Theoretical Mechanics Exam 2 Resources

Exam on Tuesday Apr. 3rd.

Topics (i think):

forces of constraint

Lagrange Multipliers

generalized coordinates?

Hamiltonian

2024-01-26 Calc. of Variations

Central Forces

2024-03-01 Non-Inertial reference frames

Rigid Bodies

Inertia tensor

Notes: (i think? not positive)

2024-01-26 Calc. of Variations

2024-01-30 Calc. of Variations Continued

2024-02-02 General Coordinates Continued

2024-02-06 Hamilton and Conservation

2024-02-09 Angular and Hamiltonian

2024-02-16 Central Forces

2024-02-23 Central forces cont.

2024-03-01 Non-Inertial reference frames

2024-03-15 Rigid Bodies Cont.

2024-03-19 Rotating Rigid Bodies

2024-03-22 Parallel Axis Theorem

TMECH Exam 2 back exam:

https://drive.google.com/file/d/1KQP2U31L1ObIGDyZkiN4ml7hPKDD41q3/view?usp=sharing

Equations:

1. Taylor Series

$$(x+h)=(x){+}^{\prime }(x)h{+}^{\prime \prime }(x)\frac{h^2}{2!}+\dots$$

2. Small Angle Approximation

$$\tan \theta \approx \sin \theta \approx \theta$$ $$\cos \theta \approx 1$$

3. . Kinetic energy formulas

$$T_{\text{rolling}}=\frac{1}{2}I{\omega }^2$$ $$T_{\text{linear}}=\frac{1}{2}m{v}^2$$

4. Potential Energy formulas

$$U_{\text{ gravity}}=mgh\text{ (depends on coordinate system)}$$ $$U_{\text{spring}}=\frac{1}{2}k\Delta x^2\text{ (hookian spring)}$$

5. Harmonic Oscillator diffeq

$$\ddot{\theta }=-{\omega }^2\theta$$

6. Lagrangian

$$L=T-U$$

7. Euler-Lagrange Equation

$$\frac{d}{dt}\frac{\partial L}{\partial {\dot{q}}_i}-\frac{\partial L}{\partial q_i}=0$$

8. Modified Lagrangian

$$\tilde{L}=L-\lambda f$$ $$f=\text{ force }$$

9. Hamiltonian

$$H=\sum _i{\dot{q}}_j\frac{\partial L}{\partial {\dot{q}}_j}-L$$

Sometimes equals

$$H=T+U$$

Hamilton’s Equations of motion

$${\dot{q}}_k=\frac{\partial H}{\partial p_k}$$

and

$$-{\dot{p}}_k=\frac{\partial H}{\partial q_k}$$

10. Centrifugal Potential

$$U_{cf}=\frac{l}{2\mu r^2}$$

11. Effective Potential

$$U_{eff}=U+U_{cf}$$

12. Discrete Inertia Tensor

$$I_{ij}=\sum _{\alpha }{m}_{\alpha }\left({\delta }_{ij}\sum _k{x}_{\alpha ,k}^2-x_{\alpha ,i}{x}_{\alpha ,j}\right)$$

14. Continuous Inertia Tensor

$$I_{ij}={\int }_{v}p(\vec{r})\left({\delta }_{ij}\sum _k{x}_k^2-x_i{x}_j\right)dV$$

15. Eigenvalue and Eigenvector Relation

$$(\mathbf{A}-\lambda \mathbf{I})\cdot \vec{\mathbf{v}}=0$$

and

$$\det (\mathbf{A}-\lambda \mathbf{I})=0$$