Flashcards
Classical Mechanics
Cards in this deck (19)
- 1Generalized Forces\[Q_j=\sum_i\vec{F}_i\cdot\frac{\partial\vec{r}_i}{\partial q_j}\]
\[Q_j\equiv-\frac{\partial V}{\partial q_j}\]
\[Q_j=-\frac{\partial U}{\partial q_j}+\frac{d}{dt}\Big(\frac{\partial U}{\partial\dot{q}_j}\Big)\] - 2\(m\) Equations of Motion Including Lagrange Multipliers\[\frac{d}{dt}\Big(\frac{\partial L}{\partial\dot{q}_k}\Big)-\frac{\partial L}{\partial q_k}+\sum_{\alpha=1}^{m}\lambda_{\alpha}\frac{\partial f_{\alpha}}{\partial q_k}=0\]
- 3Equation for Generalized Forces of Constraint in Terms of Lagrange Multipliers\[-\sum_{\alpha=1}^m\lambda_{\alpha}\frac{\partial f_{\alpha}}{\partial q_k}=Q_k\]
- 4Cyclic Coordinates\[\frac{\partial L}{\partial q_i}=0\]
\(\rightarrow \) \(q_i\) is cyclic. - 5Constant Energy FunctionThe Lagrangian has no explicit time dependence. Energy is conserved.
- 6How to Construct the Hamiltonian via the Lagrangian1.) \(L=T-V\)
2.) \(p_i=\partial L/\partial\dot{q}_i\)
3.) \(H=\sum_ip_i\dot{q}_i-L\)
4.) Invert \(p_i=\partial L/\partial\dot{q}_i\) to obtain \(\dot{q}_i\) in terms of \(q_i\), \(p_i\), \(t\)
5.) Eliminate \(\dot{q}_i\) from \(H\) in step (3)
6.) Get \(H(q_i,p_i,t)\) - 7Conditions for H=T+V=E1.) When equations of constraint do not explicitly depend on time.
2.) When the potential depends only on the coordinates. - 8Hamilton's Equations\[\dot{p}_i=-\frac{\partial H}{\partial q_i}\]
\[\dot{q}_i=\frac{\partial H}{\partial p_i}\] - 9Lagrangian for a Central Force Problem\[L=(1/2)m(\dot{r}^2+r^2\dot{\theta}^2)-V(r)\]
- 10Effective Potential and Effective Force for a Central Force Problem\[V'(r)=V(r)+\frac{l^2}{2mr^2}\]
\[f'(r)=-\frac{\partial V'(r)}{\partial r}\] - 11The Conditions for a Stable Circular Orbit\[f'(r)=-\frac{\partial V'(r)}{\partial r}\Big|_{r=r_0}\]
\[\frac{\partial^2V'(r)}{\partial r^2}\Big|_{r=r_0}>0\] - 12Equilibrium Points when Using the Effective PotentialThe equilibrium points \(q_0\) are the points where:
\[\frac{\partial V_{eff}(q_0)}{\partial q}=0\] - 13Stability of Equilibrium when Using the Effective PotentialThe equilibrium is stable at \(q_0\) if:
\[\frac{\partial^2V_{eff}(q_0)}{\partial q^2}>0\] - 14Chasle's TheoremWe can split \(T\) into translational motion and rotational motion:
\[T=(1/2)Mv^2+(1/2)I\omega^2\] - 15Moment of Inertial of a Solid Sphere\[I=\frac{2}{5}MR^2\]
- 16Moment of Inertial of a Spherical Shell\[I=\frac{2}{3}MR^2\]
- 17Moment of Inertial of a Rod Rotating about its Center\[I=\frac{1}{12}ML^2\]
- 18Moment of Inertial of a Rod Rotating about its End\[I=\frac{1}{3}ML^2\]
- 19Euler Equations of Motion\[I_1\dot{\omega}_1-\omega_2\omega_3(I_2-I_3)=N_1\]
\[I_2\dot{\omega}_2-\omega_3\omega_1(I_3-I_1)=N_2\]
\[I_3\dot{\omega}_3-\omega_1\omega_2(I_1-I_2)=N_3\]