# Variational Methods

The basic problem of the calculus of variations is to determine the function that extremizes a functional . In general, there can be more than one independent variable and the integrand can depend on several functions and their higher derivatives.

The extremal functions are solutions of the Euler(–Lagrange) equations that are obtained by setting the first variational derivatives of the functional with respect to each function equal to zero. Since many ordinary and partial differential equations that occur in physics and engineering can be derived as the Euler equations for appropriate functionals, variational methods are of general utility.

VariationalD[f,u[x],x],VariationalD[f,u[x,y,…],{x,y,…}] | |

give the first variational derivative of the functional defined by the integrand f, where f depends on one function u and one independent variable x or several independent variables x, y, … | |

VariationalD[f,{u[x,y,…],v[x,y,…],…},{x,y,…}] | |

give a list of the first variational derivatives of the functional defined by the integrand f, where f depends on several functions u, v, … and several independent variables x, y, … | |

EulerEquations[f,u[x],x],EulerEquations[f,u[x,y,…],{x,y,…}] | |

give the Euler equation for the integrand f, where f depends on one function u and one independent variable x or several independent variables x, y, … | |

EulerEquations[f,{u[x,y,…],v[x,y,…],…},{x,y,…}] | |

give a list of the Euler equations for the integrand f, where f depends on several functions u, v, … and several independent variables x, y, … |

First variational derivatives and Euler equations.

VariationalD gives the first variational derivatives of a functional defined by the integrand . may depend on several functions u, v, w, …; their derivatives of arbitrary order; and variables x, y, z, …. EulerEquations returns the Euler(–Lagrange) equations given the integrand . Again, may depend on several functions u, v, w, …; their derivatives of arbitrary order; and variables x, y, z, ….

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FirstIntegrals[f,u[x],x],FirstIntegrals[f,{u[x],v[x],…},x] | |

give first integrals when the integrand f is independent of one or more of , or independent of x | |

FirstIntegral[u] | first integral associated with the variable u (appears in the output of FirstIntegrals) |

When there is only one independent variable x, FirstIntegrals gives conserved quantities in the following cases: (1) if f does not depend on a coordinate u explicitly, it is referred to as an ignorable coordinate and the corresponding Euler equation possesses an obvious first integral (a conserved generalized momentum), and (2) if f depends on u, v, … and their first derivatives only and has no explicit x dependence, FirstIntegrals also returns the first integral corresponding to the Hamiltonian.

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The Ritz variational principle affords a powerful technique for the approximate solution of (1) eigenvalue problems where is an operator and is a weight function and (2) problems of the form where is a positive definite operator and is given. A judicious choice for the trial function that satisfies boundary conditions and depends on variational parameters , , ... must be given in both cases. For (1) VariationalBound[{f,g},u[x,y,…],{{x,x_{min},x_{max}},{y,y_{min},y_{max}},…}, extremizes where and . The result is an upper bound on the corresponding eigenvalue and optimal values for the parameters. For (2) VariationalBound[f,u[x,y,…], extremizes the functional with and yields the value of the functional and the optimal parameters. VariationalBound can also be used to extremize general functionals given appropriate trial functions. NVariationalBound performs the same functions as VariationalBound numerically. It uses the internal function FindMinimum and has the same options and input format for parameters.

VariationalBound[{f,g},u[x,y,…],{{x,x_{min},x_{max}},{y,y_{min},y_{max}},…},u_{t},{a,a_{min},a_{max}},{b,b_{min},b_{max}},…] | |

give an upper bound for the eigenvalue and the optimal values of a, b, … in the range | |

VariationalBound[f,u[x,y,…],{{x,x_{min},x_{max}},{y,y_{min},y_{max}},…},u_{t},{a,a_{min},a_{max}},{b,b_{min},b_{max}},…] | |

give the value of the functional and optimal values of a, b, … | |

NVariationalBound[{f,g},u[x,y,…],{{x,x_{min},x_{max}},{y,y_{min},y_{max}},…},u_{t},{a,a_{0},a_{min},a_{max}},{b,b_{0},b_{min},b_{max}},…] | |

evaluate numerically an upper bound for the eigenvalue and the optimal values of a, b, … in the range given initial values , , ... | |

NVariationalBound[f,u[x,y,…],{{x,x_{min},x_{max}},{y,y_{min},y_{max}},…},u_{t},{a,a_{0},a_{min},a_{max}},{b,b_{0},b_{min},b_{max}},…] | |

evaluate numerically the value of the functional and optimal values of a, b, … given initial values, , … |

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