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In//Lagrangian//mechanics,//the//trajectory//of//a//system//of//particles//is//derived//by//solving//the//Lagrange//equations//in//one//of//two//forms,//either//the//Lagrange//equations//of//the//first//kind,//which//treat//constraints//explicitly//as//extra//equations,//often//using//Lagrange//multipliers;//or//the//Lagrange//equations//of//the//second//kind,//which//incorporate//the//constraints//directly//by//judicious//choice//of//generalized//coordinates.//In//each//case,//a//mathematical//function//called//the//Lagrangian//is//a//function//of//the//generalized//coordinates,//their//time//derivatives,//and//time,//and//contains//the//information//about//the//dynamics//of//the//system.
No//new//physics//is//introduced//in//Lagrangian//mechanics//compared//to//Newtonian//mechanics.//Newton's//laws//can//include//non-conservative//forces//like//friction;//however,//they//must//include//constraint//forces//explicitly//and//are//best//suited//to//Cartesian//coordinates.//Lagrangian//mechanics//is//ideal//for//systems//with//conservative//forces//and//for//bypassing//constraint//forces//in//any//coordinate//system.//Dissipative//and//driven//forces//can//be//accounted//for//by//splitting//the//external//forces//into//a//sum//of//potential//and//non-potential//forces,//leading//to//a//set//of//modified//Euler-Lagrange//(EL)//equations.//Generalized//coordinates//can//be//chosen//by//convenience,//to//exploit//symmetries//in//the//system//or//the//geometry//of//the//constraints,//which//may//simplify//solving//for//the//motion//of//the//system.//Lagrangian//mechanics//also//reveals//conserved//quantities//and//their//symmetries//in//a//direct//way,//as//a//special//case//of//Noether's//theorem.Lagrangian//mechanics//is//important//not//just//for//its//broad//applications,//but//also//for//its//role//in//advancing//deep//understanding//of//physics.//Although//Lagrange//only//sought//to//describe//classical//mechanics//in//his//treatise//Mécanique//analytique,//William//Rowan//Hamilton//later//developed//Hamilton's//principle//that//can//be//used//to//derive//the//Lagrange//equation//and//was//later//recognized//to//be//applicable//to//much//of//fundamental//theoretical//physics//as//well,//particularly//quantum//mechanics//and//the//theory//of//relativity.//It//can//also//be//applied//to//other//systems//by//analogy,//for//instance//to//coupled//electric//circuits//with//inductances//and//capacitances.Lagrangian//mechanics//is//widely//used//to//solve//mechanical//problems//in//physics//and//when//Newton's//formulation//of//classical//mechanics//is//not//convenient.//Lagrangian//mechanics//applies//to//the//dynamics//of//particles,//field

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