example of an ODE with time-independent coefficients in $\mathbb{R}^3$ such that there does not exist an Euler-Lagrange equation
$$\frac{\partial L}{\p …
- 20.6k
15
votes
6
answers
3k
views
Maxwell equations as Euler-Lagrange equation without electromagnetic potential
In (mathematical) physics many equations of motion can be interpreted as Euler-Lagrange (EL) equations. The Maxwell equation for electromagnetic (EM) field (say in vacuum and in absence of charges) se …
- 20.6k
10
votes
3
answers
804
views
Rigorous justification that overdetermined systems do not have a solution
There is the following well known and very useful heuristic principle: Assume one has a natural map from the space of $k$-tuples of functions in $n$ variables into the space of $K$-tuples of functions …
- 20.6k
3
votes
2
answers
289
views
Classification of Lagrangians with given Euler-Lagrange equations
In (mathematical) physics the equations of motion of a system of particles are often interpreted as Euler-Lagrange equations for appropriate Lagrangian $L=L(x,\dot x,t)$ where $x$ is a collection of v …
- 20.6k
9
votes
1
answer
719
views
Why the least action principle is always (?) used in this particular form?
The least action principle in (mathematical) physics says the following. Given a system, e.g. collection of particles, whose motion satisfies a known system of differential equations (of second order …
- 20.6k