import streamlit as st

# Set the page title
st.title("PHY 504: Advanced Mechanics")

# Course Details
st.markdown("""
## Course Details
- **Course Title**: Advanced Mechanics  
- **Credits**: 3  
- **Prerequisites**: PHY 404G, MA 214  
- **Instructor**: [Instructor Name]  
- **Office Hours**: [Office Hours]  

## Course Description
This course extends the foundational principles learned in classical mechanics, focusing on advanced topics such as the dynamics of particles, rigid bodies, Lagrangian and Hamiltonian mechanics, constrained motions, and small oscillations. Applications will include modern techniques used in the study of chaotic systems and continuous media.

## Course Objectives
By the end of this course, students will be able to:
- Understand and apply Lagrangian and Hamiltonian mechanics.
- Analyze mechanical systems with constraints.
- Solve problems involving small oscillations and rigid body dynamics.
- Apply advanced mechanics to real-world problems and chaotic systems.

---
""")

# Weekly Outline with Problems

# Week 1-2: Newtonian Mechanics
with st.expander("**Week 1-2: Review of Newtonian Mechanics and Introduction to Variational Principles**"):
    st.markdown("""
    ### Topics Covered
    - Newton’s Laws of Motion.
    - Application of Newton's laws to basic systems.
    - Introduction to the Principle of Least Action.
    - Derivation of Euler-Lagrange equations.
    - Application of calculus of variations to mechanics.

    ### Problems
    1. Derive the equation of motion for a simple pendulum using Newton’s laws.  
       (Topic: Newton’s Laws and pendulum system.)
    2. Solve for the trajectory of a projectile under gravity (without air resistance).  
       (Topic: Application of Newton’s second law to free fall.)
    3. Apply the principle of least action to derive the Euler-Lagrange equation.  
       (Topic: Introduction to the Principle of Least Action.)
    4. Derive the equation of motion for a mass-spring system using the variational principle.  
       (Topic: Variational principles applied to simple systems.)
    5. Apply the variational principle to a charged particle in an electric field.  
       (Topic: Principle of least action applied to a charged particle.)
    6. Analyze a system of two masses connected by a spring using Newtonian mechanics.  
       (Topic: Multi-particle system using Newton’s laws.)
    7. Derive the Euler-Lagrange equation for a free particle in Cartesian coordinates.  
       (Topic: Euler-Lagrange equations for simple systems.)
    8. Prove that for a particle moving under a central force, the angular momentum is conserved.  
       (Topic: Conservation laws in mechanics.)
    9. Solve the brachistochrone problem using calculus of variations.  
       (Topic: Application of calculus of variations.)
    10. Apply Newton’s second law to a system of three interacting masses and derive their equations of motion.  
        (Topic: Newton’s second law applied to interacting systems.)
    """)

# Week 3: Lagrangian Mechanics
with st.expander("**Week 3: Lagrangian Mechanics: Generalized Coordinates, Lagrange's Equations**"):
    st.markdown("""
    ### Topics Covered
    - Generalized coordinates and degrees of freedom.
    - Derivation of Lagrange’s equations from the principle of least action.
    - Solving mechanical problems using Lagrange’s equations.
    - Applications to systems with multiple degrees of freedom.

    ### Problems
    11. Derive the equations of motion for a double pendulum using Lagrange’s equations.  
        (Topic: Lagrangian mechanics applied to a double pendulum.)
    12. Solve the equations of motion for a simple pendulum using the Lagrangian method.  
        (Topic: Lagrange’s equations applied to a simple pendulum.)
    13. Apply Lagrange’s equations to a bead sliding on a rotating hoop.  
        (Topic: Generalized coordinates and Lagrange’s equations.)
    14. Analyze the motion of a particle in polar coordinates using Lagrange’s equations.  
        (Topic: Lagrangian mechanics in polar coordinates.)
    15. Derive the Lagrangian for a charged particle in a magnetic field.  
        (Topic: Lagrangian mechanics for a charged particle.)
    16. Solve the equations of motion for a mass sliding down a frictionless incline.  
        (Topic: Lagrangian mechanics for constrained systems.)
    17. Use Lagrange’s method to describe the motion of a particle constrained to move on a paraboloid.  
        (Topic: Application of Lagrangian mechanics to constrained systems.)
    18. Calculate the kinetic and potential energies of a particle in cylindrical coordinates.  
        (Topic: Generalized coordinates and energy calculations.)
    19. Derive the Lagrangian for a simple Atwood machine and solve the equations of motion.  
        (Topic: Lagrangian mechanics applied to constrained motion.)
    20. Apply Lagrangian mechanics to describe the motion of a particle in a harmonic potential.  
        (Topic: Lagrangian mechanics in oscillatory systems.)
    """)

# Week 4: Constraints
with st.expander("**Week 4: Constraints: Types, Generalized Forces**"):
    st.markdown("""
    ### Topics Covered
    - Holonomic and non-holonomic constraints.
    - Virtual work and generalized forces.
    - Lagrange multipliers.
    - Application of constraints to mechanical systems.

    ### Problems
    21. Analyze the motion of a block on an inclined plane with friction using virtual work.  
        (Topic: Virtual work and constraints.)
    22. Solve the problem of a pendulum attached to a sliding mass (non-holonomic constraint).  
        (Topic: Non-holonomic constraints in mechanical systems.)
    23. Use the Lagrange multiplier technique to solve the motion of a particle on a sphere.  
        (Topic: Lagrange multipliers for constrained motion.)
    24. Derive the generalized forces for a rolling disc on a rough surface.  
        (Topic: Generalized forces and non-holonomic constraints.)
    25. Solve the constrained motion of a particle in a tube rotating about a vertical axis.  
        (Topic: Holonomic constraints in rotating systems.)
    26. Analyze the motion of a particle in a cone using holonomic constraints.  
        (Topic: Constrained systems with holonomic constraints.)
    27. Calculate the virtual work done on a system with both holonomic and non-holonomic constraints.  
        (Topic: Virtual work in mixed constraint systems.)
    28. Use generalized coordinates to solve the problem of a particle constrained to move along a curved surface.  
        (Topic: Generalized coordinates for constrained systems.)
    29. Analyze the motion of a double pulley system using Lagrange’s equations.  
        (Topic: Lagrangian mechanics applied to pulley systems.)
    30. Solve for the motion of a charged particle in a magnetic field under a non-holonomic constraint.  
        (Topic: Non-holonomic constraints in electromagnetic systems.)
    """)

# Week 5: Rigid Body Motion
with st.expander("**Week 5: Rigid Body Motion: Euler Angles and Equations**"):
    st.markdown("""
    ### Topics Covered
    - Kinematics of rigid bodies.
    - Euler’s angles and their relation to rotational motion.
    - Euler’s equations of motion for rigid bodies.
    - Dynamics of rotating systems.

    ### Problems
    31. Solve for the rotational motion of a symmetric top without precession.  
        (Topic: Rigid body dynamics of symmetric tops.)
    32. Derive Euler’s equations for the motion of a free rigid body.  
        (Topic: Euler’s equations in rigid body motion.)
    33. Use Euler angles to describe the rotation of a rigid body about a fixed point.  
        (Topic: Rigid body kinematics using Euler angles.)
    34. Analyze the motion of a gyroscope using Euler’s equations.  
        (Topic: Dynamics of a gyroscope using Euler’s equations.)
    35. Derive the equations of motion for a spinning top under gravity.  
        (Topic: Rotational dynamics of a spinning top.)
    36. Solve for the rotational dynamics of a rolling sphere on a plane.  
        (Topic: Rolling motion in rigid body dynamics.)
    37. Calculate the angular momentum of a rigid body rotating about a fixed axis.  
        (Topic: Angular momentum in rigid body motion.)
    38. Analyze the precession of a gyroscope and solve for its angular velocity.  
        (Topic: Precession in gyroscopes.)
    39. Solve for the nutation of a symmetric top under external forces.  
        (Topic: Nutation in rigid body dynamics.)
    40. Apply Euler’s equations to describe the motion of a satellite in orbit.  
        (Topic: Application of rigid body dynamics to satellites.)
    """)

# Week 6: Hamiltonian Mechanics
with st.expander("**Week 6: Hamiltonian Mechanics and Canonical Transformations**"):
    st.markdown("""
    ### Topics Covered
    - Hamiltonian formulation of mechanics.
    - Canonical equations of motion.
    - Canonical transformations and their generating functions.
    - Application of Hamilton’s equations to complex systems.

    ### Problems
    41. Derive the Hamiltonian for a harmonic oscillator and solve the equations of motion.  
        (Topic: Hamiltonian mechanics for oscillatory systems.)
    42. Calculate the Hamiltonian for a free particle in Cartesian coordinates.  
        (Topic: Hamiltonian formulation for free particles.)
    43. Solve for the motion of a charged particle in an electromagnetic field using the Hamiltonian formalism.  
        (Topic: Hamiltonian mechanics in electromagnetic systems.)
    44. Derive the canonical equations of motion for a particle moving in a central potential.  
        (Topic: Canonical equations for central force problems.)
    45. Perform a canonical transformation and verify that it preserves Hamilton’s equations.  
        (Topic: Canonical transformations in Hamiltonian mechanics.)
    46. Solve the equations of motion for a pendulum using Hamilton’s formalism.  
        (Topic: Hamiltonian mechanics applied to the simple pendulum.)
    47. Derive the Hamiltonian for a two-body problem and solve for the relative motion.  
        (Topic: Hamiltonian mechanics in two-body problems.)
    48. Apply a generating function to transform the coordinates and momenta of a system.  
        (Topic: Generating functions in canonical transformations.)
    49. Solve the problem of a particle moving in a one-dimensional potential using the Hamiltonian method.  
        (Topic: Hamiltonian mechanics for one-dimensional systems.)
    50. Use the Hamilton-Jacobi equation to solve for the motion of a particle in a potential well.  
        (Topic: Hamilton-Jacobi theory in potential wells.)
    """)

# Week 7: Poisson Brackets and Hamilton-Jacobi Theory
with st.expander("**Week 7: Poisson Brackets and Hamilton-Jacobi Theory**"):
    st.markdown("""
    ### Topics Covered
    - Introduction to Poisson brackets and their properties.
    - Application of Poisson brackets to constants of motion.
    - Hamilton-Jacobi theory and its applications.
    - Action-angle variables and their role in integrable systems.

    ### Problems
    51. Calculate the Poisson brackets for the coordinates and momenta of a particle in Cartesian coordinates.  
        (Topic: Poisson brackets in Cartesian systems.)
    52. Show that angular momentum components form a Lie algebra using Poisson brackets.  
        (Topic: Poisson brackets in angular momentum systems.)
    53. Use Poisson brackets to check if a given quantity is conserved in a dynamical system.  
        (Topic: Conservation laws using Poisson brackets.)
    54. Solve the Hamilton-Jacobi equation for a free particle.  
        (Topic: Hamilton-Jacobi theory applied to free particles.)
    55. Apply the Hamilton-Jacobi method to the Kepler problem.  
        (Topic: Hamilton-Jacobi theory in central force systems.)
    56. Derive the generating functions for a canonical transformation using Poisson brackets.  
        (Topic: Generating functions in Poisson bracket systems.)
    57. Solve for the motion of a harmonic oscillator using the Hamilton-Jacobi equation.  
        (Topic: Hamilton-Jacobi theory in oscillatory systems.)
    58. Prove that the Poisson bracket of two constants of motion is zero.  
        (Topic: Poisson bracket properties in conserved systems.)
    59. Calculate the Poisson bracket for a particle in a central potential.  
        (Topic: Poisson brackets in central force problems.)
    60. Use the Hamilton-Jacobi method to analyze the motion of a particle in a time-dependent potential.  
        (Topic: Hamilton-Jacobi theory in time-dependent systems.)
    """)

# Week 8: Action-Angle Variables
with st.expander("**Week 8: Action-Angle Variables**"):
    st.markdown("""
    ### Topics Covered
    - Introduction to action-angle variables.
    - Application to periodic motion and integrable systems.
    - Adiabatic invariants.
    - Use in analyzing systems with slowly varying parameters.

    ### Problems
    61. Calculate the action and angle variables for a simple harmonic oscillator.  
        (Topic: Action-angle variables in oscillatory systems.)
    62. Use action-angle variables to solve the problem of a pendulum oscillating near equilibrium.  
        (Topic: Action-angle variables in pendulum systems.)
    63. Derive the action-angle variables for a particle moving in a central potential.  
        (Topic: Action-angle variables in central force systems.)
    64. Solve for the action-angle variables in the case of a Kepler orbit.  
        (Topic: Action-angle variables in planetary motion.)
    65. Analyze adiabatic invariants in a slowly varying potential.  
        (Topic: Adiabatic invariants in systems with slow variations.)
    66. Solve for the action variable in a system with periodic motion.  
        (Topic: Action variables in periodic motion.)
    67. Apply action-angle variables to describe the motion of a charged particle in a magnetic field.  
        (Topic: Action-angle variables in electromagnetic systems.)
    68. Calculate the action-angle variables for a one-dimensional box potential.  
        (Topic: Action-angle variables in confined systems.)
    69. Use action-angle variables to analyze the motion of a particle in a quartic potential.  
        (Topic: Action-angle variables in non-linear potentials.)
    70. Solve for the motion of a planet in an elliptical orbit using action-angle variables.  
        (Topic: Action-angle variables in orbital mechanics.)
    """)

# Week 9-10: Small Oscillations and Normal Modes
with st.expander("**Week 9-10: Small Oscillations and Normal Modes**"):
    st.markdown("""
    ### Topics Covered
    - Small oscillations and linearization of the equations of motion.
    - Normal mode analysis.
    - Eigenvalue problems in mechanics.
    - Application to coupled oscillators and vibrational modes.

    ### Problems
    71. Derive the normal modes of a system of two coupled pendulums.  
        (Topic: Normal modes in coupled systems.)
    72. Calculate the frequencies of small oscillations for a three-mass spring system.  
        (Topic: Small oscillations in multi-mass systems.)
    73. Solve the equations of motion for a vibrating string using normal mode analysis.  
        (Topic: Normal modes in continuous systems.)
    74. Analyze the small oscillations of a double pendulum.  
        (Topic: Small oscillations in multi-degree systems.)
    75. Derive the normal mode frequencies for a molecule modeled as a system of coupled masses.  
        (Topic: Normal mode analysis in molecular dynamics.)
    76. Solve for the normal modes of oscillation in a system of two coupled oscillators.  
        (Topic: Coupled oscillators and normal modes.)
    77. Use normal mode analysis to describe the vibrations of a rectangular membrane.  
        (Topic: Normal modes in vibrating membranes.)
    78. Solve for the eigenfrequencies of a chain of coupled oscillators.  
        (Topic: Normal mode analysis in linear oscillator chains.)
    79. Analyze the small oscillations of a particle in a double well potential.  
        (Topic: Small oscillations in potential wells.)
    80. Calculate the normal modes of oscillation for a coupled mass-spring system.  
        (Topic: Normal modes in mass-spring systems.)
    """)

# Week 11-12: Chaotic Systems
with st.expander("**Week 11-12: Chaotic Systems: An Introduction**"):
    st.markdown("""
    ### Topics Covered
    - Introduction to chaos in dynamical systems.
    - Sensitivity to initial conditions.
    - Phase space analysis and Poincaré sections.
    - Lyapunov exponents and chaotic attractors.

    ### Problems
    81. Solve the logistic map and analyze the onset of chaos.  
        (Topic: Chaos theory in simple maps.)
    82. Use Poincaré sections to analyze the motion of a double pendulum.  
        (Topic: Poincaré sections in chaotic systems.)
    83. Solve the equations of motion for the damped driven pendulum and analyze chaotic behavior.  
        (Topic: Chaos in driven systems.)
    84. Apply the Lyapunov exponent to determine the chaotic nature of a dynamical system.  
        (Topic: Lyapunov exponents in chaotic systems.)
    85. Solve for the motion of a particle in a double-well potential and analyze chaotic trajectories.  
        (Topic: Chaos in double-well potentials.)
    86. Analyze the sensitivity to initial conditions in the Lorenz system.  
        (Topic: Sensitivity to initial conditions in chaotic systems.)
    87. Calculate the fractal dimension of a chaotic attractor.  
        (Topic: Fractal dimension analysis in chaotic systems.)
    88. Analyze the bifurcations in a driven nonlinear oscillator.  
        (Topic: Bifurcation theory in chaotic systems.)
    89. Solve the equations of motion for a kicked rotor and analyze its chaotic behavior.  
        (Topic: Chaotic motion in periodically kicked systems.)
    90. Use phase space plots to describe the transition from regular to chaotic motion.  
        (Topic: Phase space analysis in chaotic systems.)
    """)

# Week 13-14: Advanced Topics in Mechanics
with st.expander("**Week 13-14: Advanced Topics in Mechanics (Fluid Dynamics, Plasma Physics)**"):
    st.markdown("""
    ### Topics Covered
    - Fluid dynamics and the Navier-Stokes equations.
    - Continuity equation and Bernoulli’s theorem.
    - Plasma physics and the magnetohydrodynamic (MHD) approximation.
    - Applications to astrophysical and engineering systems.

    ### Problems
    91. Derive the Euler equation for inviscid fluid flow and solve for a simple flow pattern.  
        (Topic: Euler equations in fluid dynamics.)
    92. Apply Bernoulli’s equation to describe fluid flow through a nozzle.  
        (Topic: Bernoulli’s theorem in fluid flow.)
    93. Analyze the motion of a fluid using the continuity equation.  
        (Topic: Continuity equation in fluid dynamics.)
    94. Solve for the velocity field in a rotating fluid.  
        (Topic: Fluid dynamics in rotating systems.)
    95. Derive the equations of motion for a charged particle in a plasma.  
        (Topic: Plasma physics and particle motion.)
    96. Analyze the stability of a plasma using magnetohydrodynamics (MHD).  
        (Topic: Stability analysis in plasma systems.)
    97. Solve the Navier-Stokes equations for a laminar flow problem.  
        (Topic: Navier-Stokes equations in laminar flow.)
    98. Calculate the drag force on a sphere moving through a viscous fluid.  
        (Topic: Viscous forces in fluid dynamics.)
    99. Derive the dispersion relation for waves in a plasma.  
        (Topic: Wave dynamics in plasma systems.)
    100. Solve the problem of shock waves in a compressible fluid using fluid dynamics principles.  
        (Topic: Shock waves in fluid mechanics.)
    """)

# Week 15: Review and Final Exam
with st.expander("**Week 15: Review and Final Exam Preparation**"):
    st.markdown("""
    ### Topics Covered
    - Comprehensive review of all course material.
    - In-class problem-solving.
    - Review of advanced problem sets.
    """)

# Textbooks Section
st.markdown("""
## Textbooks
- **Classical Mechanics" by Herbert Goldstein, Charles Poole, and John Safko  
- **Mechanics: Volume 1 (Course of Theoretical Physics)" by L.D. Landau and E.M. Lifshitz
""")