理论力学 计算题(4)

?Fy?Fx?a12??a21 即?y?x?Fx?F?a13?z?a31 ?z?x

?Fy?z?a23??Fz?a32 ?y即势能存在aij满足条件是：a12?a21 a13?a31 a23?a32

?V.............(1)?x?V由Fy?a21x?a22y?a23z??............(2)

?y?VFz?a31x?a32y?a33z??.............(3)?zFx?a11x?a12y?a13z??（1）式积分得V??1a11x2?a12yx?a13zx?f(y,z)........(4) 2（4）式对y偏微分=（2）式得

?V?f(y,z)??a12x???a12x?a22y?a23z ?y?y即

?f(y,z)??a22y?a23z...........(5) ?y1a22y2?a23zy?g(z)............(6) 21122（6）式代入（4）式得V??a11x?a12yx?a13zx??a22y?a23zy?g(z)........(7)

22（5）式积分得f(y,z)??（7）式对z偏微分=（3）式得

?V?g(z)??a13x?a23y???a13x?a23y?a33z ?z?z?g(z)即??a33z...........(8)

?z12（8）式积分得g(z)??a33z?c............(9)

2（9）式代入（4）式得

111V??a11x2?a12yx?a13zx??a22y2?a23zy?a33z2?c........(10)

222取x?0,y?0,z?0,V?0 则c?0得势能为

111V??a11x2?a12yx?a13zx??a22y2?a23zy?a33z2222

1??(a11x2?a22y2?a33z2?2a12xy?2a23zy?2a31zx)2????某力场的力矢为F?xi?yj?zk

试证明该力场是否为保守力场，若为保守力场，求出其势能函数。

?i??解：由于??r??xFx

?j??yFy?k???Fz?Fy????Fx?Fz????i??????z??y?z???z?xFz????Fy?Fx???j????x??y??k?0 ???故力场为保守力场

??VF???x.............(1)?x?x??V?由?Fy???y.............(2)

?y??F???V?z..............(3)z??z?x2?f?y,z?????4? 积分（1）式得V??2y2?V?f?y,z????y 积分得f(y,z)???g?z?????5? （4）式对y偏微分=（2）式得

?y?y2x2y2??g?z?????6? 代（5）入（4）得V??22z2?V?g?z?（6）式对z偏微分=（3）式得???z 积分得g?z????c????7?

2?z?zx2y2z2???c 代（7）入（6）得V??222x2y2z2?? 取x?0,y?0,z?0,V?0 则c?0得势能函数为V??222???有一质点在xy平面上运动，质点受到的力为F?(x?y)i?(x?y)j，质点在平面上由点

?A（1,0）沿直线运动到点B（1,1）,求力F所作的功

解法1：由功的定义计算

W??BA??BBF?dr??(Fxdx?Fydy)??(x?y)dx?(x?y)dy

AA又x?1,dx?0

所以

W??BA??BB1F?dr??(Fxdx?Fydy)??(x?y)dx?(x?y)dy??(1?y)dyAA011?(y?y2)1?022B

解法2：由功的定义计算

??B(1,0)(1,1)112(1,1),0)W??F?dr??(Fxdx?Fydy)??(x?y)dx??(x?y)dy?(x2?xy)((1?(xy?y)(1,0)1,0)AA(1,0)(1,0)2211?1??22或

??BB(1,1)11W??F?dr??(Fxdx?Fydy)??(x?y)dx?(x?y)dy??d(x2?xy?y2)AAA(1,0)22

1212(1,1)1111?(x?xy?y)(1,0)?(?1?)??222222B解法3：由保守力性质计算

???ijk??????Fz?Fy??r??????x?y?z??y?zFxFyFz???(0?0)i?(0?0)j?(1?1)k?0????Fx?Fz??i???z??x??????Fy?Fx???j????x??y??k ????故力场F为保守力场

??VF???x?y.............(1)?x?x??V??x?y.............(2) ?Fy???y??F???V?0..............(3)z??z?x2?xy?f?y?????4? 积分（1）式得V??2（4）式对y偏微分=（2）式得

?V?f?y???x???x?y ?y?y