Basic Examples (2)
Basic Examples
(2)
The Euler equations for the arc length in two dimensions yields a straight line:
∫s
In[2]:=
1+
,y[x],x2
y'[x]
Out[2]=
-[x]0
′′
y
3/2
(1+[x])
2
′
y
In[3]:=
DSolve[%,y[x],x]
Out[3]=
{{y[x]C[1]+xC[2]}}
———
A simple pendulum has the Lagrangian m+mgrcos(θ):
1
2
2
r
2
θ
In[2]:=
1
2
2
r
2
θ'[t]
Out[2]=
-mr(gSin[θ[t]]+r[t])0
′′
θ
The solution to the pendulum equation can be expressed using the function :
In[3]:=
DSolve[%,θ[t],t]
Solve::ifun:Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.
Out[3]=
θ[t]-2JacobiAmplitude(2g+rC[1]+4gtC[2]+2rtC[1]C[2]+2g+rC[1]),,θ[t]2JacobiAmplitude(2g+rC[1]+4gtC[2]+2rtC[1]C[2]+2g+rC[1]),
1
2
r
2
t
2
t
2
C[2]
2
C[2]
4g
2g+rC[1]
1
2
r
2
t
2
t
2
C[2]
2
C[2]
4g
2g+rC[1]
Scope (4)
Scope
(4)
Applications (3)
Applications
(3)