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Problems of Analytical Mechanics

Analytical mechanics is a highly mathematical form of analysis that helps physicists and engineers bridge the gap between the 18th-century and 20th-century laws of motion, energy and momentum. It removes some of the constraints placed on Newton's theories of motion of a body, and opens up an endless array of coordinates by which these particles can travel. The field is subdivided into Lagrangian mechanics and Hamiltonian mechanics, opposite in approach to solving analytical mechanics problems, but both coming to a similar endpoint.

Equations
- Analytical mechanics does not offer a specific solution, just a step in the right direction.
  
  While analytical mechanics creates a simplified solution to a complex problem, the solutions to analytical mechanics problems are always another equation, not a specific answer. To actually solve the equation, many iterations are needed to narrow down and pinpoint the actual values, assuming they exist. For this reason, analytical mechanics is often tied in with computational analysis, which is a form of solving scientific problems with mathematical models. Analytical mechanics is the link between Newton's theories of motion and the world of quantum mechanics, where particles behave as both matter and energy, usually at the atomic level.
Vantages
- Different approaches to analytical mechanics provide insight into different values of inertia.
  
  Analytical mechanics creates a general set of coordinates that are not necessarily fixed at any one point. The analyst has the choice as to where the coordinates are placed based on the type of motion, the number of external forces acting on a body, and the number of degrees of freedom within the system. By changing the coordination location to fit the situation, the resultant equations are much less complicated than they would otherwise be. Setting up the equations to solve the problem is where the discipline splits. The more popular and more appropriate approach, Lagrangian mechanics, relies on the positions and velocities of particles, while Hamiltonian mechanics relies on positions and momentum. Also, Lagrange's principles hold the conservation of potential and kinetic energy intact, while Hamilton's principles conserve momentum.
Pendulum
- Calculating where you will land is extremely difficult.
  
  A brave person attached to a bungee cord jumps off an arch bridge that spans a river. The jumper lunges out and away from the bridge, and as the bungee cord tightens and snaps him back up, he begins swinging and bouncing. If he gets scared and cuts the cord 12 seconds after the jump, where would his friends have to be stationed in a boat to pick him up within 1 second of him splashing into the river? Analytical mechanics can only narrow down the variables you will need to compute the contact point, making it easier for less complex physics equations to finish the problem.
4x4 Example
- Flipping a truck uphill involves computational mechanics supported by analytical equations.
  
  A 3,800-pound 4-by-4 truck with large mud-bog tires is out in the fields four-wheeling. The truck is equipped with coil-mounted springs and shock absorbers. If the truck drives up a 45-degree hill and flies up off the ground, what is the maximum angle the truck could land at on its back tires and bounce without flipping the truck over? Analytical mechanics cannot solve this problem alone, but can provide the groundwork for calculating the variables at hand.

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