左傳‧昭公十四年

冬，十二月，蒲餘侯茲夫殺莒公子意恢，郊公奔齊，公子鐸逆庚與於齊，齊隰黨，公子鉏，送之，有賂田。

晉邢侯與雍子爭鄐田，久而無成，士景伯如楚，叔魚攝理，韓宣子命斷舊獄，罪在雍子，雍子納其女於叔魚，叔魚蔽罪邢侯，邢侯怒，殺叔魚，與雍子於朝，宣子問其罪於叔向，叔向曰，三人同罪，施生戮死，可也，雍子自知其罪，而賂以買直，鮒也鬻獄，邢侯專殺，其罪一也。已惡而掠美為昏，貪以敗官為墨，殺人不忌為賊。夏書曰，昏墨賊殺，皋陶之刑也，請從之，乃施邢侯，而尸雍子，與叔魚於市，仲尼曰，叔向，古之遺直也，治國制刑，不隱於親，三數叔魚之惡，不為末減，曰，義也夫，可謂直矣，平丘之會，數其賄也，以寬衛國，晉不為暴，歸魯季孫，稱其詐也，以寬魯國，晉不為虐，邢侯之獄，言其貪也，以正刑書，晉不為頗，三言而除，三惡加三利，殺親益榮，猶義也夫。

有言，恐已生︰勿掠人之美成語。

原想假借『廣義力』

Generalized forces

Generalized forces find use in Lagrangian mechanics, where they play a role conjugate to generalized coordinates. They are obtained from the applied forces, F_i, i=1,…, n, acting on a system that has its configuration defined in terms of generalized coordinates. In the formulation of virtual work, each generalized force is the coefficient of the variation of a generalized coordinate.

Virtual work

Generalized forces can be obtained from the computation of the virtual work, δW, of the applied forces.^[1]^:265

The virtual work of the forces, F_i, acting on the particles P_i, i=1,…, n, is given by

$\displaystyle \delta W=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot \delta \mathbf {r} _{i}$

where δr_i is the virtual displacement of the particle P_i.

Generalized coordinates

Let the position vectors of each of the particles, r_i, be a function of the generalized coordinates, q_j, j=1,…,m. Then the virtual displacements δr_i are given by

$\displaystyle \delta \mathbf {r} _{i}=\sum _{j=1}^{m}{\frac {\partial \mathbf {r} _{i}}{\partial q_{j}}}\delta q_{j},\quad i=1,\ldots ,n,$

where δq_j is the virtual displacement of the generalized coordinate q_j.

The virtual work for the system of particles becomes

$\displaystyle \delta W=\mathbf {F} _{1}\cdot \sum _{j=1}^{m}{\frac {\partial \mathbf {r} _{1}}{\partial q_{j}}}\delta q_{j}+\ldots +\mathbf {F} _{n}\cdot \sum _{j=1}^{m}{\frac {\partial \mathbf {r} _{n}}{\partial q_{j}}}\delta q_{j}.$

Collect the coefficients of δq_j so that

$\displaystyle \delta W=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {r} _{i}}{\partial q_{1}}}\delta q_{1}+\ldots +\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {r} _{i}}{\partial q_{m}}}\delta q_{m}.$

Generalized forces

The virtual work of a system of particles can be written in the form

$\displaystyle \delta W=Q_{1}\delta q_{1}+\ldots +Q_{m}\delta q_{m},$

where

$\displaystyle Q_{j}=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {r} _{i}}{\partial q_{j}}},\quad j=1,\ldots ,m,$

are called the generalized forces associated with the generalized coordinates q_j, j=1,…,m.

Velocity formulation

In the application of the principle of virtual work it is often convenient to obtain virtual displacements from the velocities of the system. For the n particle system, let the velocity of each particle P_i be V_i, then the virtual displacement δr_i can also be written in the form^[2]

$\displaystyle \delta \mathbf {r} _{i}=\sum _{j=1}^{m}{\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}_{j}}}\delta q_{j},\quad i=1,\ldots ,n.$

This means that the generalized force, Q_j, can also be determined as

$\displaystyle Q_{j}=\sum _{i=1}^{n}\mathbf {F} _{i}\cdot {\frac {\partial \mathbf {V} _{i}}{\partial {\dot {q}}_{j}}},\quad j=1,\ldots ,m.$

D’Alembert’s principle

D’Alembert formulated the dynamics of a particle as the equilibrium of the applied forces with an inertia force (apparent force), called D’Alembert’s principle. The inertia force of a particle, P_i, of mass m_i is

$\displaystyle \mathbf {F} _{i}^{*}=-m_{i}\mathbf {A} _{i},\quad i=1,\ldots ,n,$