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2 Variational Problems with Fixed Boundaries
For any fixed value x ∈ [x 0 , x 1 ], The difference y(x) − y 0 (x) between an admissible function y(x) and another admissible function y 0 (x) is called the variation of
the function y(x) at y 0 (x) or variation of a function, it is written as δy, δ is called
the variational symbol, variational notation or variational operator. In this case
there is
δy = y(x) − y 0 (x) = εη(x)
(2.2.7)
where, ε is a small parameter introduced by Lagrange, but it is not the function of x;
η(x) is an arbitrary function of x. Because the admissible functions are all through
the fixed endpoints of an interval, namely their values are all equal at the endpoints
of an interval, so at the endpoints of an interval, arbitrary function η(x) satisfies
η(x 0 ) = η(x 1 ) = 0
(2.2.8)
that is
δy(x 0 ) = δy(x 1 ) = 0
(2.2.9)
Equation (2.2.9) is the condition that should be satisfied of the variation of a
function at fixed endpoints, namely fixed boundary condition.
Because the admissible function y(x) is the argument of the functional J [y(x)],
so it can also define the variational like this: the difference y(x) − y 0 (x) between an
argument y(x) and another argument y 0 (x) is called the variation of the argument
y(x) at y 0 (x).
For an arbitrary fixed value x ∈ [x 0 , x 1 ], if an admissible function y(x) and another
admissible function y 0 (x) have the zero-th order approach degree, then y(x) − y 0 (x)
is called the strong variation of the function y(x) at y 0 (x). If y(x) and y 0 (x) have the
first (or first above) approach degree, then y(x) − y 0 (x) is called the weak variation
of the function y(x) at y 0 (x). The strong variation and weak variation are generally
called variation.
The definition of the above variation can also be generalized to the situation of
multivariate function.
Obviously, from the definition of the variation, the variation δy of the function
y(x) is a function of x. Notice that the difference between the variation δy of a
function and the increment y of a function, as shown in Fig. 2.5, the variation
δy of a function is the difference εη(x) between two different functions y(x) and
y 0 (x) when the independent variable x is fixed, the function changes; The increment
y of a function y(x) is due to the independent variable x has an increment such
that the same function y(x) creates an increment, the function is still the original
function. This also shows that the role of the variational symbol δ is which represents
a tiny change of the function when the corresponding independent variable x has a
determined value.
Because in variation the independent variable x remains unchanged, the change
of function form causes a change of function value, so in variational operation the
2 Variational Problems with Fixed Boundaries
For any fixed value x ∈ [x 0 , x 1 ], The difference y(x) − y 0 (x) between an admissible function y(x) and another admissible function y 0 (x) is called the variation of
the function y(x) at y 0 (x) or variation of a function, it is written as δy, δ is called
the variational symbol, variational notation or variational operator. In this case
there is
δy = y(x) − y 0 (x) = εη(x)
(2.2.7)
where, ε is a small parameter introduced by Lagrange, but it is not the function of x;
η(x) is an arbitrary function of x. Because the admissible functions are all through
the fixed endpoints of an interval, namely their values are all equal at the endpoints
of an interval, so at the endpoints of an interval, arbitrary function η(x) satisfies
η(x 0 ) = η(x 1 ) = 0
(2.2.8)
that is
δy(x 0 ) = δy(x 1 ) = 0
(2.2.9)
Equation (2.2.9) is the condition that should be satisfied of the variation of a
function at fixed endpoints, namely fixed boundary condition.
Because the admissible function y(x) is the argument of the functional J [y(x)],
so it can also define the variational like this: the difference y(x) − y 0 (x) between an
argument y(x) and another argument y 0 (x) is called the variation of the argument
y(x) at y 0 (x).
For an arbitrary fixed value x ∈ [x 0 , x 1 ], if an admissible function y(x) and another
admissible function y 0 (x) have the zero-th order approach degree, then y(x) − y 0 (x)
is called the strong variation of the function y(x) at y 0 (x). If y(x) and y 0 (x) have the
first (or first above) approach degree, then y(x) − y 0 (x) is called the weak variation
of the function y(x) at y 0 (x). The strong variation and weak variation are generally
called variation.
The definition of the above variation can also be generalized to the situation of
multivariate function.
Obviously, from the definition of the variation, the variation δy of the function
y(x) is a function of x. Notice that the difference between the variation δy of a
function and the increment y of a function, as shown in Fig. 2.5, the variation
δy of a function is the difference εη(x) between two different functions y(x) and
y 0 (x) when the independent variable x is fixed, the function changes; The increment
y of a function y(x) is due to the independent variable x has an increment such
that the same function y(x) creates an increment, the function is still the original
function. This also shows that the role of the variational symbol δ is which represents
a tiny change of the function when the corresponding independent variable x has a
determined value.
Because in variation the independent variable x remains unchanged, the change
of function form causes a change of function value, so in variational operation the