based on d’Alembret’s Principle or Hamilton’s Principle.

expressed in terms of the scalar kinetic and potential energies of the system.

$$ \left\{\frac d{dt}\left(\frac{\partial L}{\partial\.{q}_j}\right)-\frac{\partial L}{\partial q_j}\right\}=Q_j^{EX} $$

• Remarkable connection between symmetry 对称 and invariance 守恒 of a system under transformation.[ Noether’s theorem]

invariant transformations and Noether’s Theorem

The Lagrangian is a scalar with units of energy, which do not change if the coordinate representation is changed.

Transformations for which (the explicit equations of motion are the same for both the old and new variables) the equations of motion are invariant, are called invariant transformations.守恒变换

$$ \begin{aligned}\dot{p}j&=\frac{\partial L}{\partial q_j}+\left[\sum{k=1}^m\lambda_k\frac{\partial g_k}{\partial q_j}(\mathbf{q},t)+Q_j^{EXC}\right]\end{aligned} $$

Noether’s Theorem: “For each symmetry of the Lagrangian, there is a conserved quantity” — if the Lagrangian does not explicitly contain a particular coordinate of displacement $q_i$,

$$ \frac{\partial L}{\partial q_j}+\left[\sum_{k=1}^m\lambda_k\frac{\partial g_k}{\partial q_j}(\mathbf{q},t)+Q_j^{EXC}\right]=0 $$

then the corresponding conjugate momentum $p_i$ is conserved.

$$ \dot{p}_j=\frac{dp_j}{dt}=0 $$

Symmetries and Invariance

(Jacobi’s) Generalized Mometum

Consider a holonomic 完整system of $N$ masses under the influnce of conservative force depend on position $q_j$ but velocity $\dot q_j$.

The generalized momentum associated with coordinate $q_j$ is defined to be

$$ \frac{\partial L}{\partial\dot{q}_j}\equiv p_j $$