How to define more complicated differential expressions in Sage?
Dear Community,
i have a very hard time doing something pretty straight foreward in Sage. In mechanics, there is a Lagrange-formalism that yields the equation of motion of a system of $n$ degrees of freedom in the following way:
$$ 0=\frac{d}{dt}\left(\frac{\partial L}{\partial x'_i(t)}\right) - \frac{\partial L}{\partial x_i(t)} $$
where $L$ is the Lagrange function
$$ L=E_{Kin} (x'_{1}(t),x'_2(t)...x'_n(t); x_1(t),x_2(t)...x_n(t) ) - E _{Pot}(x_1(t),x_2(t),...,x_n(t)) $$
is the kinetic energy minus the potential energy. The kinetic energy depends on the speeds of the coordinates $x'_i(t)$, while the potential energy depends on the coordinates $x_i(t)$.
In Mathematica, the hack is
L = Ekin[x'[t]] - Epot[x[t]];
eq = 0 == D[D[L, x'[t]], t] - D[L, x[t]];
which gives
0 == Epot'[x[t]] + Ekin''[x'[t]] x''[t]
I can easily specify something for $E_{Kin}$ and $E_{Pot}$, solve the latter equation for $x''$, and give it as a right handside to some time integration scheme.
How can this be done in Sage? Assume you have complicated expressions for $E_{Pot}$ and $E_{Kin}$ which you do not want to differentiate by hand.
var('L,x,t')
x = function('x',t)
L = function('L',x,t)
allows for
diff(L,t)
but not
diff(L,x)
Thanks already!
